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Submodular containment is hard, even for networks
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Operations Research Letters 19 (1996) 95-99
Submodular containment is hard, even for networks 1 S. T h o m a s M c C o r m i c k Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, BC, Canada V6T 1Z2 Received l February 1995; revised 1 January 1996
Abstract Suppose that we have two submodular base polyhedra in the same space. What is the complexity of deciding if one is contained in the other? This paper shows that this is strongly co-NP-complete even in the case that the two base polyhedra are defined by cut capacity functions coming from two networks on the same set of nodes. This implies that (unless P = NP) there can be no polynomial algorithm for a problem in dynamic games that asks for a min-cost network that can "counter" any move in a given network.
Keywords." Submodularity; Complexity; Networks; Games
1. Introduction This paper considers the problem o f deciding if the base polyhedron of a given network on node set V is a subset o f the base polyhedron of a second network on V. This question arises as the separation routine when we try to use the Ellipsoid Algorithm (see [6]) to solve a two-person dynamic game on networks considered by Blanchini et al. [2]. Our main result is that this problem is strongly co-NP-complete. This gives a corollary that there cannot be a polynomial algorithm to solve the game unless P = NP.
I This research was partially supported by an NSERC Operating Grant, an NSERC Grant for Research Abroad, and a UBC Killam Faculty Study Leave Fellowship. This research was done while the author was visiting Laboratoire ARTEMIS IMAG at Universit6 Joseph Fourier de Grenoble, France, and developed from discussions at the Discrete Optimization Network Workshop 1 in Trento, Italy, supported by the Human Capital and Mobility Program of the EEC.
This is an interesting result for three reasons. First, the problem that it considers is a pure decision problem, and one where most peoples' first intuition is that it should have a polynomial algorithm. Second, the proof is rather different from most NP-completeness proofs. Third, our co-NP-completeness result shows that the intended application is hard via the Ellipsoid Algorithm, which is unusual. More formally, given a network with node set V, arc set A, and non-negative capacities u E R A, define its cut capacity function x : 2 v ~ R by x(S) = ~-~{uij ] i E S, j f~ S}. We assume w.l.o.g, that for all v ~ w E V, A contains both arcs v ~ w and w ~ v (since we can effectively delete an arc from A by setting its capacity to zero). Cut capacity functions are one o f the best-known examples o f submodular set functions (see [4] for a background on submodularity): Recall that a finite-valued set function f : 2 V R on ground set V is submodular if
f ( S ) + f ( T ) >>.f ( S U T) + f ( S M T) for all S, T C_ V.
0167-6377/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PH S01 6 7 - 6 3 7 7 ( 9 6 ) 0 0 0 1 6-8
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S. T. McCormick/ Operations Research Letters 19 (1996) 95-99
We call a set function f on V of network type, or network submodular, if f is the cut capacity function of some network. We use the standard notation that if x E ~v and SC_V, then x ( S ) ~ E i E s X i. We also define (s) to be the family of all 2-element subsets of S. Thus if x is a function on two-element subsets, then x((S)) = ~-~{x({v,w})l{v,w } C_S}. The base polyhedron associated with f is B f = {x E ~ z Ix(S) ~< f ( S ) , for all S C V, and x ( V ) = f ( V ) } . A feasible flow is a vector x c ~'~ that satisfies 0 ~< x ~< u. The excess of x at node i is ~-~kixki - ~-~ijxij. It is wellknown that the base polyhedron of a cut function is precisely the set of node excesses of feasible flows, sometimes called the boundary of the network. Blanchini et al. [2] consider a problem in twoperson dynamic games on networks that we will call Min-Cost Countering Network: Two players make alternate moves. The first player chooses her move from the base polyhedron of network ~ / ' f on node set V with capacities u f leading to cut function f . This move created imbalances at the nodes equal to the excesses. The second player chooses a move from the base polyhedron of network Jff.q on node set V which is subtracted from the imbalances created by the first player. The second player wants to try to zero out the imbalances. He is given per-unit costs for establishing capacity for all i ~ j E V, and wants to find a min-cost set of