The probabilistic minimum dominating set problem

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The probabilistic minimum dominating set problem Nicolas Boria 1 , Cécile Murat ∗ , Vangelis Th. Paschos Université Paris-Dauphine, PSL Research University, CNRS UMR 7243, Lamsade, 75016 Paris, France

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Article history: Received 3 June 2015 Received in revised form 13 October 2016 Accepted 24 October 2016 Available online xxxx Keywords: Complexity Approximation Probabilistic optimization Dominating set Wireless sensor networks

abstract We present a natural wireless sensor network problem, which we model as a probabilistic version of the min dominating set problem (called probabilistic min dominating set). We first show that calculation of the objective function of this general probabilistic problem is #P-complete. We then introduce a restricted version of probabilistic min dominating set and show that, this time, calculation of its objective function can be performed in polynomial time and that this restricted problem is ‘‘just’’ NP-hard, since it is a generalization of the classical min dominating set. We study the complexity of this restricted version in graphs where min dominating set is polynomial, mainly in trees and paths and then we give some approximation results for it. © 2016 Elsevier B.V. All rights reserved.

1. Introduction: wireless sensor networks and probabilistic dominating set Very frequently, in wireless sensor networks [37], one wishes to identify a subset of sensors, called ‘‘master’’ sensors, that will have a particular role in messages transmission, namely, to centralize and process messages sent by the rest of the sensors, called ‘‘slave’’ sensors, in the network. These latter sensors will be only nodes of intermediate messages transmission, while the former ones will be authorized to make several operations on messages received and will be, for this reason, better or fully equipped and preprogrammed. So, the objective for designing such a network is to identify a subset of sensors (the master sensors) such that, every other sensor is linked to some sensor in this set. In other words, one wishes to find a dominating set in the graph of sensors. Since the equipment of master sensors induces some extra cost, if this cost is the same for all master sensors, we have a minimum cardinality dominating set problem (min dominating set), while if any master sensor has its own cost, we have a minimum weight dominating set problem. Sensors can be broken down at any time but, since the network must always remain operational, once a sensor failure arrives, a new set of master sensors has to be recomputed very quickly (solution from scratch being very costly in time is proscribed). For simplicity, we deal with master sensors of uniform equipment cost (hopefully, it will be clear later that this assumption is not restrictive for the model) and we suppose that any sensor, can be broken down with some probability qi (so, it remains operational, i.e., present in the network, with probability pi = 1 − qi ) depending on its construction, proper equipment, age, etc.

∗

Corresponding author. E-mail addresses: [email protected] (N. Boria), [email protected] (C. Murat), [email protected] (V.Th. Paschos).

1 Current address: Dipartimento di Automatica e Informatica, Politecnico di Torino, Italy. This research was conducted when author was with the LAMSADE. http://dx.doi.org/10.1016/j.dam.2016.10.016 0166-218X/© 2016 Elsevier B.V. All rights reserved.

