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A new bound for the midpoint solution in minmax regret optimization with an application to the robust shortest path problem
European Journal of Operational Research 244 (2015) 739–747
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Discrete Optimization
A new bound for the midpoint solution in minmax regret optimization with an application to the robust shortest path problem ✩ André B. Chassein∗, Marc Goerigk Fachbereich Mathematik, Technische Universität Kaiserslautern, Germany
a r t i c l e
i n f o
Article history: Received 25 June 2014 Accepted 10 February 2015 Available online 17 February 2015 Keywords: Combinatorial optimization Minmax regret Robust optimization Approximation Robust shortest paths
a b s t r a c t Minmax regret optimization aims at finding robust solutions that perform best in the worst-case, compared to the respective optimum objective value in each scenario. Even for simple uncertainty sets like boxes, most polynomially solvable optimization problems have strongly NP-complete minmax regret counterparts. Thus, heuristics with performance guarantees can potentially be of great value, but only few such guarantees exist. A popular heuristic for combinatorial optimization problems is to compute the midpoint solution of the original problem. It is a well-known result that the regret of the midpoint solution is at most 2 times the optimal regret. Besides some academic instances showing that this bound is tight, most instances reveal a way better approximation ratio. We introduce a new lower bound for the optimal value of the minmax regret problem. Using this lower bound we state an algorithm that gives an instance-dependent performance guarantee for the midpoint solution that is at most 2. The computational complexity of the algorithm depends on the minmax regret problem under consideration; we show that our sharpened guarantee can be computed in strongly polynomial time for several classes of combinatorial optimization problems. To illustrate the quality of the proposed bound, we use it within a branch and bound framework for the robust shortest path problem. In an experimental study comparing this approach with a bound from the literature, we find a considerable improvement in computation times. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Robust optimization is a paradigm for optimization under uncertainty, that has been receiving increasing attention over the last two decades. While many variants of robust optimization exist, most aim at optimizing the worst-case performance of a solution without knowledge of a probability distribution over the uncertain input data. For general surveys on robust optimization, see Ben-Tal, Ghaoui, and Nemirovski (2009), Ben-Tal and Nemirovski (2002), Goerigk and Schöbel (2013). One of the best-known approaches in robust optimization is to minimize the maximum difference in the objective value of the robust solution over all scenarios, compared to the best possible objective value achievable in each scenario. This approach is usually
✩ Effort sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-13-1-3066. The US Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright notation thereon. ∗ Corresponding author. Tel.: +49 6312054824. E-mail address: [email protected] (A. B. Chassein).
http://dx.doi.org/10.1016/j.ejor.2015.02.023 0377-2217/© 2015 Elsevier B.V. All rights reserved.
known as minmax regret, see Kouvelis and Yu (1997), Aissi, Bazgan, and Vanderpooten (2009). However, due to its min-max-min structure, these problems are typically very hard to solve, both from a theoretical and from a practical point of view (for theoretical complexity, we refer to Aissi, Bazgan, and Vanderpooten (2007); Aissi et al. (2009); Averbakh and Lebedev (2005), while practical complexity is demonstrated, e.g., for the minmax regret spanning tree problem (Kasperski, Maku´ chowski, & Zielinski, 2012; Montemanni, 2006; Yaman, Karasan, ¸ & Pınar, 2001)). Thus, heuristic algorithms with performance guarantees are highly desirable; however, only few such heuristics for minmax regret versions of general combinatorial problems exist. To the best of our knowledge, the heuristic with the current best approximation ratio of 2 for interval data is the midpoint algorithm (Conde, 2012; Kasperski ´ & Zielinski, 2006). While there exist academic instances that show the tightness of this approximation ratio, the midpoint solution shows a considerably better performance in practice. In this paper we present an instance-dependent approach to determine an upper bound on the approximation ratio that is always at most 2. While this approach can be applied to any combinatorial optimization problem, we discuss
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problem types where this bound can be computed in strongly polynomial time. Furthermore, we analyze the performance of the proposed bound within a branch and bound (BB) framework for the robust shortest path problem. Comparing the bound with the one proposed in Montemanni, Gambardella, and Donati (2004), we find considerable improvements in computation times. The remainder of this paper is structured as follows. In Section 2, we briefly recapitulate the formal definition of the minmax regret problem, and introduce necessary notation. The main part of this paper is the lower bound on minmax regret problems introduced in Section 3, which is then used to derive stronger upper bounds on the approximation ratio of the midpoint solution. To this end, an auxiliary problem needs to be solved, which is described in detail in Section 4. We proceed to discuss how this bound can be computed for the shortest path problem, the minimum spanning tree problem, the assignment problem, and the minimum s − t cut problem in the same section. Section 5 describes how to apply the analysis to partially fixed solutions, and how this can be used within a branch and bound algorithm. In Section 6 we present computational data to compare our bound to other approaches. Finally, we conclude this work and point out further research questions in Section 7. 2. Minmax regret problems and notation Many combinatorial optimization problems (e.g., shortest path, min cut, minimum spanning tree, . . . ) can be represented as
min x∈X
n
ci xi ,
(P )
i=1
where X ⊆ {0, 1}n =: Bn denotes the set of feasible solutions, and ci the costs of element i. We refer to (P ) as the classic optimization problem. In contrast to classic optimization where one assumes to know the set of feasible solutions and the cost function exactly, in robust optimization it is assumed to have uncertainty in the set of feasible solutions and/or the cost function. In this paper, as usual for minmax regret optimization, we consider the case where the set of feasible solutions is known and only the cost function is uncertain. That is, instead of considering only one specific cost vector c we assume to have an uncertainty set U ⊆ Rn of different possible cost vectors c. There are several ways to reformulate such an uncertain optimization problem to a well-defined problem formulation, which is also known as the robust counterpart (Goerigk & Schöbel, 2013). Here we consider the approach of minmax regret optimization. The regret is defined as the difference between the realized objective value and the best possible objective value over all scenarios. Formally, the regret problem can be formulated as
min max x∈X
c∈U
n
ci xi −
val∗c
,
(R)
i=1
where val∗c := miny∈X ni=1 ci yi denotes the best possible objective value in scenario c ∈ U . As frequently done for such problems, we consider uncertainty sets U which are the Cartesian product of intervals, i.e., n
U := × [ci , ci ]. i=1
Unfortunately, the minmax regret problem has a higher computational complexity than the original optimization problem in most cases. For example, the minmax regret problem of the shortest path problem, which is a polynomially solvable problem for a positive cost vector c, is strongly NP-complete for interval uncertainty (Averbakh & Lebedev, 2004). A frequently used scenario for the interval uncertainty is the midpoint scenario, where every entry of the cost vector ci is just the
average of the lower bound ci and the upper bound ci . This average is denoted as cˆi , i.e.,
cˆ :=
1 (c + c). 2
As NP-complete problems are often hard to solve exactly, one can try to find approximate optimal solutions in polynomial time. One such approach is to use the midpoint solution
xmid := arg min x∈X
n
cˆi xi ,
i=1
which can be computed by solving the midpoint scenario of the classic optimization problem of type (P ). The midpoint solution is always a ´ 2−approximation (Kasperski & Zielinski, 2006). In other words, the regret of the midpoint solution xmid is always not more than twice the regret of the optimal solution of (R). There are problem instances where this approximation guarantee is tight. But for a lot of instances the midpoint solution has smaller regret than twice the optimum regret, and is often even an optimal solution to (R). As the 2−approximation result is tight we can of course not improve this result for all instances, but we can improve the approximation guarantee for specific instances with only small computational effort. To do so, we first show an easy technique to find a lower bound for the optimal value of the minmax regret problem, and then use the lower bound obtained this way to improve the 2−approximation guarantee depending on the instance. Furthermore, one consequence of our new analysis method is the already proven 2−approximation guarantee for the midpoint solution. To simplify the presentation we introduce some more notations. We abbreviate N := {1, . . . , n}, and denote by val(x, c) := ni=1 ci xi the objective value of a solution x for the cost vector c. Furthermore, denote by opt (c) = arg minx∈X ni=1 ci xi any optimal solution of the classic optimization problem for a given cost vector c, and by
Reg(x) = max(val(x, c) − val∗c ) c∈U
the (maximum) regret for a solution x ∈ X . We write OPT for minx∈X Reg(x). Note that we do not use any probability distribution in the above definition of the minmax regret problem. Still, we make use of probability distributions in the following analysis. To this end, denote by P a probability distribution over the set U , and by EP (·) the expected value with respect to P. We follow the convention to denote random variables with capital letters (e.g., we write C instead of c to denote random cost vectors). We specify the elements that are chosen in a solution x ∈ X by x = {i ∈ N : xi = 1}. Furthermore, we define the worst-case scenario of x as cx , where
cix :=
ci ci
if i ∈ x . if i ∈ /x
This definition extends to vectors cS ∈ U , where S is any subset of N . One directly finds the following result, which is also shown in Aissi et al. (2009). Lemma 2.1. For all x ∈ X , we have that Reg(x) = val(x, cx ) − val∗cx . 3. A general lower bound for minmax regret problems The following observation is crucial for the later analysis. Observation 3.1. Let any probability distribution P over the uncertainty set U be given. Then,
Reg(x) = max val(x, c) − val∗c ≥ EP (val(x, C ) − val∗C ) c∈U
for all x ∈ X .
A. B. Chassein, M. Goerigk / European Journal of Operational Research 244 (2015) 739–747
Theorem 3.5. The midpoint solution xmid is a λ−approximation for the minmax regret problem (R), where
Lemma 3.2. For any fixed probability distribution P, we have:
OPT ≥ min val(x, EP (C )) − x∈X
(
EP val∗C
)
λ=
Proof.
OPT = min max val(x, c) − val∗c x∈X
≥ min EP (val(x, C ) −
val(xmid , cxmid ) − val∗cxmid
val(xmid , cˆ) − minS⊆N
1 2
(val∗cS + val∗cS )
)
(1)
Proof. From the preceding argumentation follows that
)
(2)
OPT ≥ val(xmid , cˆ) −
= min val(x, EP (C )) − EP (val∗C )
(3)
x∈X
= min EP (val(x, C )) − x∈X
(
EP val∗C
x∈X
Inequality (1) follows from Observation 3.1, and Equalities (2) and (3) use the linearity of the expectation and the objective function. Notation 3.3. We call a probability distribution P centered if EP (Ci ) = cˆi for all i = 1, . . . , n. Using Lemma 3.2, we get for all centered probability distributions P a lower bound LB(P ) for the optimal value of the regret problem. Definition 3.4. Given a centered probability distribution P, we set
LB(P ) := val(xmid , cˆ) − EP (val∗C ) Note that LB(P ) is indeed a lower bound of OPT, as
OPT ≥ min val(x, EP (C )) − EP (val∗C ) x∈X
1 min(val∗cS + val∗cS ). 2 S⊆N
≤ val(opt (cxmid ), cxmid ) + val(xmid , cxmid ).