capacities ug for the network Jffg leading to cut function g such that B y C_B g. Requiring that g's base polyhedron contains f ' s base polyhedron says that the second player can counter any move by the first player, i.e., can reduce the imbalances to zero. This application motivates us to consider the problem of Submodular Containment: Given two submodular functions f and g on the same ground set V, decide i f B f C B g. Since every constraint in a submodular base polyhedron is tight [4], deciding i f B f C_B g is equivalent to deciding if f ( S ) <~g(S) for all S C_ V. For general submodular functions there appears to be no better way to do this than to check all 2 tzl sets S, which is not polynomial. However, for submodular functions which have a more compact representation, especially those arising from networks, it might seem that there is some hope for finding an efficient algorithm for Submodular Containment. As well, [2] is concerned with dynamic
games on networks, so this is a natural specialization to consider anyway. Thus we restrict our attention to Network Submodular Containment, where f and g are restricted to be the cut functions of two networks on the same node set. The result in this paper is: Theorem 1.1. Suppose that f and g are the cut capacity functions o f networks .Ark = ( V,A, u k ), k = f , g. Then it is strongly NP-complete to decide if B f ~ B °, i.e., Network Submodular Containment is strongly co-NP-complete (and so (general) Submodular Containment is also strongly co-NP-complete). An intuitive way to see this is that one way to try to verify if f ( S ) ~< g(S) for all S C_ V is to mins(g(S) f ( S ) ) . If this minimum is non-negative then B y is indeed contained in Bg; else the minimizing S provides a proof that B f is not contained in Bg. Unfortunately, a difference of submodular functions like g - f is generally not submodular, even for network submodular functions, and minimizing non-submodular set functions is usually quite hard. Note that the complexity of Network Submodular Containment is intimately related to the complexity of Min-Cost Countering Network. If we try to solve Min-Cost Countering Network (which is a linear program with an exponential number of constraints) via the Ellipsoid Algorithm (see Grftschel et al. [6] for details), then the required separation routine is Network Submodular Containment. Therefore a corollary of Theorem 1.1 is: Corollary 1.2. There is no polynomial algorithm for Min-Cost Countering Network (unless P = NP). Although Submodular Containment is a natural problem to consider in this context, showing that MinCost Countering Network is hard by going through the Ellipsoid Algorithm is rather indirect. It would be interesting to have a proof that directly shows that Min-Cost Countering Network is NP-complete. Network f f is a compact representation of the 2lZl values of f , but our proof will use an alternate compact representation for network submodular functions developed by Tomizawa and Fujishige [7] (see also [4, Section 3.6] for an exposition in English). If f is any set function on 2 7, then for v E V we
S.T. McCormick/Operations Research Letters 19 (1996) 95-99 define f l ( v ) = f({v}), and for {v,w} E (v) we define f 2 ( { v , w } ) = f ( v ) + f ( w ) - f ( { v , w } ) . I f f is a cut capacity function these work out to be f l ( v ) = Y~weV-v U~w, and f 2( {v, w}) = (U~w+ U~v), and these O([ VI2) values completely determine f via (1) When a general set function satisfies (1), we say that it is o f order 2. Note that it can take exponential time to check ifa set function given by an evaluation oracle is of order 2, since we can "hide" a perturbation of an order 2 set function in any single subset (see [3]). However, all set functions we consider hereafter are either cut capacity functions (which are certainly of order 2), or will be constructed so that they satisfy (1) (which are of order 2 by definition). Now, given a set function f of order 2, define a transportation network ~"f with left nodes V, each with supply f l ( v ) , right nodes (2r), each with demand f 2 ( { v , w } ) , and arc v ~ {w,x} if and only if v E {w,x}. In order for ~-'-f to have a feasible flow we must clearly have non-negativity of supplies and demands: f l ( v ) >~ 0 for all vE V
and
f 2 ( { v , w } ) >~0 for all {v,w} E (v);
(2)
sum of supplies equals sum of demands: fl(w) = f2 ((2V)) ;
(3)
and for every subset S of supply nodes there must be enough capacity in the demand nodes which are neighbors of S to satisfy the total demand in S: f l ( S ) ~ ~-~{f2( {v,w} ) [ v or w belongs to S} for all S C_ V.