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Following the possible failures, we must be able to propose quickly a new solution that is a dominating set for the subnetwork. Since nodes of the initially computed dominating set (called a priori dominating set) have already been equipped, we take in a first time, the track of this solution, i.e., the part of the solution in the surviving (present) sub-network. If it is feasible, then no corrective action is necessary. On the other hand, if the remaining subset is no more a dominating set, then we have to modify it with recourse (i.e., with an additional cost per elementary modification) in order to obtain a dominating set of present sub-network. To equip or program a new sensor has a significant cost, because of obligation to work in emergency. So, the supplementary equipments induced by the recourse will be more expensive. In this paper we handle such a model for min dominating set, called probabilistic min dominating set. The objective of which consists of determining an a priori dominating set through a graph G = (V , E ) of n sensors, with probabilities Pr[V ] = (p1 , . . . , pn ) for sensor performing well. We consider recourse models that take into account the modifications to the a priori dominating set in order to obtain an a posteriori solution, feasible for the subgraph effectively present. Without loss of generality, we assume that equipment costs for a sensor selected in the a priori solution are fixed to 1 and equipment costs for a sensor added in the recourse are fixed to α > 1, for any added sensor. The goal is to minimize the total expected cost. 2. Preliminaries The min dominating set problem including stochastic elements has not been very extensively studied. In [1], authors handle the connected dominating set in graphs where the weights associated to vertices are stochastic. The feasibility of the solution is not a goal there. In our context, the sensors’ presence is the only stochastic element taken into account, and the difficulty comes from the fact that an initial solution (an a priori dominating set) does not necessarily remain feasible after the failure of some of its elements. Such a setting is analogous, for example, to the probabilistic travelling salesman problem with time windows in which each customer has a probability pi of requiring a service on a given day and an a priori tour must be modified for the given day. This problem was studied in [40], following the so-called a priori optimization introduced by Jaillet for the probabilistic travelling salesman problem in [20,21]. Here, we also use the a priori optimization setting for the min dominating set problem. To our knowledge, it is the first time that such a recourse model is proposed for the dominating set. Here, we need to obtain a feasible a posteriori solution for each given subgraph and, for doing this, we need to modify the a priori solution. This context differs from the one of chance constrained approach for which we cannot modify this a priori solution and where the goal is to propose an a priori solution that guarantees, with a fixed probability, that the a posteriori solution (obtained without recourse) will be feasible. Such a model is proposed in [28,4,5] for wireless sensor networks, where sensors can fail: the aim is to assign transmission powers to the nodes of a wireless sensor network in such a way that connectivity should be guaranteed with a given level of reliability, while the total cost is minimized. Note that neither a priori optimization, nor chance constrained approach has ever been applied for the dominating set problem. The framework of probabilistic combinatorial optimization that we adopt in this paper was introduced by [20,7]. In [2,7–10, 20–23], restricted versions of routing and network-design probabilistic minimization problems (in complete graphs) have been studied under the robustness model dealt here (called a priori optimization). In [3,11,12,15], the analysis of the probabilistic minimum travelling salesman problem, originally presented in [7,20], has been revisited in order to propose new efficient resolution. In [16,17], authors introduce a generalization of the probabilistic travelling salesman with time constraints and study two types of recourse. Several other combinatorial problems have been also handled in the probabilistic combinatorial optimization framework, with or without recourse, including minimum colouring [32,14], maximum independent set and minimum vertex cover [30,31], longest path [29], Steiner tree problems [35,36], minimum spanning tree [9,13]. The paper is organized as follows. In the next section, we introduce the probabilistic min dominating set-problem. We consider 2-stage modification strategies that in the first stage take the track of the a priori solution in the surviving subgraph and in the second stage, if this track is infeasible, they complete it in some way into a feasible solution. We establish the expression of the total expected cost associated to probabilistic min dominating set and show that, even under relatively simple and natural second stage completions (as for example greedily entering some non-dominated vertices in final solution), the calculation of its objective function is #P-complete. We then focus on a very simple (almost ‘‘silly’’) modification strategy under which probabilistic min dominating set is in NP (Section 3.3). For this version, we study in Section 4, polynomial cases, in particular when input graphs are paths, cycles or trees and we propose polynomial time algorithms for these cases. Finally, in Section 5, we give some approximation results for both the cases of identical sensor probabilities and of distinct probabilities. The main results in Section 4 imply that probabilistic min dominating set is polynomial in trees with degrees bounded by O(log n) and in general trees assuming identical probabilities. It remains however open if the problem is polynomial in general trees with distinct vertex-probabilities, which seems to be a difficult problem. The approximability upper bounds given in Section 5 are quite far from those known for the classical min dominating set problem. For instance, although min dominating set is approximable within ratio O(log n), probabilistic min dominating set is approximable within ratios ∆ + 1 for the recourse model and ∆ − ln ∆, for a simplified version of it 2 where α = 1, in the case of identical sensors, and within ratio ∆ /ln ∆ when heterogeneous sensors are assumed, where ∆ denotes the maximum degree of the input graph. This might be due to the fact that probabilistic min dominating set (even in its simplified form) seems to be much harder than its deterministic counterpart.

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Fig. 1. A graph with a dominating set.