2val(xmid , cˆ) − val(xmid , cxmid ) = 2
= val(xmid , cˆ) − EP (val∗C ) = LB(P ).
P∈P
= min P∈P
=
2
How to solve problem (S ) is further analyzed in Section 4. We denote by LB∗ the lower bound that is given by an optimal solution of (S ). We are now in the position to state the main result of this paper.
ci
i∈xmid
ci + ci − ci 2
ci
= val(xmid , cxmid ) We reconsider inequality (5). By adding val(xmid , cxmid ) on the left hand side and 2val(xmid , cˆ) − val(xmid , cxmid ) on the right hand side we get:
min(val∗cS + val∗cS ) + val(xmid , cxmid ) S⊆N
≤ val(opt (cxmid ), cxmid ) + val(xmid , cxmid ) + 2val(xmid , cˆ) − val(xmid , cxmid ) By rearranging these terms we get:
val(xmid , cxmid ) − val(opt (cxmid ), cxmid ) ≤ 2val(xmid , cˆ) − min(val∗cS + val∗cS )
(6)
S⊂N
From Lemma 2.1 we know that the left hand side of (6) equals Reg(xmid ). If Reg(xmid ) = 0, the midpoint solution is already the optimal solution and we are done, as 0 is a lower bound for all regret problems. If Reg(xmid ) > 0 we use (4) and (6) to show that OPT > 0.
2OPT ≥ 2val(xmid , cˆ) − min(val∗cS + val∗cS ) S⊂N
≥ Reg(xmid ) > 0 Hence we conclude Reg(xmid ) = 0 ⇐⇒ OPT = 0. Assuming that Reg(xmid ) > 0, we know that val(xmid , cˆ) − minS⊂N 12 (val∗cS + val∗S ) > 0 and we can divide it on both sides of
λ := (S )
cˆi −
i∈xmid
Inequality (6) to get
1 (val∗c1 + val∗c2 ) 2
1 = min(val∗cS + val∗cS ) 2 S⊆N
i∈xmid
min EP (val∗C ) ≤ min EP (val∗C )
(5)
The first inequality follows from the fact that the left hand side is the minimum of a set valued function over all possible sets S, and the right hand side is just the same function evaluated for one specific set (the set xmid ). The second inequality holds as val(opt (cxmid ), cxmid ) ≤ val(xmid , cxmid ). Note that 2val(xmid , cˆ) − val(xmid , cxmid ) = val(xmid , cxmid ), as seen by the following equations.
=
c
≤
S⊆N
x∈X
timization problem can be simplified to
Reg(xmid ) OPT
min(val∗cS + val∗cS ) ≤ val∗cxmid + val∗cxmid
i∈xmid
To get the best such bound, one has to find a centered probability distribution that minimizes EP (val∗C ). Due to the linearity of the objective function in c we know that val∗c ≥ val∗c for all c ∈ U . Hence, it follows that EP (val∗C ) ≥ val∗c for all probability distributions P. As a consequence we also have LB(P ) ≤ val(xmid , cˆ) − val∗c . This gives us an immediate upper bound for the lower bound. Note that this upper bound is not necessarily a lower bound for (R). Denote by P the set of all centered probability distributions on U . We consider the problem of finding the best possible lower bound, which can be considered as the optimization problem minP∈P EP (val∗C ). To solve this problem efficiently we restrict the solution space to the special class P of probability distributions defined as follows. We say a scenario in U is an extreme scenario, if and only if ci ∈ {ci , ci } for all i ∈ N . P are all probability distributions that have only two outcomes c1 and c2 , which are both extreme scenarios and equally likely. As the probability distributions are centered, it follows that the scenarios c1 and c2 must be complementary in the sense that if ci1 = ci , it follows that ci2 = ci and vice versa for all i ∈ N . Hence, we can identify every probability distribution P ∈ P by the elements S = {i | ci1 = ci } ⊆ N that attain their upper bound cost in the first scenario. As before, denote by cS the extreme scenario where ciS = ci ∀i ∈ S and ciS = ci ∀i ∈ S. The probability distribution P ∈ P identified with the set S therefore has the nice property that EP (val∗C ) = 0.5(val∗c1 + val∗c2 ) = 0.5(val∗cS + val∗S ). Hence, the op-
(4)
To verify the approximation property we have to show that λ. To this end, we estimate:
= min val(x, cˆ) − EP (val∗C )
P∈P
≤ 2.
Furthermore, we have that OPT = 0 ⇐⇒ Reg(xmid ) = 0.
c∈U
val∗C
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c
val(xmid , cxmid ) − val(opt (cxmid ), cxmid )
val(xmid , cˆ) − minS⊂N
1 2
(val∗cS + val∗cS )
≤ 2.
From Lemma 2.1 and Inequality (4) follows the desired approximation guarantee
Reg(xmid ) ≤ λ. OPT
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Theorem 3.5 motivates the analysis of the optimization problem (S ). If the set xmid solves the optimization problem (S ) to optimality, we get
However, for the purpose of the following analysis, the problem structure becomes more obvious by writing
min(
min
S⊆N
val∗cS
+
val∗cS
) = val(opt (c
xmid
), c
xmid
) + val(opt (c
xmid
), c
xmid
)
= val(opt (cxmid ), cxmid ) + val(xmid , cxmid ).
(7)
Observation 3.6. If the set xmid solves the optimization problem (S ) to optimality, then λ = 2. Proof. λ = 2 is a direct consequence of Eq. (7) and the fact that 2val(xmid , cˆ) − val(xmid , cxmid ) = val(xmid , cxmid ):
val(xmid , cxmid ) − val(opt (cxmid ), cxmid )
val(xmid , cˆ) − minS⊂N
(val∗cS + val∗cS ) val(xmid , cxmid ) − val(opt (cxmid ), cxmid ) = val(xmid , cˆ) − 0.5val(opt (cxmid ), cxmid ) − 0.5val(xmid , cxmid ) val(xmid , cxmid ) − val(opt (cxmid ), cxmid ) = =2 0.5val(xmid , cxmid ) − 0.5val(opt (cxmid ), cxmid ) 1 2
c
optimization problem for the cost vector cx to obtain val∗x , as we c have already computed val∗c = val∗x . In the next section we consider c problem (S ) in more detail. 4. Solving the lower bound problem (S ) 4.1. General reformulations As before, let an arbitrary combinatorial optimization problem of type (P ) be given. The purpose of this section is to compute the best possible lower bound LB∗ . To solve problem (S ), we need to find a partition of all elements from N into the two sets S and S. We can represent problem (S ) in the following way.
min
n
xi (ci + zi (ci − ci )) +
i=1
n
yi (ci − zi (ci − ci ))
(8)
i=1
s.t. x, y ∈ X
(9)
n
z∈B
(10)
The partition of the set N is modeled via the variables zi , which determine if an element is in S or in S, respectively. Additionally to the z variables, we compute two feasible solutions x, y ∈ X such that the total sum of objective values is minimized. Note that the formulation (8)–(10) is nonlinear. As only binary variables are involved, it can easily be linearized using the following problem formulation, where z indicates which elements are used in both solutions x and y.
min
n
(ci xi + ci yi + (ci − ci )zi )
i=1
s.t. zi ≥ xi + yi − 1 x, y ∈ X z ∈ Bn
∀i ∈ N
s.t. x, y ∈ X with fi : {0, 1, 2} → R,
⎧ ⎪ ⎨0 fi (u) = ci ⎪ ⎩ ci + ci
if u = 0 if u = 1 if u = 2
In the next sections we show how to solve this problem formulation efficiently for the shortest path, the minimum spanning tree, the assignment, and the minimum s − t cut problem. Note that the minmax regret counterpart of all these problems is strongly NP-complete (Aissi, Bazgan, & Vanderpooten, 2005, 2008; Averbakh & Lebedev, 2004). 4.2. The shortest path problem
If, on the other hand, xmid is not the optimal solution, there is a possibility to improve the approximation guarantee. As a general heuristic for the problem (S ) we can first solve the classic optimization problem for the cost vector c, denote the solution that is obtained in this way by x , and then use the partition (x ,x ) for (S ), where x is the complement of x . To evaluate the objective function val∗x + val∗x we just have to solve one additional classic c
fi (xi + yi )
i=1
we can use this equality to tighten the proof of Theorem 3.5 and conclude that λ = 2 in this case
λ=
n
Let a graph G = (V, E, c) be given, with node set V, edge set E, and a cost function c : E → R+ , as well as two nodes s, t ∈ V. The goal is to find the shortest path from s to t, where the length of a path is the sum of the costs ce over every edge e of the path. We can describe the shortest path problem as a combinatorial optimization problem over the ground set E with feasible set X = {x | x represents a s − t path}. To solve (S), the idea is to create a new multigraph G = (V, E , c ) with the same node set V and the edge set E = {e = (i, j), e∗ = (i, j) | ∀e = (i, j) ∈ E} containing the old edge set E as well as one additional parallel edge e∗ for every original edge e ∈ E. The cost function c : E → R+ is defined to be ce for every edge e ∈ E and ce for all copied versions e∗ . Problem (S ) reduces to finding two edge-disjoint s − t paths in G , as can be seen by the following relationship between two paths in G with the cost function f as defined in Section 4.1, and two edgedisjoint s − t paths in G . If an edge e and the copied version e∗ is used on both paths, this edge pair contributes costs of size ce + ce to the objective function, which are exactly the same costs as paid in problem (S ) for using an edge twice. If only one edge of the edge pair e and e∗ is used, it will always be edge e in an optimal solution due to its lower costs, and hence the contribution in the objective function is ce , which are again the same costs as paid in problem (S ) for using an edge once. If no edge of an edge pair is used, no costs accrue in both model formulations. The problem of finding two edge-disjoint paths in G can be represented as a minimum cost flow problem or can be solved directly using, e.g., Suurballe’s algorithm (Suurballe & Tarjan, 1984). Theorem 4.1. LB∗ can be computed in strongly polynomial time for the shortest path problem. 4.3. The minimum spanning tree problem Let again a graph G = (V, E, c) be given with node set V, edge set E, and a cost function c : E → R+ . The goal is to find a spanning tree with minimal costs where the cost of a tree are just the sum over the costs ce over every edge e in the tree. We can describe the minimum spanning tree problem as a combinatorial optimization problem over the ground set E with feasible set X = {x | x represents a spanning tree}. We can reutilize the idea used for the shortest path problem (see Section 4.2) and create the same auxiliary multigraph G = (V, E , c ). But instead of searching for two edge-disjoint s − t paths, we search for two edge-disjoint minimum spanning trees in G . This problem can be solved efficiently as shown in Roskind and Tarjan (1985). Theorem 4.2. LB∗ can be computed in strongly polynomial time for the minimum spanning tree problem.