(4)
The three conditions (2)-(4) are also sufficient for there to be a feasible flow in )--f [ 1]. Fix such a feasible flow and define U~w to be the flow on the arc v {v, w} of )'-f. Then it can be seen that f is the cut capacity function for the network with capacities u.
Theorem 1.3 (Tomizawaand Fujishige [7]). A function f : 2 v --* ~ o f order 2 is a network submodular
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set function if and only if its associated f i and f 2 satisfy (2)-(4). For set functions of order 2, conditions (2)--(4) can be checked with O(1 V[2) evaluations of f followed by a network feasibility computation. Thus (as noted in [4]) this theorem gives a polynomial algorithm to decide if a submodular function f of order 2 given by an evaluation oracle is of network type (and if f is network submodular, a network representing f can then be constructed in polynomial time from the feasible flow). If one is willing to restrict Min-Cost Countering Network to networks Jt/'° with B y = B g, then Theorem 1.3 yields a polynomial algorithm [7]. (This is because B f = B g if and only i f f 1 = gl and f 2 = g2, so we can find a min-cost # by solving the above transportation problem.) However, the following example shows that in general, the cost of a network ~A/~gwith B f C B g can be arbitrarily smaller than the cost of a least-cost network JI/~g with B y --- B0: Define Jl/f by V = {1,2,3}, and uf2 = 1, u f = 0 otherwise. Suppose that the cost of establishing capacity is M on arc 1 ~ 2 and zero on all other arcs. Then the unique network X a with B f = / ~ is ,/~f, with cost M. However, the network ~ g with u~3 = ug2 = 1 and u~ = 0 otherwise has B y C B a at cost zero. Now define h = g - f , h 1=gl _ f l , and h 2 = g 2 _ f 2 and formally consider the transportation network )--h with supplies h 1 and demands h 2. Then deciding if B f ~ B g is equivalent to deciding if there is a subset S C_ V such that hi(S) < h2( (s)); we call such a subset a violating subset. Note that since both )--0 and ~'-f satisfy (3), SOmust j-h. Since f ~< 9 is equivalent t ° h >/0, it is tempting to think that it is also equivalent to h 1 t> 0 and h 2 >>.O. Indeed, it is true that f ~< g implies that h I >1 0 since if hl({v}) < 0, then g({v}) < f({v}). Thus from now on we shall restrict our attention to pairs f , g with h I /> 0. However, the following examples show that, even assuming that h 1 >/0, f ~< g is not equivalent to h 2 ~> 0 in either direction: Consider the network 3rh with left nodes {1,2,3} and right nodes {12, 13,23}. With hi(i) = e, i = 1,2,3 and h2(12) = 3e, h2(13) = h2(23) = 0, then {1,2} is a violating set even though we have h i /> 0 and h 2 >>,O. With h1(2) --- e, h i ( l ) = hi(3) = 0 and h2(12) = h2(23) = e, h2(13) = - e , we have no violating sets even though h 1 ~- 0 and h 2 ~ O.