Algorithm 1 The modification strategy M. Input: A graph G = (V , E ), an a priori dominating set D and a set V ′ ⊆ V Output: A dominating set D′ for G[V ′ ] given the dominating set D, order the vertices of V in such a way that {v1 , v2 , . . . , vn−|D| } (i.e., the first n − |D| vertices) are the vertices of V \ D and {vn−|D|+1 , . . . vn } are the vertices of D; D′ := ∅; for i := n − |D| + 1, . . . , n do if vi ∈ V ′ then D′ := D′ ∪ {vi }; for i := 1, . . . , n − |D| do if vi ∈ V ′ and Γ (vi ) ∩ D′ = ∅ then D′ := D′ ∪ {vi }; return D′ . 3. Probabilistic dominating set 3.1. Problem definition It is clear that given a network of sensors, identifying master set of them is equivalent to determining a dominating set in the associated graph where vertices are the sensors of the network and, for any linked pair of them, an edge links the corresponding vertices. The min dominating set problem is formally defined as follows. Let G = (V , E ) be a connected graph defined on a set V of vertices with a set E ⊆ V × V of edges. A vertex-set D is said to be a dominating set of G if, for any v ∈ V \ D, v has at least one neighbour in D. In the min dominating set problem, the objective is to determine a minimum-size dominating set in G. The decision version of min dominating set problem is one of the first 21 NP-complete problems [25] and remains NP-complete even in bipartite graphs, while it is polynomial in trees. In the sensor network problem handled in the paper, we associate a probability pi with every vertex vi ∈ V (the probability that sensor i remains operational). So, given a set D of master sensors, if some sensors of D fail, we need to adapt D to the surviving network in order to maintain it operational at any time. If recourse is not allowed the surviving subset could no longer be a dominating set. For example, consider the graph of Fig. 1 for which the set D = {v1 , v5 , v7 } is dominating. If we consider V ′ = {v1 , v2 , v3 , v6 , v7 , v9 } as the set of present vertices, then D′ = D ∩ V ′ = {v1 , v7 } is no longer a dominating set of G′ the subgraph induced by V ′ . Since min dominating set is NP-complete, we do not have sufficient time to compute ex novo another optimal solution when sensor failures occur. So, we propose a recourse induced by a modification strategy, denoted by M, such that when given a dominating set D of G and a subgraph G′ = G[V ′ ] induced by a set V ′ ⊆ V (the surviving sensors), it transforms D into a set D′ that is a dominating set for G′ . Such a strategy could be, for example, one specified in Algorithm 1, where Γ (v) denotes the set of neighbours of vertex v ∈ V . For instance, with respect to the graph of Fig. 1, the second stage completion of D′ performed by Algorithm 1 would enter in it vertices v3 and v6 . Consider now a graph G = (V , E ), a probability pi for every vertex vi ∈ V , a dominating set D of G and the modification strategy M just specified. The functional of M (associated with the a priori solution D) representing the total expected cost can be expressed by:

E(G, D, M) =



Pr V ′ val D′

 

 

(1)

V ′ ⊆V

where Pr[V ′ ] is the probability of V ′ (i.e., the probability that only the vertices of V ′ will not be broken down), D′ is the dominating set returned by M for G[V ′ ] and val(D′ ) is its cost. The probabilistic min dominating set problem consists of determining the optimal a priori master set D∗ which minimizes the functional given in (1). In other words, for our wireless sensor network design problem, the a priori solution D∗

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has the property that, under the recourse M described above, the solution constructed on the surviving network minimizes in average the additional cost needed in order that the network remains operational. In general several computational-mathematical problems can appear in such an issue. The first one is to write the functional down in an analytical closed form: we handle this issue in Section 3.2. If such an expression for the functional is possible, a second issue is to prove that its value (i.e., the objective function of the corresponding probabilistic problem) is polynomially computable; this is neither trivial nor senseless (see, for example, [17,27,41]), since the summation for the functional in a graph of order n, includes 2n terms, one for each subgraph of G. So, polynomiality of the computation of the functional is, in general, not immediate. We now write down (1) in a more explicit way. Starting from this generic expression for the functional and denoting by wi , the weight associated with vertex vi (wi = 1, if vi ∈ D, and wi = α > 1, otherwise), one gets:

E(G, D, M) =



  

Pr V ′ ·

vi ∈V

V ′ ⊆V

=



=



wi · 1vi ∈D′ =

 ′

Pr V

vi ∈D

pi + α ·

Pr V ′ · wi · 1vi ∈D′

 

vi ∈V V ′ ⊆V

· 1vi ∈V ′ + α ·

vi ∈D V ′ ⊆V



 

Pr V ′ · 1vi ∈D′

 

vi ∈V \D V ′ ⊆V



  Pr vi ∈ D′ .