A. B. Chassein, M. Goerigk / European Journal of Operational Research 244 (2015) 739–747
4.4. The assignment problem In the assignment problem we are given a complete bipartite graph G = (V, E, c) with a bipartition of the node set in equally sized sets V1 ∪ V2 = V and a cost function c : E → R+ . The goal is to find a perfect matching of minimum total cost. It is a well known result that the assignment problem can be transformed to a min cost flow problem on an auxiliary graph G = (V , E , c , u ) (Ahuja, Magnanti, & Orlin, 1993). The idea is to enlarge the graph by introducing one new source node s and one new sink node t to the graph and to connect s with all vertices of V1 and t with all vertices of V2 . The capacity of all edges is set to 1 and the cost of all edges incident to s or t is set to 0. We define an auxiliary graph G = (V , E , c , u ) similar to G by using the same methodology as well as the idea used for the shortest path or minimum spanning tree problem. As before, we introduce one new source node s and one new sink node t to the graph and connect them with the vertices of V1 and V2 . Further, we create for every edge e ∈ E a parallel copy e∗ . The costs of all connector edges are set to 0 and to ce for all edges e ∈ E and to ce for all copies e∗ . The capacity of every connector edge is set to 2 and the capacity of every edge e ∈ E and their corresponding copy e∗ is set to 1. We can translate a flow with flow value 2|V1 | in G to two assignments. Note that due to the capacity constraints of the connecting edges and the required flow value, two units of flow leave every node of V1 and enter every node of V2 . We call two nodes v and u connected if flow is sent from v to u or from u to v. If both units of flow leaving a node of v1 ∈ V1 are sent through an edge e = (v1 , v2 ) and their corresponding copy e∗ = (v1 , v2 ), v1 is assigned to v2 in both assignments. After removing all nodes assigned in that way, we have that every node of V1 is connected to two different nodes of V2 and vice versa. We now pick an arbitrary not yet assigned node v1 ∈ V1 that is connected to v2 and v2 ∈ V2 . We assign v1 to v2 in the first assignment and delete the flow sent from v1 to v2 . By doing this we reduce the number of nodes that are connected to v2 from 2 to 1. Let v1 be the node that is still connected to node v2 . We assign v1 to v2 in the second assignment and delete the flow sent from v1 to v2 . By doing this we reduce the number of nodes that are connected to v1 from 2 to 1. Hence, we know again which assignment pair we have to pick for the first assignment. By repeating this argumentation we generate a cycle, see Fig. 1. Every edge picked with his original orientation in this cycle repre-
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sents a part for the first assignment. All other edges picked represent a part of the second assignment. If there are still not yet assigned nodes left, we can just start the process again by picking a new not yet assigned node v∗1 ∈ V1 . The argumentation about the costs is the same as in the shortest path or minimum spanning tree problem. If a specific assignment is made twice, two units of flow must be sent through edge e and their corresponding copy e∗ contributing ce + ce to the costs. If a specific assignment is made only once, in an optimal solution the unit of flow will always be sent over the cheaper edge e contributing ce to the costs. Theorem 4.3. LB∗ can be computed in strongly polynomial time for the assignment problem. 4.5. The minimum s − t cut problem Let a graph G = (V, E, c) be given with node set V, arc set E, a cost function c : E → R+ as well as two nodes s, t ∈ V. The goal is to find an s − t cut separating s and t with minimal costs. We can describe the minimum s − t cut problem as a combinatorial optimization problem by using the following well known mixed-integer programming formulation (Lawler, 1976). The variable de indicates if arc e is used in the s − t cut.
min
ce de
e∈E
s.t. de − pi + pj ≥ 0
∀e = (i, j) ∈ E
ps − pt ≥ 1 d ∈ B|E| p ∈ R| V | We transform the graph G to the graph G = (V , E , c ) by replacing each arc with a pair of arcs. We introduce for every arc e = (i, j) ∈ E a new node ve as well as two new arcs e1 = (i, ve ) and e2 = (ve , j). The node set V = V ∪ {ve | ∀e ∈ E} consists of all nodes from V combined with all new nodes. The arc set E = {e1 , e2 | ∀e ∈ E} is the set of all new generated arcs. The new cost function c is defined as, c (e1 ) = ce and c (e2 ) = ce . It can be seen thatthe following problem is equivalent
A
a
A
a
B
b
B
b
C
c
C
c
D
d
D
d
E
e
E
e
F
f
F
f
Fig. 1. Translate flow to two assignments: Every black edge on the left represents one unit of flow. This flow is used to generate two assignments shown on the right side as red (dashed) and blue (solid) assignment. The red assignment is given by (A, a), (B, b), (C, d), (D, c), (E, e), (F, f ) and the blue assignment by (A, a), (B, d), (C, c), (D, b), (E, f ), (F, e). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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to (S ) for the min s − t cut problem:
min
scheme. Let a node in the branching tree be given, and let this node be characterized by the set X of feasible solutions. Then they use the following lower bound on the regret of this node:
de ce + d˜e ce
e∈E
s.t. de − pi + pve ≥ 0
∀e = (i, j) ∈ E
LBold (X ) = SP (c, IN(X ), OUT (X )) − SP (cOUT (X ), ∅, ∅)
d˜e − pve + pj ≥ 0
∀e = (i, j) ∈ E
where SP (c, A, B) is the value of the shortest s − t path including all edges in A and excluding all edges of B for the cost vector c. Applying our approach we get as lower bound for a distribution P .
p s − pt ≥ 2 d, d˜ ∈ B|E |
p ∈ R| V |
Furthermore, as the coefficient matrix for the above problem is totally unimodular, we may solve the LP relaxation instead. As the dual of the LP relaxation is a min-cost flow problem, we have shown the next theorem. Theorem 4.4. LB∗ can be computed in strongly polynomial time for the minimum s − t cut problem. 5. Adapting the lower bound for branch and bound algorithms To find the optimal solution of NP-complete minmax regret problems one might resort on branch and bound algorithms. An important ingredient of a successful branch and bound algorithm is an effective and easy to compute lower bound for every node in the branch and bound tree. As every node in a branch and bound tree defines a restriction for the set of feasible solutions, we have to show how the lower bound introduced in this paper can be computed, if parts of the solutions are already fixed, to make it usable in this context. 5.1. Including fixed variables Assume a branch and bound node defines a restricted feasible set X ⊂ X . Hence the minmax regret problem in this node reduces to
min max val(x, c) − val∗c x∈X
c∈U
Denote by IN(X ) = {i ∈ N | xi = 1 ∀x ∈ X } the set of all ground elements that have to be part of every feasible solution in X and by OUT (X ) = {i ∈ N | xi = 0 ∀x ∈ X } the set of all ground elements that must not be part of any feasible solution in X . Using Lemma 2.1 we can restrict the set U to the set U = {c ∈ U | ci = ci ∀i ∈ IN(X ) ∧ ci = ci ∀i ∈ OUT (X )} without changing the problem. Next we use the argumentation as before, but instead of letting P be an arbitrary distribution over U , we choose an arbitrary distribution P over U .
min max val(x, c) − val∗c = min max val(x, c) − val∗c x∈X
c∈U
x∈X
c∈U
≥ min EP (val(x, C ) − val∗C ) x∈X
= min EP (val(x, C )) − EP (val∗C ) x∈X
= min val(x, EP (C )) − EP (val∗C ) x∈X
Note that if the solution is completely fixed, i.e., X = {x }, then also U = {c } reduces to a set with only one element. In this case the probability distribution P has only one outcome – the vector c – and, hence, the lower bound is equal to Reg(x ), which is an upper bound of the optimal regret value. Therefore, the gap between lower and upper bound vanishes for this branch and bound node.
LBnew (X ) = min val(x, EP (C )) − EP (val∗C ) x∈X = ce − cˆe + SP (cˆ, IN(X ), OUT (X )) − EP (val∗C ) e∈IN(X )
The problem of choosing a good distribution P can again be reformulated as a min cost flow problem or can be solved directly with Suurballe’s algorithm applied to an accordingly adjusted instance, see Section 4.2. Again, we create for each arc e a parallel copy e∗ , but instead of assigning cost of ce to e and ce to e∗ for all arcs, we assign ∀e ∈ IN(X ) (∀e ∈ OUT (X )) costs of ce (ce ) to arc e and his copy e∗ . For all other arcs e ∈ / IN(X ) ∪ OUT (X ) we use the common cost assignment. Hence the effort of computing LBnew (X ) is roughly the same as for three shortest path computations, which is still reasonable compared to the two shortest path computations needed to compute LBold (X ). A direct comparison of both bounds is presented in Appendix B. 6. Experiments We illustrate the strength of the proposed lower bound using the shortest path problem as an example. 6.1. Instances and computational environment In Montemanni et al. (2004) two different types of randomly generated graph classes are considered. We consider the same graph classes but decided to modify the cost structure of the first graph type to gain more flexibility for the experiments. The graph family R-n-c-d-δ consists of randomly generated graphs such that each graph has n nodes and an approximate arc density of δ . The start vertex s is the first node of the graph and the end vertex t is the last node of the graph. The second graph family K-n-c-d-w are layered graphs. Every layer is completely connected to the next layer. The start node s is connected to the first layer and the last layer is connected to the end node t. The complete graph consists of n nodes and every layer of w nodes. To generate the costs, a random number m is sampled from the interval [1, c]. The lower bound cost ce is sampled uniformly from [(1 − d)m, (1 + d)m] and the upper bound cost ce is chosen uniformly from the interval [c, (1 + d)m]. Thus, we can control the cost variability using the parameter d. All experiments were conducted on an Intel Core i5-3470 processor, running at 3.20 GHz, 8 GB RAM under Windows 7. We used CPLEX version 12.6. for comparisons. Algorithms were implemented with Visual Studio 2010 Express using C# 5.0 in the .NET framework v. 4.5. Only one core was used for all computations. 6.2. Value of the lower bound
5.2. Branch and bound algorithm for the minmax regret shortest path problem As before, we assume we are given a graph G = (V, E, c) with node set V, edge set E, a cost function c : E → R+ as well as two nodes s, t ∈ V. In Montemanni et al. (2004) a branch and bound algorithm for the minmax regret shortest path problem is presented. We refer to Montemanni et al. (2004) for a detailed description of the branching
In this experiment we compute the improved lower bound for the approximation ratio of the midpoint solution. We implemented Suurballe’s algorithm to solve the lower bound problem (S ). The results are given by Table 1. The first column λ gives the improved approximation guarantee, while column Tabs gives the absolute time in seconds needed to solve Suurballe’s algorithm. The last column Trel gives the ratio of times needed to compute the lower and the upper bound.