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S. T. McCormick/Operations Research Letters 19 (1996) 95-99
2. Proof o f the theorem
Since we are going to show a co-NP-completeness result, we will reduce an NP-complete problem to the complement o f Network Submodular Containment, i.e., deciding if there is a violating set for h = g - f . The problem that we shall reduce to the complement o f Network Submodular Containment is a restricted form of Clique: the input is an undirected graph G = (N, E), and an integer k, 0 < k < [NI, and we are to decide if G has a clique o f size at least k. We will sometimes talk about the set o f co-ed#es E = (~) - E. We can restrict ourselves w.l.o.g, to graphs arising from the reduction that Garey and Johnson [5] use to prove the NP-completeness o f Clique. It is easy to verify that these graphs have the property that for 0 ~< 1 -%
2ha + 2~ - 2mb - 2nil n(n - 1) - 2m
(5)
Next, we do not want any (k - 1)-cliques to be violating (ignore node 0 for the moment), i.e., ( k - 1)a/> (k~l)b, or k-2b.
a ~> ~
(6)
Note that (6) also implies that no/-clique with I ~< k 1 can be violating. Next, we do want a k-clique to be k violating (still ignoring node 0), i.e., ka < (2)b, or a < ~--~b.
(7)
Putting (6) and (7) together we get l ( k - 2)b ~< a < ½ ( k - 1)b. Thus we can define a = (½k - l)b. Finally,
we want to ensure that for 0 <,%l <,%n - k , no set o f k + l nodes which does not include a k-clique can cause a violation (still ignoring node 0). Recall that any such set o f k + l nodes must include at least I + 1 co-edges. Thus we want ( k + l ) a >/((k2+t ) - ( l + 1 ) ) b + ( l + 1)c, or (substituting the value of a) c ~< ~ ( 2 - k - l).
(8)
Define c = ½b(2 - n), which is the smallest right-hand side over all l -%
~ IS'l~,
(9)
then the amount o f violation for S p will be at least as large as the violation for S, and we will be done. But choosing ~t so that ~ >/(n - 1)fl = (n - 1)(A + ~)/n will ensure (9). This implies that • > / A ( n - 1 ). Choosing ~ = max{0,A(n - 1)} satisfies (9) and ~t >/0 as required. We have not yet fixed the value o f b; it just gives a positive scaling to the other parameters, so we can arbitrarily set b --- 1. We have now fixed all five parameters so that there is a violating set for h if and only if there is a clique o f size k in G. It remains to show that this h can be expressed as g - f for two network submodular functions g and f . Let f come from the network which is the complete directed graph with capacity d on every arc, for a d to be determined soon. We can then compute that the amount o f slack (the fight-hand side minus the left-hand side) in (4) for this f is at least nd for all S ~ 0, V. Thus we can certainly choose d large enough so that when we compute g = f + h for the h defined above, then g will also satisfy (4) and (2). Since both f and h satisfy (3), g does also, and so by Theorem (3), g is also a network submodular function (and so a network representation o f this g can be computed in polynomial time). All numbers involved in this reduction are polynomial functions o f n and m, so our problem is strongly NP-complete.
S. T. McCormick/Operations Research Letters 19 (1996) 95-99
Acknowledgements I thank Franca Rinaldi for telling me about this problem, for helpful discussions, and for pointing out the relevance of [4, Section 3.6]. I also thank Jean Fonlupt for arranging my participation at the Trento DONet Workshop, Satoru Iwata and a referee for pointing out errors in my original version of Theorem 1.3, and Maurice Queyranne for pointing out [3].
[3] [4]
[5]
[6]
References [7] [1] R.K. Ahuja, T.L. Magnanti and J.B. Orlin, Network Flows." Theory, Algorithms and Applications, Prentice-Hall, New York, NY, 1993. [2] F. Blanchini, F. Rinaldi and W. Ukovich, "A network design problem for a distribution system with uncertain demands",
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Research Report 27/93/RR, Dipartimento di Matematica ed Informatica, University of Udine, Italy, 1994; to appear in SIAM J. Optim. W.H. Cunningham, "Minimum cuts, modular functions, and matroid polyhedra", Networks 15, 205-215 (1985). S. Fujishige, Submodular Functions and Optimization, Annals of Discrete Mathematics, Vol. 47, North-Holland, Amsterdam, 1991. M.R. Garey and D.S. Johnson, Computers and Intractability, A Guide to the Theory ofNP-completeness, W.H. Freeman and Company, New York, 1979. M. Grrtschel, L. Lovfisz and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, Berlin, 1988. N. Tomizawa and S. Fujishige, "Theory ofhyperspace (VIII) - on the structures of hypermatroids of network type", Papers of the Technical Group on Circuits and Systems, Institute of Electronics and Communication Engineers of Japan, CAS8162, 1981 (in Japanese).