A. B. Chassein, M. Goerigk / European Journal of Operational Research 244 (2015) 739–747 Table 1 Average performance of the new approximation guarantee. Tabs is given in seconds. Graph class
λ
Tabs
Trel
R-10-1000-0.25-1.0 R-100-1000-0.25-0.5 R-500-1000-0.25-0.5 R-1000-1000-0.25-0.5 R-10000-1000-0.25-0.01 R-10000-1000-0.25-0.05 K-102-1000-1.0-2 K-402-1000-1.0-10 K-1002-1000-1.0-10 K-2002-1000-1.0-10 K-4002-1000-1.0-10
1.220 1.330 1.539 1.494 1.692 1.530 1.895 1.640 1.631 1.621 1.625
<0.001 0.001 0.013 0.036 0.204 0.503 <0.001 0.003 0.007 0.011 0.031
1.50 1.57 1.56 1.67 1.27 1.65 1.58 1.55 1.39 1.52 1.59
Every experiment is done on 100 graphs randomly generated from the corresponding graph class, and presented values are averaged. The average approximation guarantees are always below 2, showing the potential of the new lower bound. It is also interesting to observe that the improved approximation guarantee does not decrease with an increasing instance size. But we have to mention that the new approximation guarantee is weaker for K-graphs than for R-graphs. Nevertheless, good improvements can be achieved for K-graphs if the edge costs have high variability. To compute the lower bound we have to do one shortest path computation (to find the value of the midpoint path) as well as one run of Suurballe’s algorithm. The computation of the upper bound needs two shortest path computations, one to find the worst case value of the midpoint path and one to compute the value of the regret path. As expected, the computational effort of Suurballe’s algorithm is comparable to two shortest path computations, which results in a time ratio of circa 1.5. 6.3. Branch and bound algorithms We compare the performance of five different algorithms. The first three algorithms use the branching scheme suggested in Montemanni
10
R−x−1000−0.25−0.1
5
104
10
BBold BBnew
104
BBboth BB*new
10
3
10
2
CPX
10
3
10
2
300
400
500
600
10 0.01
(a) R−graphs with different numbers of nodes.
105 10
4
10
K−x−1000−0.25−5
105
BBold BBnew BBboth
10 10
2
10
101 0
0.05
0.1
0.25
0.5
1.0
10
142
10 0.01
4
0.05
0.1
0.25
0.4
3
162
182
202
(d) K−graphs with different numbers of nodes.
222
K−202−1000−x−5
K−202−1000−0.25−x
105
BBold BBnew BBboth
BBold BBnew BBboth
4
10
*
BBnew CPX
BBnew CPX
3
10
102
1
100 0.01
0.5
(c) R−graphs with different arc densities.
102
1
100 122
10
*
BBnew CPX
102 10
CPX
3
(b) R−graphs with different cost variabilities.
*
3
BBboth BB*new
CPX
0
200
BBold BBnew
104
BBboth
101
0
100
R−500−1000−0.25−x
5
10
BBold BBnew BB*new
101 10
et al. (2004). The only difference of these algorithms is the method to compute the lower bound. In the first algorithm BBold we implemented the lower bound LBold and in the second algorithm BBnew we use our lower bound LBnew (see Section 5.2). The third algorithm BBboth computes both bounds in every branching node and chooses the best one. In this branching scheme a branch and bound node represents a path from node s to some other node v. To enlarge this path the authors of Montemanni et al. (2004) choose the shortest path P from v to t with respect to the worst case cost. This makes sense as they have to compute the path P for calculating their lower bound LBnew anyways. The fourth algorithm BB∗new is again a branch and bound algorithm using lower bound LBnew but instead of computing the worst case path P in every branch and bound node to complete the path we compute the shortest path with respect to the midpoint costs. In this way we save one shortest path computation to compute LBnew . For the fifth algorithm CPX we use the mixed-integer programming formulation of the minmax regret problem (see Appendix A) and solve it using CPLEX. We consider two different experiments. For smaller instances we compare the time needed for each algorithm to solve the problem to optimality, while for larger instances we compare the gaps between the best lower bound and the best upper bound after a time limit of 30 seconds. Every experiment is done on 100 graphs randomly generated from the corresponding graph class. The results of the first experiment are presented in Fig. 2. The computation times on the y-axis are given in milliseconds. For general R-graphs we see that all three branch and bound algorithms have almost the same performance except for BBold that shows a bad performance for instances with very high cost variability. Further we observe that CPX is clearly outperformed by BBnew on R-graphs. It is also worth mentioning that BBold is a bit faster than BBnew on these instances where the quality of the lower bound seems to be less important than the time needed to compute the bound. (See Appendix B for a direct comparison of the quality of LBold and LBnew .) This seems to be different for K-graphs, as, except for some small instances, BBold is clearly outperformed by BBnew . In general
R−500−1000−x−0.1
5
745
1
10
0.1
0.15
0.2
0.25
0.3
(e) K−graphs with different cost variabilities.
Fig. 2. Average computation times in milliseconds.
100
5
10
20
40
50
(f) K−graphs with different layer sizes.
100
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We showed that any probability distribution over U can be used to compute a lower bound for the minmax regret problem. The solution space of the lower bound problem (S ) is a strongly restricted space of probability distributions. For further research it would be interesting to investigate lower bound problems that are built on wider spaces of probability distributions. Additional research should be made to find an algorithm that has an approximation guarantee that is strictly less than 2 or to show some inapproximability results for minmax regret problems.
BBold BBnew
80.0
BBboth BB*new
60.0
CPX
40.0 20.0
Appendix A. Mixed-integer programming formulation for the robust shortest path problem
30 K0 -1 .9 -0
0 -1 .9 -0
00
00
0 -1
0 -1
02
02
20 K1
0 .0 -0 -1
00
1
0 .0 -0 .5 -0
00
10 0-
10 0-
00
00
0 -1
0 -1
R
R
Fig. 3. Average gap after 30 seconds time limit (in percent). Note that if CPLEX is not able to state an integrality gap for one instance within 30 seconds, we assigned a gap of 100 percent to this instance.
the branch and bound algorithms show a weaker performance on K-graphs than on R-graphs as they are outperformed by CPX. One exception is Fig. 2(f). The increasing layer width reduces on the one hand the problem difficulty for the branch and bound algorithms, as the branching depth is reduced with increasing layer size, and on the other hand increases the number of arcs in the graph which explains the worse performance of CPX, as an increasing number of arcs increases the size of the IP formulation. It is also interesting to note that the algorithm BBboth is outperformed on all instances by BBnew . Hence, we see that in general it is not worth computing LBold if one has already computed LBnew for a branch and bound algorithm. The performances of BBnew and BB∗new are very similar. Therefore, we cannot state clearly which is the better one. Fig. 3 shows the results of the second experiment. These results confirm the statements that are made for the first experiment. On K-graphs the use of branch and bound algorithms is questionable, as they are clearly outperformed by CPX. Nevertheless, the more difficult computation of LBnew pays off especially for complicated instances of both graph classes. For R-graphs the use of branch and bound algorithms seems to be reasonable particularly for instances that are too big to be handled by CPLEX. 7. Conclusions and further research In this work we further refined the analysis of the most widely researched general approximation algorithm to minmax regret optimization: the midpoint solution. By underestimating the worst-case over all scenarios using a purpose-constructed probability distribution, we find a new proof for the well-known result that the midpoint solution is a 2-approximation. However, the proof technique yields further insight, as it can be used to construct approximation guarantees that are dependent on the given instance. To this end, an optimization problem over probability distributions needs to be solved to gain the strongest possible estimate. We show that this problem can easily be solved for many well-known combinatorial optimization problems. Finally, the analysis was further complemented using experimental data for randomly created uncertain shortest path problems. Our results suggest that the new performance guarantee improves the already known guarantee of 2 for most instances and can be computed very fast. The knowledge about the better performance guarantee and the lower bound can be used to considerably improve existing branch and bound algorithms.
Given a directed graph G = (V, E) with interval uncertainties for all edge costs ce ∈ [ce , ce ] ∀e ∈ E. The problem of finding an s − t path with minimal regret can be formulated as an mixed-integer programming formulation.
min
c e xe − yt
e∈E
s.t. yj ≤ yi + ce + (ce − ce )xe ⎧ ⎪ ⎨ 1 if v = s xe − xe = −1 if v = t ⎪ ⎩ e∈δ + (v) e∈δ − (v) 0 else
∀e = (i, j) ∈ E ∀v ∈ V
ys = 0
∀v ∈ V ∀e ∈ E
yv ≥ 0 xe ∈ B
The variable x indicates which edges are part of the robust s − t path and the vertex potential y is used to include the value of the shortest s − t path if the worst case scenario for the robust path is realized. For a detailed explanation of the mixed integer formulation we refer to Karasan, ¸ Pinar, and Yaman (2001) and Aissi et al. (2009). Appendix B. Comparison of lower bounds We did the following experiment to compare the quality of LBold and LBnew directly. We generated 1000 graphs of type K-202-10000.25-5 and solved each instance using algorithm BBboth . In each where UB is the upper branching node we compute the gap UB−LB UB bound of the corresponding branching node and LB is either LBold or LBnew . To visualize how the gap evolves during the branching algorithm we assign to every gap value the length of the path that is fixed in the corresponding branch and bound node. The plotted value in Fig. 5 shows the average value of all gaps corresponding to the same path length. Fig. 4 shows the average number of branch and bound nodes per fixed path length. Fig. 5 clearly indicates that LBnew dominates LBold for every fixed path length. But note that LBnew is not better than LBold for all branch and bound nodes of the branching tree. 30 25
Number Nodes
0.0
20 15 10 5 0
0
5
10
15
20
25
30
35
40
Fixed Path Length Fig. 4. Average number of nodes per fixed path length.
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0.60
LBold LBnew
0.50
Gap
0.40 0.30 0.20 0.10 0.00
0
5
10
15
20
25
30
35
40
Fixed Path Length Fig. 5. Average gap for fixed path length.
References Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network flows: Theory, algorithms, and applications. Prentice-Hall, Inc. Aissi, H., Bazgan, C., & Vanderpooten, D. (2005). Complexity of the min–max and min–max regret assignment problems. Operations Research Letters, 33(6), 634– 640. Aissi, H., Bazgan, C., & Vanderpooten, D. (2007). Approximation of min–max and min– max regret versions of some combinatorial optimization problems. European Journal of Operational Research, 179(2), 281–290. Aissi, H., Bazgan, C., & Vanderpooten, D. (2008). Complexity of the min–max (regret) versions of min cut problems. Discrete Optimization, 5(1), 66–73. Aissi, H., Bazgan, C., & Vanderpooten, D. (2009). Min–max and min–max regret versions of combinatorial optimization problems: A survey. European Journal of Operational Research, 197(2), 427–438. Averbakh, I., & Lebedev, V. (2004). Interval data minmax regret network optimization problems. Discrete Applied Mathematics, 138(3), 289–301.
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Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Discrete Optimization
A new bound for the midpoint solution in minmax regret optimization with an application to the robust shortest path problem ✩ André B. Chassein∗, Marc Goerigk Fachbereich Mathematik, Technische Universität Kaiserslautern, Germany
a r t i c l e
i n f o
Article history: Received 25 June 2014 Accepted 10 February 2015 Available online 17 February 2015 Keywords: Combinatorial optimization Minmax regret Robust optimization Approximation Robust shortest paths
a b s t r a c t Minmax regret optimization aims at finding robust solutions that perform best in the worst-case, compared to the respective optimum objective value in each scenario. Even for simple uncertainty sets like boxes, most polynomially solvable optimization problems have strongly NP-complete minmax regret counterparts. Thus, heuristics with performance guarantees can potentially be of great value, but only few such guarantees exist. A popular heuristic for combinatorial optimization problems is to compute the midpoint solution of the original problem. It is a well-known result that the regret of the midpoint solution is at most 2 times the optimal regret. Besides some academic instances showing that this bound is tight, most instances reveal a way better approximation ratio. We introduce a new lower bound for the optimal value of the minmax regret problem. Using this lower bound we state an algorithm that gives an instance-dependent performance guarantee for the midpoint solution that is at most 2. The computational complexity of the algorithm depends on the minmax regret problem under consideration; we show that our sharpened guarantee can be computed in strongly polynomial time for several classes of combinatorial optimization problems. To illustrate the quality of the proposed bound, we use it within a branch and bound framework for the robust shortest path problem. In an experimental study comparing this approach with a bound from the literature, we find a considerable improvement in computation times. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Robust optimization is a paradigm for optimization under uncertainty, that has been receiving increasing attention over the last two decades. While many variants of robust optimization exist, most aim at optimizing the worst-case performance of a solution without knowledge of a probability distribution over the uncertain input data. For general surveys on robust optimization, see Ben-Tal, Ghaoui, and Nemirovski (2009), Ben-Tal and Nemirovski (2002), Goerigk and Schöbel (2013). One of the best-known approaches in robust optimization is to minimize the maximum difference in the objective value of the robust solution over all scenarios, compared to the best possible objective value achievable in each scenario. This approach is usually
✩ Effort sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-13-1-3066. The US Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright notation thereon. ∗ Corresponding author. Tel.: +49 6312054824. E-mail address: [email protected] (A. B. Chassein).
http://dx.doi.org/10.1016/j.ejor.2015.02.023 0377-2217/© 2015 Elsevier B.V. All rights reserved.
known as minmax regret, see Kouvelis and Yu (1997), Aissi, Bazgan, and Vanderpooten (2009). However, due to its min-max-min structure, these problems are typically very hard to solve, both from a theoretical and from a practical point of view (for theoretical complexity, we refer to Aissi, Bazgan, and Vanderpooten (2007); Aissi et al. (2009); Averbakh and Lebedev (2005), while practical complexity is demonstrated, e.g., for the minmax regret spanning tree problem (Kasperski, Maku´ chowski, & Zielinski, 2012; Montemanni, 2006; Yaman, Karasan, ¸ & Pınar, 2001)). Thus, heuristic algorithms with performance guarantees are highly desirable; however, only few such heuristics for minmax regret versions of general combinatorial problems exist. To the best of our knowledge, the heuristic with the current best approximation ratio of 2 for interval data is the midpoint algorithm (Conde, 2012; Kasperski ´ & Zielinski, 2006). While there exist academic instances that show the tightness of this approximation ratio, the midpoint solution shows a considerably better performance in practice. In this paper we present an instance-dependent approach to determine an upper bound on the approximation ratio that is always at most 2. While this approach can be applied to any combinatorial optimization problem, we discuss
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problem types where this bound can be computed in strongly polynomial time. Furthermore, we analyze the performance of the proposed bound within a branch and bound (BB) framework for the robust shortest path problem. Comparing the bound with the one proposed in Montemanni, Gambardella, and Donati (2004), we find considerable improvements in computation times. The remainder of this paper is structured as follows. In Section 2, we briefly recapitulate the formal definition of the minmax regret problem, and introduce necessary notation. The main part of this paper is the lower bound on minmax regret problems introduced in Section 3, which is then used to derive stronger upper bounds on the approximation ratio of the midpoint solution. To this end, an auxiliary problem needs to be solved, which is described in detail in Section 4. We proceed to discuss how this bound can be computed for the shortest path problem, the minimum spanning tree problem, the assignment problem, and the minimum s − t cut problem in the same section. Section 5 describes how to apply the analysis to partially fixed solutions, and how this can be used within a branch and bound algorithm. In Section 6 we present computational data to compare our bound to other approaches. Finally, we conclude this work and point out further research questions in Section 7. 2. Minmax regret problems and notation Many combinatorial optimization problems (e.g., shortest path, min cut, minimum spanning tree, . . . ) can be represented as
min x∈X
n
ci xi ,
(P )
i=1
where X ⊆ {0, 1}n =: Bn denotes the set of feasible solutions, and ci the costs of element i. We refer to (P ) as the classic optimization problem. In contrast to classic optimization where one assumes to know the set of feasible solutions and the cost function exactly, in robust optimization it is assumed to have uncertainty in the set of feasible solutions and/or the cost function. In this paper, as usual for minmax regret optimization, we consider the case where the set of feasible solutions is known and only the cost function is uncertain. That is, instead of considering only one specific cost vector c we assume to have an uncertainty set U ⊆ Rn of different possible cost vectors c. There are several ways to reformulate such an uncertain optimization problem to a well-defined problem formulation, which is also known as the robust counterpart (Goerigk & Schöbel, 2013). Here we consider the approach of minmax regret optimization. The regret is defined as the difference between the realized objective value and the best possible objective value over all scenarios. Formally, the regret problem can be formulated as
min max x∈X
c∈U
n
ci xi −
val∗c
,
(R)
i=1
where val∗c := miny∈X ni=1 ci yi denotes the best possible objective value in scenario c ∈ U . As frequently done for such problems, we consider uncertainty sets U which are the Cartesian product of intervals, i.e., n
U := × [ci , ci ]. i=1
Unfortunately, the minmax regret problem has a higher computational complexity than the original optimization problem in most cases. For example, the minmax regret problem of the shortest path problem, which is a polynomially solvable problem for a positive cost vector c, is strongly NP-complete for interval uncertainty (Averbakh & Lebedev, 2004). A frequently used scenario for the interval uncertainty is the midpoint scenario, where every entry of the cost vector ci is just the
average of the lower bound ci and the upper bound ci . This average is denoted as cˆi , i.e.,
cˆ :=
1 (c + c). 2
As NP-complete problems are often hard to solve exactly, one can try to find approximate optimal solutions in polynomial time. One such approach is to use the midpoint solution
xmid := arg min x∈X
n
cˆi xi ,
i=1
which can be computed by solving the midpoint scenario of the classic optimization problem of type (P ). The midpoint solution is always a ´ 2−approximation (Kasperski & Zielinski, 2006). In other words, the regret of the midpoint solution xmid is always not more than twice the regret of the optimal solution of (R). There are problem instances where this approximation guarantee is tight. But for a lot of instances the midpoint solution has smaller regret than twice the optimum regret, and is often even an optimal solution to (R). As the 2−approximation result is tight we can of course not improve this result for all instances, but we can improve the approximation guarantee for specific instances with only small computational effort. To do so, we first show an easy technique to find a lower bound for the optimal value of the minmax regret problem, and then use the lower bound obtained this way to improve the 2−approximation guarantee depending on the instance. Furthermore, one consequence of our new analysis method is the already proven 2−approximation guarantee for the midpoint solution. To simplify the presentation we introduce some more notations. We abbreviate N := {1, . . . , n}, and denote by val(x, c) := ni=1 ci xi the objective value of a solution x for the cost vector c. Furthermore, denote by opt (c) = arg minx∈X ni=1 ci xi any optimal solution of the classic optimization problem for a given cost vector c, and by
Reg(x) = max(val(x, c) − val∗c ) c∈U
the (maximum) regret for a solution x ∈ X . We write OPT for minx∈X Reg(x). Note that we do not use any probability distribution in the above definition of the minmax regret problem. Still, we make use of probability distributions in the following analysis. To this end, denote by P a probability distribution over the set U , and by EP (·) the expected value with respect to P. We follow the convention to denote random variables with capital letters (e.g., we write C instead of c to denote random cost vectors). We specify the elements that are chosen in a solution x ∈ X by x = {i ∈ N : xi = 1}. Furthermore, we define the worst-case scenario of x as cx , where
cix :=
ci ci
if i ∈ x . if i ∈ /x
This definition extends to vectors cS ∈ U , where S is any subset of N . One directly finds the following result, which is also shown in Aissi et al. (2009). Lemma 2.1. For all x ∈ X , we have that Reg(x) = val(x, cx ) − val∗cx . 3. A general lower bound for minmax regret problems The following observation is crucial for the later analysis. Observation 3.1. Let any probability distribution P over the uncertainty set U be given. Then,
Reg(x) = max val(x, c) − val∗c ≥ EP (val(x, C ) − val∗C ) c∈U
for all x ∈ X .