Operations Research Letters 19 (1996) 95-99
Submodular containment is hard, even for networks 1 S. T h o m a s M c C o r m i c k Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, BC, Canada V6T 1Z2 Received l February 1995; revised 1 January 1996
Abstract Suppose that we have two submodular base polyhedra in the same space. What is the complexity of deciding if one is contained in the other? This paper shows that this is strongly co-NP-complete even in the case that the two base polyhedra are defined by cut capacity functions coming from two networks on the same set of nodes. This implies that (unless P = NP) there can be no polynomial algorithm for a problem in dynamic games that asks for a min-cost network that can "counter" any move in a given network.
Keywords." Submodularity; Complexity; Networks; Games
1. Introduction This paper considers the problem o f deciding if the base polyhedron of a given network on node set V is a subset o f the base polyhedron of a second network on V. This question arises as the separation routine when we try to use the Ellipsoid Algorithm (see [6]) to solve a two-person dynamic game on networks considered by Blanchini et al. [2]. Our main result is that this problem is strongly co-NP-complete. This gives a corollary that there cannot be a polynomial algorithm to solve the game unless P = NP.
I This research was partially supported by an NSERC Operating Grant, an NSERC Grant for Research Abroad, and a UBC Killam Faculty Study Leave Fellowship. This research was done while the author was visiting Laboratoire ARTEMIS IMAG at Universit6 Joseph Fourier de Grenoble, France, and developed from discussions at the Discrete Optimization Network Workshop 1 in Trento, Italy, supported by the Human Capital and Mobility Program of the EEC.
This is an interesting result for three reasons. First, the problem that it considers is a pure decision problem, and one where most peoples' first intuition is that it should have a polynomial algorithm. Second, the proof is rather different from most NP-completeness proofs. Third, our co-NP-completeness result shows that the intended application is hard via the Ellipsoid Algorithm, which is unusual. More formally, given a network with node set V, arc set A, and non-negative capacities u E R A, define its cut capacity function x : 2 v ~ R by x(S) = ~-~{uij ] i E S, j f~ S}. We assume w.l.o.g, that for all v ~ w E V, A contains both arcs v ~ w and w ~ v (since we can effectively delete an arc from A by setting its capacity to zero). Cut capacity functions are one o f the best-known examples o f submodular set functions (see [4] for a background on submodularity): Recall that a finite-valued set function f : 2 V R on ground set V is submodular if
f ( S ) + f ( T ) >>.f ( S U T) + f ( S M T) for all S, T C_ V.