A. B. Chassein, M. Goerigk / European Journal of Operational Research 244 (2015) 739–747
Theorem 3.5. The midpoint solution xmid is a λ−approximation for the minmax regret problem (R), where
Lemma 3.2. For any fixed probability distribution P, we have:
OPT ≥ min val(x, EP (C )) − x∈X
(
EP val∗C
)
λ=
Proof.
OPT = min max val(x, c) − val∗c x∈X
≥ min EP (val(x, C ) −
val(xmid , cxmid ) − val∗cxmid
val(xmid , cˆ) − minS⊆N
1 2
(val∗cS + val∗cS )
)
(1)
Proof. From the preceding argumentation follows that
)
(2)
OPT ≥ val(xmid , cˆ) −
= min val(x, EP (C )) − EP (val∗C )
(3)
x∈X
= min EP (val(x, C )) − x∈X
(
EP val∗C
x∈X
Inequality (1) follows from Observation 3.1, and Equalities (2) and (3) use the linearity of the expectation and the objective function. Notation 3.3. We call a probability distribution P centered if EP (Ci ) = cˆi for all i = 1, . . . , n. Using Lemma 3.2, we get for all centered probability distributions P a lower bound LB(P ) for the optimal value of the regret problem. Definition 3.4. Given a centered probability distribution P, we set
LB(P ) := val(xmid , cˆ) − EP (val∗C ) Note that LB(P ) is indeed a lower bound of OPT, as
OPT ≥ min val(x, EP (C )) − EP (val∗C ) x∈X
1 min(val∗cS + val∗cS ). 2 S⊆N
≤ val(opt (cxmid ), cxmid ) + val(xmid , cxmid ).
2val(xmid , cˆ) − val(xmid , cxmid ) = 2
= val(xmid , cˆ) − EP (val∗C ) = LB(P ).
P∈P
= min P∈P
=
2
How to solve problem (S ) is further analyzed in Section 4. We denote by LB∗ the lower bound that is given by an optimal solution of (S ). We are now in the position to state the main result of this paper.
ci
i∈xmid
ci + ci − ci 2
ci
= val(xmid , cxmid ) We reconsider inequality (5). By adding val(xmid , cxmid ) on the left hand side and 2val(xmid , cˆ) − val(xmid , cxmid ) on the right hand side we get:
min(val∗cS + val∗cS ) + val(xmid , cxmid ) S⊆N
≤ val(opt (cxmid ), cxmid ) + val(xmid , cxmid ) + 2val(xmid , cˆ) − val(xmid , cxmid ) By rearranging these terms we get:
val(xmid , cxmid ) − val(opt (cxmid ), cxmid ) ≤ 2val(xmid , cˆ) − min(val∗cS + val∗cS )
(6)
S⊂N
From Lemma 2.1 we know that the left hand side of (6) equals Reg(xmid ). If Reg(xmid ) = 0, the midpoint solution is already the optimal solution and we are done, as 0 is a lower bound for all regret problems. If Reg(xmid ) > 0 we use (4) and (6) to show that OPT > 0.
2OPT ≥ 2val(xmid , cˆ) − min(val∗cS + val∗cS ) S⊂N
≥ Reg(xmid ) > 0 Hence we conclude Reg(xmid ) = 0 ⇐⇒ OPT = 0. Assuming that Reg(xmid ) > 0, we know that val(xmid , cˆ) − minS⊂N 12 (val∗cS + val∗S ) > 0 and we can divide it on both sides of
λ := (S )
cˆi −
i∈xmid
Inequality (6) to get
1 (val∗c1 + val∗c2 ) 2
1 = min(val∗cS + val∗cS ) 2 S⊆N
i∈xmid
min EP (val∗C ) ≤ min EP (val∗C )
(5)
The first inequality follows from the fact that the left hand side is the minimum of a set valued function over all possible sets S, and the right hand side is just the same function evaluated for one specific set (the set xmid ). The second inequality holds as val(opt (cxmid ), cxmid ) ≤ val(xmid , cxmid ). Note that 2val(xmid , cˆ) − val(xmid , cxmid ) = val(xmid , cxmid ), as seen by the following equations.
=
c
≤
S⊆N
x∈X
timization problem can be simplified to
Reg(xmid ) OPT
min(val∗cS + val∗cS ) ≤ val∗cxmid + val∗cxmid
i∈xmid
To get the best such bound, one has to find a centered probability distribution that minimizes EP (val∗C ). Due to the linearity of the objective function in c we know that val∗c ≥ val∗c for all c ∈ U . Hence, it follows that EP (val∗C ) ≥ val∗c for all probability distributions P. As a consequence we also have LB(P ) ≤ val(xmid , cˆ) − val∗c . This gives us an immediate upper bound for the lower bound. Note that this upper bound is not necessarily a lower bound for (R). Denote by P the set of all centered probability distributions on U . We consider the problem of finding the best possible lower bound, which can be considered as the optimization problem minP∈P EP (val∗C ). To solve this problem efficiently we restrict the solution space to the special class P of probability distributions defined as follows. We say a scenario in U is an extreme scenario, if and only if ci ∈ {ci , ci } for all i ∈ N . P are all probability distributions that have only two outcomes c1 and c2 , which are both extreme scenarios and equally likely. As the probability distributions are centered, it follows that the scenarios c1 and c2 must be complementary in the sense that if ci1 = ci , it follows that ci2 = ci and vice versa for all i ∈ N . Hence, we can identify every probability distribution P ∈ P by the elements S = {i | ci1 = ci } ⊆ N that attain their upper bound cost in the first scenario. As before, denote by cS the extreme scenario where ciS = ci ∀i ∈ S and ciS = ci ∀i ∈ S. The probability distribution P ∈ P identified with the set S therefore has the nice property that EP (val∗C ) = 0.5(val∗c1 + val∗c2 ) = 0.5(val∗cS + val∗S ). Hence, the op-
(4)
To verify the approximation property we have to show that λ. To this end, we estimate:
= min val(x, cˆ) − EP (val∗C )
P∈P
≤ 2.
Furthermore, we have that OPT = 0 ⇐⇒ Reg(xmid ) = 0.
c∈U
val∗C
741
c
val(xmid , cxmid ) − val(opt (cxmid ), cxmid )
val(xmid , cˆ) − minS⊂N
1 2
(val∗cS + val∗cS )
≤ 2.
From Lemma 2.1 and Inequality (4) follows the desired approximation guarantee
Reg(xmid ) ≤ λ. OPT
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A. B. Chassein, M. Goerigk / European Journal of Operational Research 244 (2015) 739–747
Theorem 3.5 motivates the analysis of the optimization problem (S ). If the set xmid solves the optimization problem (S ) to optimality, we get
However, for the purpose of the following analysis, the problem structure becomes more obvious by writing
min(
min
S⊆N
val∗cS
+
val∗cS
) = val(opt (c
xmid
), c
xmid
) + val(opt (c
xmid
), c
xmid
)
= val(opt (cxmid ), cxmid ) + val(xmid , cxmid ).
(7)
Observation 3.6. If the set xmid solves the optimization problem (S ) to optimality, then λ = 2. Proof. λ = 2 is a direct consequence of Eq. (7) and the fact that 2val(xmid , cˆ) − val(xmid , cxmid ) = val(xmid , cxmid ):
val(xmid , cxmid ) − val(opt (cxmid ), cxmid )
val(xmid , cˆ) − minS⊂N
(val∗cS + val∗cS ) val(xmid , cxmid ) − val(opt (cxmid ), cxmid ) = val(xmid , cˆ) − 0.5val(opt (cxmid ), cxmid ) − 0.5val(xmid , cxmid ) val(xmid , cxmid ) − val(opt (cxmid ), cxmid ) = =2 0.5val(xmid , cxmid ) − 0.5val(opt (cxmid ), cxmid ) 1 2
c
optimization problem for the cost vector cx to obtain val∗x , as we c have already computed val∗c = val∗x . In the next section we consider c problem (S ) in more detail. 4. Solving the lower bound problem (S ) 4.1. General reformulations As before, let an arbitrary combinatorial optimization problem of type (P ) be given. The purpose of this section is to compute the best possible lower bound LB∗ . To solve problem (S ), we need to find a partition of all elements from N into the two sets S and S. We can represent problem (S ) in the following way.
min
n
xi (ci + zi (ci − ci )) +
i=1
n
yi (ci − zi (ci − ci ))
(8)
i=1
s.t. x, y ∈ X
(9)
n
z∈B
(10)
The partition of the set N is modeled via the variables zi , which determine if an element is in S or in S, respectively. Additionally to the z variables, we compute two feasible solutions x, y ∈ X such that the total sum of objective values is minimized. Note that the formulation (8)–(10) is nonlinear. As only binary variables are involved, it can easily be linearized using the following problem formulation, where z indicates which elements are used in both solutions x and y.