0167-6377/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PH S01 6 7 - 6 3 7 7 ( 9 6 ) 0 0 0 1 6-8
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S. T. McCormick/ Operations Research Letters 19 (1996) 95-99
We call a set function f on V of network type, or network submodular, if f is the cut capacity function of some network. We use the standard notation that if x E ~v and SC_V, then x ( S ) ~ E i E s X i. We also define (s) to be the family of all 2-element subsets of S. Thus if x is a function on two-element subsets, then x((S)) = ~-~{x({v,w})l{v,w } C_S}. The base polyhedron associated with f is B f = {x E ~ z Ix(S) ~< f ( S ) , for all S C V, and x ( V ) = f ( V ) } . A feasible flow is a vector x c ~'~ that satisfies 0 ~< x ~< u. The excess of x at node i is ~-~kixki - ~-~ijxij. It is wellknown that the base polyhedron of a cut function is precisely the set of node excesses of feasible flows, sometimes called the boundary of the network. Blanchini et al. [2] consider a problem in twoperson dynamic games on networks that we will call Min-Cost Countering Network: Two players make alternate moves. The first player chooses her move from the base polyhedron of network ~ / ' f on node set V with capacities u f leading to cut function f . This move created imbalances at the nodes equal to the excesses. The second player chooses a move from the base polyhedron of network Jff.q on node set V which is subtracted from the imbalances created by the first player. The second player wants to try to zero out the imbalances. He is given per-unit costs for establishing capacity for all i ~ j E V, and wants to find a min-cost set of capacities ug for the network Jffg leading to cut function g such that B y C_B g. Requiring that g's base polyhedron contains f ' s base polyhedron says that the second player can counter any move by the first player, i.e., can reduce the imbalances to zero. This application motivates us to consider the problem of Submodular Containment: Given two submodular functions f and g on the same ground set V, decide i f B f C B g. Since every constraint in a submodular base polyhedron is tight [4], deciding i f B f C_B g is equivalent to deciding if f ( S ) <~g(S) for all S C_ V. For general submodular functions there appears to be no better way to do this than to check all 2 tzl sets S, which is not polynomial. However, for submodular functions which have a more compact representation, especially those arising from networks, it might seem that there is some hope for finding an efficient algorithm for Submodular Containment. As well, [2] is concerned with dynamic
games on networks, so this is a natural specialization to consider anyway. Thus we restrict our attention to Network Submodular Containment, where f and g are restricted to be the cut functions of two networks on the same node set. The result in this paper is: Theorem 1.1. Suppose that f and g are the cut capacity functions o f networks .Ark = ( V,A, u k ), k = f , g. Then it is strongly NP-complete to decide if B f ~ B °, i.e., Network Submodular Containment is strongly co-NP-complete (and so (general) Submodular Containment is also strongly co-NP-complete). An intuitive way to see this is that one way to try to verify if f ( S ) ~< g(S) for all S C_ V is to mins(g(S) f ( S ) ) . If this minimum is non-negative then B y is indeed contained in Bg; else the minimizing S provides a proof that B f is not contained in Bg. Unfortunately, a difference of submodular functions like g - f is generally not submodular, even for network submodular functions, and minimizing non-submodular set functions is usually quite hard. Note that the complexity of Network Submodular Containment is intimately related to the complexity of Min-Cost Countering Network. If we try to solve Min-Cost Countering Network (which is a linear program with an exponential number of constraints) via the Ellipsoid Algorithm (see Grftschel et al. [6] for details), then the required separation routine is Network Submodular Containment. Therefore a corollary of Theorem 1.1 is: Corollary 1.2. There is no polynomial algorithm for Min-Cost Countering Network (unless P = NP). Although Submodular Containment is a natural problem to consider in this context, showing that MinCost Countering Network is hard by going through the Ellipsoid Algorithm is rather indirect. It would be interesting to have a proof that directly shows that Min-Cost Countering Network is NP-complete. Network f f is a compact representation of the 2lZl values of f , but our proof will use an alternate compact representation for network submodular functions developed by Tomizawa and Fujishige [7] (see also [4, Section 3.6] for an exposition in English). If f is any set function on 2 7, then for v E V we
S.T. McCormick/Operations Research Letters 19 (1996) 95-99 define f l ( v ) = f({v}), and for {v,w} E (v) we define f 2 ( { v , w } ) = f ( v ) + f ( w ) - f ( { v , w } ) . I f f is a cut capacity function these work out to be f l ( v ) = Y~weV-v U~w, and f 2( {v, w}) = (U~w+ U~v), and these O([ VI2) values completely determine f via (1) When a general set function satisfies (1), we say that it is o f order 2. Note that it can take exponential time to check ifa set function given by an evaluation oracle is of order 2, since we can "hide" a perturbation of an order 2 set function in any single subset (see [3]). However, all set functions we consider hereafter are either cut capacity functions (which are certainly of order 2), or will be constructed so that they satisfy (1) (which are of order 2 by definition). Now, given a set function f of order 2, define a transportation network ~"f with left nodes V, each with supply f l ( v ) , right nodes (2r), each with demand f 2 ( { v , w } ) , and arc v ~ {w,x} if and only if v E {w,x}. In order for ~-'-f to have a feasible flow we must clearly have non-negativity of supplies and demands: f l ( v ) >~ 0 for all vE V
and
f 2 ( { v , w } ) >~0 for all {v,w} E (v);
(2)
sum of supplies equals sum of demands: fl(w) = f2 ((2V)) ;
(3)
and for every subset S of supply nodes there must be enough capacity in the demand nodes which are neighbors of S to satisfy the total demand in S: f l ( S ) ~ ~-~{f2( {v,w} ) [ v or w belongs to S} for all S C_ V.
(4)
The three conditions (2)-(4) are also sufficient for there to be a feasible flow in )--f [ 1]. Fix such a feasible flow and define U~w to be the flow on the arc v {v, w} of )'-f. Then it can be seen that f is the cut capacity function for the network with capacities u.
Theorem 1.3 (Tomizawaand Fujishige [7]). A function f : 2 v --* ~ o f order 2 is a network submodular
97
set function if and only if its associated f i and f 2 satisfy (2)-(4). For set functions of order 2, conditions (2)--(4) can be checked with O(1 V[2) evaluations of f followed by a network feasibility computation. Thus (as noted in [4]) this theorem gives a polynomial algorithm to decide if a submodular function f of order 2 given by an evaluation oracle is of network type (and if f is network submodular, a network representing f can then be constructed in polynomial time from the feasible flow). If one is willing to restrict Min-Cost Countering Network to networks Jt/'° with B y = B g, then Theorem 1.3 yields a polynomial algorithm [7]. (This is because B f = B g if and only i f f 1 = gl and f 2 = g2, so we can find a min-cost # by solving the above transportation problem.) However, the following example shows that in general, the cost of a network ~A/~gwith B f C B g can be arbitrarily smaller than the cost of a least-cost network JI/~g with B y --- B0: Define Jl/f by V = {1,2,3}, and uf2 = 1, u f = 0 otherwise. Suppose that the cost of establishing capacity is M on arc 1 ~ 2 and zero on all other arcs. Then the unique network X a with B f = / ~ is ,/~f, with cost M. However, the network ~ g with u~3 = ug2 = 1 and u~ = 0 otherwise has B y C B a at cost zero. Now define h = g - f , h 1=gl _ f l , and h 2 = g 2 _ f 2 and formally consider the transportation network )--h with supplies h 1 and demands h 2. Then deciding if B f ~ B g is equivalent to deciding if there is a subset S C_ V such that hi(S) < h2( (s)); we call such a subset a violating subset. Note that since both )--0 and ~'-f satisfy (3), SOmust j-h. Since f ~< 9 is equivalent t ° h >/0, it is tempting to think that it is also equivalent to h 1 t> 0 and h 2 >>.O. Indeed, it is true that f ~< g implies that h I >1 0 since if hl({v}) < 0, then g({v}) < f({v}). Thus from now on we shall restrict our attention to pairs f , g with h I /> 0. However, the following examples show that, even assuming that h 1 >/0, f ~< g is not equivalent to h 2 ~> 0 in either direction: Consider the network 3rh with left nodes {1,2,3} and right nodes {12, 13,23}. With hi(i) = e, i = 1,2,3 and h2(12) = 3e, h2(13) = h2(23) = 0, then {1,2} is a violating set even though we have h i /> 0 and h 2 >>,O. With h1(2) --- e, h i ( l ) = hi(3) = 0 and h2(12) = h2(23) = e, h2(13) = - e , we have no violating sets even though h 1 ~- 0 and h 2 ~ O.