min
n
(ci xi + ci yi + (ci − ci )zi )
i=1
s.t. zi ≥ xi + yi − 1 x, y ∈ X z ∈ Bn
∀i ∈ N
s.t. x, y ∈ X with fi : {0, 1, 2} → R,
⎧ ⎪ ⎨0 fi (u) = ci ⎪ ⎩ ci + ci
if u = 0 if u = 1 if u = 2
In the next sections we show how to solve this problem formulation efficiently for the shortest path, the minimum spanning tree, the assignment, and the minimum s − t cut problem. Note that the minmax regret counterpart of all these problems is strongly NP-complete (Aissi, Bazgan, & Vanderpooten, 2005, 2008; Averbakh & Lebedev, 2004). 4.2. The shortest path problem
If, on the other hand, xmid is not the optimal solution, there is a possibility to improve the approximation guarantee. As a general heuristic for the problem (S ) we can first solve the classic optimization problem for the cost vector c, denote the solution that is obtained in this way by x , and then use the partition (x ,x ) for (S ), where x is the complement of x . To evaluate the objective function val∗x + val∗x we just have to solve one additional classic c
fi (xi + yi )
i=1
we can use this equality to tighten the proof of Theorem 3.5 and conclude that λ = 2 in this case
λ=
n
Let a graph G = (V, E, c) be given, with node set V, edge set E, and a cost function c : E → R+ , as well as two nodes s, t ∈ V. The goal is to find the shortest path from s to t, where the length of a path is the sum of the costs ce over every edge e of the path. We can describe the shortest path problem as a combinatorial optimization problem over the ground set E with feasible set X = {x | x represents a s − t path}. To solve (S), the idea is to create a new multigraph G = (V, E , c ) with the same node set V and the edge set E = {e = (i, j), e∗ = (i, j) | ∀e = (i, j) ∈ E} containing the old edge set E as well as one additional parallel edge e∗ for every original edge e ∈ E. The cost function c : E → R+ is defined to be ce for every edge e ∈ E and ce for all copied versions e∗ . Problem (S ) reduces to finding two edge-disjoint s − t paths in G , as can be seen by the following relationship between two paths in G with the cost function f as defined in Section 4.1, and two edgedisjoint s − t paths in G . If an edge e and the copied version e∗ is used on both paths, this edge pair contributes costs of size ce + ce to the objective function, which are exactly the same costs as paid in problem (S ) for using an edge twice. If only one edge of the edge pair e and e∗ is used, it will always be edge e in an optimal solution due to its lower costs, and hence the contribution in the objective function is ce , which are again the same costs as paid in problem (S ) for using an edge once. If no edge of an edge pair is used, no costs accrue in both model formulations. The problem of finding two edge-disjoint paths in G can be represented as a minimum cost flow problem or can be solved directly using, e.g., Suurballe’s algorithm (Suurballe & Tarjan, 1984). Theorem 4.1. LB∗ can be computed in strongly polynomial time for the shortest path problem. 4.3. The minimum spanning tree problem Let again a graph G = (V, E, c) be given with node set V, edge set E, and a cost function c : E → R+ . The goal is to find a spanning tree with minimal costs where the cost of a tree are just the sum over the costs ce over every edge e in the tree. We can describe the minimum spanning tree problem as a combinatorial optimization problem over the ground set E with feasible set X = {x | x represents a spanning tree}. We can reutilize the idea used for the shortest path problem (see Section 4.2) and create the same auxiliary multigraph G = (V, E , c ). But instead of searching for two edge-disjoint s − t paths, we search for two edge-disjoint minimum spanning trees in G . This problem can be solved efficiently as shown in Roskind and Tarjan (1985). Theorem 4.2. LB∗ can be computed in strongly polynomial time for the minimum spanning tree problem.
A. B. Chassein, M. Goerigk / European Journal of Operational Research 244 (2015) 739–747
4.4. The assignment problem In the assignment problem we are given a complete bipartite graph G = (V, E, c) with a bipartition of the node set in equally sized sets V1 ∪ V2 = V and a cost function c : E → R+ . The goal is to find a perfect matching of minimum total cost. It is a well known result that the assignment problem can be transformed to a min cost flow problem on an auxiliary graph G = (V , E , c , u ) (Ahuja, Magnanti, & Orlin, 1993). The idea is to enlarge the graph by introducing one new source node s and one new sink node t to the graph and to connect s with all vertices of V1 and t with all vertices of V2 . The capacity of all edges is set to 1 and the cost of all edges incident to s or t is set to 0. We define an auxiliary graph G = (V , E , c , u ) similar to G by using the same methodology as well as the idea used for the shortest path or minimum spanning tree problem. As before, we introduce one new source node s and one new sink node t to the graph and connect them with the vertices of V1 and V2 . Further, we create for every edge e ∈ E a parallel copy e∗ . The costs of all connector edges are set to 0 and to ce for all edges e ∈ E and to ce for all copies e∗ . The capacity of every connector edge is set to 2 and the capacity of every edge e ∈ E and their corresponding copy e∗ is set to 1. We can translate a flow with flow value 2|V1 | in G to two assignments. Note that due to the capacity constraints of the connecting edges and the required flow value, two units of flow leave every node of V1 and enter every node of V2 . We call two nodes v and u connected if flow is sent from v to u or from u to v. If both units of flow leaving a node of v1 ∈ V1 are sent through an edge e = (v1 , v2 ) and their corresponding copy e∗ = (v1 , v2 ), v1 is assigned to v2 in both assignments. After removing all nodes assigned in that way, we have that every node of V1 is connected to two different nodes of V2 and vice versa. We now pick an arbitrary not yet assigned node v1 ∈ V1 that is connected to v2 and v2 ∈ V2 . We assign v1 to v2 in the first assignment and delete the flow sent from v1 to v2 . By doing this we reduce the number of nodes that are connected to v2 from 2 to 1. Let v1 be the node that is still connected to node v2 . We assign v1 to v2 in the second assignment and delete the flow sent from v1 to v2 . By doing this we reduce the number of nodes that are connected to v1 from 2 to 1. Hence, we know again which assignment pair we have to pick for the first assignment. By repeating this argumentation we generate a cycle, see Fig. 1. Every edge picked with his original orientation in this cycle repre-
743
sents a part for the first assignment. All other edges picked represent a part of the second assignment. If there are still not yet assigned nodes left, we can just start the process again by picking a new not yet assigned node v∗1 ∈ V1 . The argumentation about the costs is the same as in the shortest path or minimum spanning tree problem. If a specific assignment is made twice, two units of flow must be sent through edge e and their corresponding copy e∗ contributing ce + ce to the costs. If a specific assignment is made only once, in an optimal solution the unit of flow will always be sent over the cheaper edge e contributing ce to the costs. Theorem 4.3. LB∗ can be computed in strongly polynomial time for the assignment problem. 4.5. The minimum s − t cut problem Let a graph G = (V, E, c) be given with node set V, arc set E, a cost function c : E → R+ as well as two nodes s, t ∈ V. The goal is to find an s − t cut separating s and t with minimal costs. We can describe the minimum s − t cut problem as a combinatorial optimization problem by using the following well known mixed-integer programming formulation (Lawler, 1976). The variable de indicates if arc e is used in the s − t cut.
min
ce de
e∈E
s.t. de − pi + pj ≥ 0
∀e = (i, j) ∈ E
ps − pt ≥ 1 d ∈ B|E| p ∈ R| V | We transform the graph G to the graph G = (V , E , c ) by replacing each arc with a pair of arcs. We introduce for every arc e = (i, j) ∈ E a new node ve as well as two new arcs e1 = (i, ve ) and e2 = (ve , j). The node set V = V ∪ {ve | ∀e ∈ E} consists of all nodes from V combined with all new nodes. The arc set E = {e1 , e2 | ∀e ∈ E} is the set of all new generated arcs. The new cost function c is defined as, c (e1 ) = ce and c (e2 ) = ce . It can be seen thatthe following problem is equivalent
A
a
A
a
B
b
B
b
C
c
C
c
D
d
D
d
E
e
E
e
F
f
F
f
Fig. 1. Translate flow to two assignments: Every black edge on the left represents one unit of flow. This flow is used to generate two assignments shown on the right side as red (dashed) and blue (solid) assignment. The red assignment is given by (A, a), (B, b), (C, d), (D, c), (E, e), (F, f ) and the blue assignment by (A, a), (B, d), (C, c), (D, b), (E, f ), (F, e). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
744
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to (S ) for the min s − t cut problem:
min
scheme. Let a node in the branching tree be given, and let this node be characterized by the set X of feasible solutions. Then they use the following lower bound on the regret of this node:
de ce + d˜e ce
e∈E
s.t. de − pi + pve ≥ 0
∀e = (i, j) ∈ E
LBold (X ) = SP (c, IN(X ), OUT (X )) − SP (cOUT (X ), ∅, ∅)
d˜e − pve + pj ≥ 0
∀e = (i, j) ∈ E
where SP (c, A, B) is the value of the shortest s − t path including all edges in A and excluding all edges of B for the cost vector c. Applying our approach we get as lower bound for a distribution P .
p s − pt ≥ 2 d, d˜ ∈ B|E |
p ∈ R| V |
Furthermore, as the coefficient matrix for the above problem is totally unimodular, we may solve the LP relaxation instead. As the dual of the LP relaxation is a min-cost flow problem, we have shown the next theorem. Theorem 4.4. LB∗ can be computed in strongly polynomial time for the minimum s − t cut problem. 5. Adapting the lower bound for branch and bound algorithms To find the optimal solution of NP-complete minmax regret problems one might resort on branch and bound algorithms. An important ingredient of a successful branch and bound algorithm is an effective and easy to compute lower bound for every node in the branch and bound tree. As every node in a branch and bound tree defines a restriction for the set of feasible solutions, we have to show how the lower bound introduced in this paper can be computed, if parts of the solutions are already fixed, to make it usable in this context. 5.1. Including fixed variables Assume a branch and bound node defines a restricted feasible set X ⊂ X . Hence the minmax regret problem in this node reduces to
min max val(x, c) − val∗c x∈X
c∈U
Denote by IN(X ) = {i ∈ N | xi = 1 ∀x ∈ X } the set of all ground elements that have to be part of every feasible solution in X and by OUT (X ) = {i ∈ N | xi = 0 ∀x ∈ X } the set of all ground elements that must not be part of any feasible solution in X . Using Lemma 2.1 we can restrict the set U to the set U = {c ∈ U | ci = ci ∀i ∈ IN(X ) ∧ ci = ci ∀i ∈ OUT (X )} without changing the problem. Next we use the argumentation as before, but instead of letting P be an arbitrary distribution over U , we choose an arbitrary distribution P over U .
min max val(x, c) − val∗c = min max val(x, c) − val∗c x∈X
c∈U
x∈X
c∈U
≥ min EP (val(x, C ) − val∗C ) x∈X
= min EP (val(x, C )) − EP (val∗C ) x∈X
= min val(x, EP (C )) − EP (val∗C ) x∈X
Note that if the solution is completely fixed, i.e., X = {x }, then also U = {c } reduces to a set with only one element. In this case the probability distribution P has only one outcome – the vector c – and, hence, the lower bound is equal to Reg(x ), which is an upper bound of the optimal regret value. Therefore, the gap between lower and upper bound vanishes for this branch and bound node.