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S. T. McCormick/Operations Research Letters 19 (1996) 95-99
2. Proof o f the theorem
Since we are going to show a co-NP-completeness result, we will reduce an NP-complete problem to the complement o f Network Submodular Containment, i.e., deciding if there is a violating set for h = g - f . The problem that we shall reduce to the complement o f Network Submodular Containment is a restricted form of Clique: the input is an undirected graph G = (N, E), and an integer k, 0 < k < [NI, and we are to decide if G has a clique o f size at least k. We will sometimes talk about the set o f co-ed#es E = (~) - E. We can restrict ourselves w.l.o.g, to graphs arising from the reduction that Garey and Johnson [5] use to prove the NP-completeness o f Clique. It is easy to verify that these graphs have the property that for 0 ~< 1 -%
2ha + 2~ - 2mb - 2nil n(n - 1) - 2m
(5)
Next, we do not want any (k - 1)-cliques to be violating (ignore node 0 for the moment), i.e., ( k - 1)a/> (k~l)b, or k-2b.
a ~> ~
(6)
Note that (6) also implies that no/-clique with I ~< k 1 can be violating. Next, we do want a k-clique to be k violating (still ignoring node 0), i.e., ka < (2)b, or a < ~--~b.
(7)
Putting (6) and (7) together we get l ( k - 2)b ~< a < ½ ( k - 1)b. Thus we can define a = (½k - l)b. Finally,
we want to ensure that for 0 <,%l <,%n - k , no set o f k + l nodes which does not include a k-clique can cause a violation (still ignoring node 0). Recall that any such set o f k + l nodes must include at least I + 1 co-edges. Thus we want ( k + l ) a >/((k2+t ) - ( l + 1 ) ) b + ( l + 1)c, or (substituting the value of a) c ~< ~ ( 2 - k - l).
(8)
Define c = ½b(2 - n), which is the smallest right-hand side over all l -%
~ IS'l~,
(9)
then the amount o f violation for S p will be at least as large as the violation for S, and we will be done. But choosing ~t so that ~ >/(n - 1)fl = (n - 1)(A + ~)/n will ensure (9). This implies that • > / A ( n - 1 ). Choosing ~ = max{0,A(n - 1)} satisfies (9) and ~t >/0 as required. We have not yet fixed the value o f b; it just gives a positive scaling to the other parameters, so we can arbitrarily set b --- 1. We have now fixed all five parameters so that there is a violating set for h if and only if there is a clique o f size k in G. It remains to show that this h can be expressed as g - f for two network submodular functions g and f . Let f come from the network which is the complete directed graph with capacity d on every arc, for a d to be determined soon. We can then compute that the amount o f slack (the fight-hand side minus the left-hand side) in (4) for this f is at least nd for all S ~ 0, V. Thus we can certainly choose d large enough so that when we compute g = f + h for the h defined above, then g will also satisfy (4) and (2). Since both f and h satisfy (3), g does also, and so by Theorem (3), g is also a network submodular function (and so a network representation o f this g can be computed in polynomial time). All numbers involved in this reduction are polynomial functions o f n and m, so our problem is strongly NP-complete.
S. T. McCormick/Operations Research Letters 19 (1996) 95-99
Acknowledgements I thank Franca Rinaldi for telling me about this problem, for helpful discussions, and for pointing out the relevance of [4, Section 3.6]. I also thank Jean Fonlupt for arranging my participation at the Trento DONet Workshop, Satoru Iwata and a referee for pointing out errors in my original version of Theorem 1.3, and Maurice Queyranne for pointing out [3].
[3] [4]
[5]
[6]
References [7] [1] R.K. Ahuja, T.L. Magnanti and J.B. Orlin, Network Flows." Theory, Algorithms and Applications, Prentice-Hall, New York, NY, 1993. [2] F. Blanchini, F. Rinaldi and W. Ukovich, "A network design problem for a distribution system with uncertain demands",
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