LBnew (X ) = min val(x, EP (C )) − EP (val∗C ) x∈X = ce − cˆe + SP (cˆ, IN(X ), OUT (X )) − EP (val∗C ) e∈IN(X )
The problem of choosing a good distribution P can again be reformulated as a min cost flow problem or can be solved directly with Suurballe’s algorithm applied to an accordingly adjusted instance, see Section 4.2. Again, we create for each arc e a parallel copy e∗ , but instead of assigning cost of ce to e and ce to e∗ for all arcs, we assign ∀e ∈ IN(X ) (∀e ∈ OUT (X )) costs of ce (ce ) to arc e and his copy e∗ . For all other arcs e ∈ / IN(X ) ∪ OUT (X ) we use the common cost assignment. Hence the effort of computing LBnew (X ) is roughly the same as for three shortest path computations, which is still reasonable compared to the two shortest path computations needed to compute LBold (X ). A direct comparison of both bounds is presented in Appendix B. 6. Experiments We illustrate the strength of the proposed lower bound using the shortest path problem as an example. 6.1. Instances and computational environment In Montemanni et al. (2004) two different types of randomly generated graph classes are considered. We consider the same graph classes but decided to modify the cost structure of the first graph type to gain more flexibility for the experiments. The graph family R-n-c-d-δ consists of randomly generated graphs such that each graph has n nodes and an approximate arc density of δ . The start vertex s is the first node of the graph and the end vertex t is the last node of the graph. The second graph family K-n-c-d-w are layered graphs. Every layer is completely connected to the next layer. The start node s is connected to the first layer and the last layer is connected to the end node t. The complete graph consists of n nodes and every layer of w nodes. To generate the costs, a random number m is sampled from the interval [1, c]. The lower bound cost ce is sampled uniformly from [(1 − d)m, (1 + d)m] and the upper bound cost ce is chosen uniformly from the interval [c, (1 + d)m]. Thus, we can control the cost variability using the parameter d. All experiments were conducted on an Intel Core i5-3470 processor, running at 3.20 GHz, 8 GB RAM under Windows 7. We used CPLEX version 12.6. for comparisons. Algorithms were implemented with Visual Studio 2010 Express using C# 5.0 in the .NET framework v. 4.5. Only one core was used for all computations. 6.2. Value of the lower bound
5.2. Branch and bound algorithm for the minmax regret shortest path problem As before, we assume we are given a graph G = (V, E, c) with node set V, edge set E, a cost function c : E → R+ as well as two nodes s, t ∈ V. In Montemanni et al. (2004) a branch and bound algorithm for the minmax regret shortest path problem is presented. We refer to Montemanni et al. (2004) for a detailed description of the branching
In this experiment we compute the improved lower bound for the approximation ratio of the midpoint solution. We implemented Suurballe’s algorithm to solve the lower bound problem (S ). The results are given by Table 1. The first column λ gives the improved approximation guarantee, while column Tabs gives the absolute time in seconds needed to solve Suurballe’s algorithm. The last column Trel gives the ratio of times needed to compute the lower and the upper bound.
A. B. Chassein, M. Goerigk / European Journal of Operational Research 244 (2015) 739–747 Table 1 Average performance of the new approximation guarantee. Tabs is given in seconds. Graph class
λ
Tabs
Trel
R-10-1000-0.25-1.0 R-100-1000-0.25-0.5 R-500-1000-0.25-0.5 R-1000-1000-0.25-0.5 R-10000-1000-0.25-0.01 R-10000-1000-0.25-0.05 K-102-1000-1.0-2 K-402-1000-1.0-10 K-1002-1000-1.0-10 K-2002-1000-1.0-10 K-4002-1000-1.0-10
1.220 1.330 1.539 1.494 1.692 1.530 1.895 1.640 1.631 1.621 1.625
<0.001 0.001 0.013 0.036 0.204 0.503 <0.001 0.003 0.007 0.011 0.031
1.50 1.57 1.56 1.67 1.27 1.65 1.58 1.55 1.39 1.52 1.59
Every experiment is done on 100 graphs randomly generated from the corresponding graph class, and presented values are averaged. The average approximation guarantees are always below 2, showing the potential of the new lower bound. It is also interesting to observe that the improved approximation guarantee does not decrease with an increasing instance size. But we have to mention that the new approximation guarantee is weaker for K-graphs than for R-graphs. Nevertheless, good improvements can be achieved for K-graphs if the edge costs have high variability. To compute the lower bound we have to do one shortest path computation (to find the value of the midpoint path) as well as one run of Suurballe’s algorithm. The computation of the upper bound needs two shortest path computations, one to find the worst case value of the midpoint path and one to compute the value of the regret path. As expected, the computational effort of Suurballe’s algorithm is comparable to two shortest path computations, which results in a time ratio of circa 1.5. 6.3. Branch and bound algorithms We compare the performance of five different algorithms. The first three algorithms use the branching scheme suggested in Montemanni
10
R−x−1000−0.25−0.1
5
104
10
BBold BBnew
104
BBboth BB*new
10
3
10
2
CPX
10
3
10
2
300
400
500
600
10 0.01
(a) R−graphs with different numbers of nodes.
105 10
4
10
K−x−1000−0.25−5
105
BBold BBnew BBboth
10 10
2
10
101 0
0.05
0.1
0.25
0.5
1.0
10
142
10 0.01
4
0.05
0.1
0.25
0.4
3
162
182
202
(d) K−graphs with different numbers of nodes.
222
K−202−1000−x−5
K−202−1000−0.25−x
105
BBold BBnew BBboth
BBold BBnew BBboth
4
10
*
BBnew CPX
BBnew CPX
3
10
102
1
100 0.01
0.5
(c) R−graphs with different arc densities.
102
1
100 122
10
*
BBnew CPX
102 10
CPX
3
(b) R−graphs with different cost variabilities.
*
3
BBboth BB*new
CPX
0
200
BBold BBnew
104
BBboth
101
0
100
R−500−1000−0.25−x
5
10
BBold BBnew BB*new
101 10
et al. (2004). The only difference of these algorithms is the method to compute the lower bound. In the first algorithm BBold we implemented the lower bound LBold and in the second algorithm BBnew we use our lower bound LBnew (see Section 5.2). The third algorithm BBboth computes both bounds in every branching node and chooses the best one. In this branching scheme a branch and bound node represents a path from node s to some other node v. To enlarge this path the authors of Montemanni et al. (2004) choose the shortest path P from v to t with respect to the worst case cost. This makes sense as they have to compute the path P for calculating their lower bound LBnew anyways. The fourth algorithm BB∗new is again a branch and bound algorithm using lower bound LBnew but instead of computing the worst case path P in every branch and bound node to complete the path we compute the shortest path with respect to the midpoint costs. In this way we save one shortest path computation to compute LBnew . For the fifth algorithm CPX we use the mixed-integer programming formulation of the minmax regret problem (see Appendix A) and solve it using CPLEX. We consider two different experiments. For smaller instances we compare the time needed for each algorithm to solve the problem to optimality, while for larger instances we compare the gaps between the best lower bound and the best upper bound after a time limit of 30 seconds. Every experiment is done on 100 graphs randomly generated from the corresponding graph class. The results of the first experiment are presented in Fig. 2. The computation times on the y-axis are given in milliseconds. For general R-graphs we see that all three branch and bound algorithms have almost the same performance except for BBold that shows a bad performance for instances with very high cost variability. Further we observe that CPX is clearly outperformed by BBnew on R-graphs. It is also worth mentioning that BBold is a bit faster than BBnew on these instances where the quality of the lower bound seems to be less important than the time needed to compute the bound. (See Appendix B for a direct comparison of the quality of LBold and LBnew .) This seems to be different for K-graphs, as, except for some small instances, BBold is clearly outperformed by BBnew . In general
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We showed that any probability distribution over U can be used to compute a lower bound for the minmax regret problem. The solution space of the lower bound problem (S ) is a strongly restricted space of probability distributions. For further research it would be interesting to investigate lower bound problems that are built on wider spaces of probability distributions. Additional research should be made to find an algorithm that has an approximation guarantee that is strictly less than 2 or to show some inapproximability results for minmax regret problems.
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the branch and bound algorithms show a weaker performance on K-graphs than on R-graphs as they are outperformed by CPX. One exception is Fig. 2(f). The increasing layer width reduces on the one hand the problem difficulty for the branch and bound algorithms, as the branching depth is reduced with increasing layer size, and on the other hand increases the number of arcs in the graph which explains the worse performance of CPX, as an increasing number of arcs increases the size of the IP formulation. It is also interesting to note that the algorithm BBboth is outperformed on all instances by BBnew . Hence, we see that in general it is not worth computing LBold if one has already computed LBnew for a branch and bound algorithm. The performances of BBnew and BB∗new are very similar. Therefore, we cannot state clearly which is the better one. Fig. 3 shows the results of the second experiment. These results confirm the statements that are made for the first experiment. On K-graphs the use of branch and bound algorithms is questionable, as they are clearly outperformed by CPX. Nevertheless, the more difficult computation of LBnew pays off especially for complicated instances of both graph classes. For R-graphs the use of branch and bound algorithms seems to be reasonable particularly for instances that are too big to be handled by CPLEX. 7. Conclusions and further research In this work we further refined the analysis of the most widely researched general approximation algorithm to minmax regret optimization: the midpoint solution. By underestimating the worst-case over all scenarios using a purpose-constructed probability distribution, we find a new proof for the well-known result that the midpoint solution is a 2-approximation. However, the proof technique yields further insight, as it can be used to construct approximation guarantees that are dependent on the given instance. To this end, an optimization problem over probability distributions needs to be solved to gain the strongest possible estimate. We show that this problem can easily be solved for many well-known combinatorial optimization problems. Finally, the analysis was further complemented using experimental data for randomly created uncertain shortest path problems. Our results suggest that the new performance guarantee improves the already known guarantee of 2 for most instances and can be computed very fast. The knowledge about the better performance guarantee and the lower bound can be used to considerably improve existing branch and bound algorithms.
Given a directed graph G = (V, E) with interval uncertainties for all edge costs ce ∈ [ce , ce ] ∀e ∈ E. The problem of finding an s − t path with minimal regret can be formulated as an mixed-integer programming formulation.
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The variable x indicates which edges are part of the robust s − t path and the vertex potential y is used to include the value of the shortest s − t path if the worst case scenario for the robust path is realized. For a detailed explanation of the mixed integer formulation we refer to Karasan, ¸ Pinar, and Yaman (2001) and Aissi et al. (2009). Appendix B. Comparison of lower bounds We did the following experiment to compare the quality of LBold and LBnew directly. We generated 1000 graphs of type K-202-10000.25-5 and solved each instance using algorithm BBboth . In each where UB is the upper branching node we compute the gap UB−LB UB bound of the corresponding branching node and LB is either LBold or LBnew . To visualize how the gap evolves during the branching algorithm we assign to every gap value the length of the path that is fixed in the corresponding branch and bound node. The plotted value in Fig. 5 shows the average value of all gaps corresponding to the same path length. Fig. 4 shows the average number of branch and bound nodes per fixed path length. Fig. 5 clearly indicates that LBnew dominates LBold for every fixed path length. But note that LBnew is not better than LBold for all branch and bound nodes of the branching tree. 30 25
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