(2)

vi ∈V \D

Indeed, let us first recall that we consider 2-stage modification strategies that in the first stage take the track of the a priori solution in the surviving subgraph and in the second stage, if this track is infeasible, they complete it in some way into a feasible solution. In other words, if a vertex vi is in D, then it will be always in D′ if it belongs to V ′ ; hence, its contribution to E(G, D, M) in (1) will be equal to 1 × pi . On the other hand, if vi ̸∈ D but vi ∈ V ′ , it will be included in D′ only if at the moment it is handled during the second stage of M, no neighbour of it in V ′ is included in D′ . 3.2. The complexity of probabilistic min dominating set The goal of this section is the study of the complexity of probabilistic min dominating set whose objective function is given by (2). As we show in what follows, computation of this function cannot be always done in polynomial time. More precisely, we will prove that calculation of the functional associated to probabilistic min dominating set under modification strategy M is #P-complete. For this, we devise a reduction from a particular version of #sat to probabilistic min dominating set. #sat, consists of calculating the number of models of a CNF formula. #sat is #P-complete even in the case, called monotone #2sat, where each clause contains only two literals appearing only positively [39]. Theorem 1. Calculating E(G, D, M) as given by (2) is #P-complete, even for α = 1. Proof. We reduce monotone #2sat to probabilistic min dominating set. For this, to any clause Ck = (xi ∨ xj ) is associated the gadget presented in Fig. 2. In all, given a formula φ over m 2-clauses on n literals, an instance G = (V , E ) for probabilistic min dominating set is built in the following way:

• V = V1 ∪ V¯ 1 ∪ U such that any xi , i = 1, . . . , n, is associated with vertices vi ∈ V1 and v¯ i ∈ V¯ 1 , and with any clause Ck , k = 1, . . . , m, is associated vertex uk ∈ U; this last set of vertices forms a clique denoted by K ; • E = E1 ∪ E2 ∪ E3 such that any literal xi , i =, . . . , n, is associated with edge e = (vi , v¯ i ) ∈ E1 , with any clause Ck = (xi ∨ xj ), k = 1, . . . , m are associated four edges of E2 : (vi , v¯ j ), (vj , v¯ i ), (uk , vi ) and (uk , vj ); E3 is the set E (K ) of edges of the clique K. In Fig. 3, we give an illustration of the transformation described just above for the monotone #2sat-instance (xi ∨ xj ) ∧ (xj ∨ xk ) ∧ (xk ∨ xl ). Vertices of V¯ 1 ∪ U are assigned with probability 1 and those of V1 with a probability 1/2. Hence, a subgraph of G will be associated with a unique subset of X (the set of variables of φ ) corresponding to a truth assignment on the literals such that if vi is present in the final graph xi will be set to TRUE while, if vi is absent, xi will be set to FALSE. A finally present subgraph (a scenario) will then be in bijection with a unique global truth assignment on φ that makes it TRUE or FALSE. Any scenario has the same occurrence-probability (1/2)n . Finally, we consider as a priori solution the dominating set defined by V1 and we consider the vertices in the following order: {¯v1 , . . . , v¯ n , u1 . . . , um , v1 , . . . , vn }. Note that for the initial graph, that is the scenario that corresponds to an assignment of all the literals with TRUE, φ is satisfied. We now examine all the possible cases with respect to the dominating set induced by the a priori solution just considered, on the gadget associated with clause Ck = (xi ∨ xj ):

• if vi and vj are both present, then application of M results in set {vi , vj } that dominates the five vertices vi , v¯ i , vj , v¯ j , uk ; • if one among vi and vj is present, say vi , application of M results in set {vi } dominating the four present vertices vi , v¯ i , v¯ j , uk ; • if, finally, vi and vj are both absent, then application of M results in a subset of vertices among v¯ i , v¯ j , uk , that depends on the other vertices of the clique K .

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Fig. 2. Transformation of Ck = (xi ∨ xj ).

Fig. 3. Transformation of the instance (xi ∨ xj ) ∧ (xj ∨ xk ) ∧ (xk ∨ xl ).

Recall (2) and apply it on a priori solution D = V1 . We have: E (G, D, M) =

 1 2

vi ∈V1

=

n



Pr v¯i ∈ D′ +





v¯ i ∈V¯ 1



+

2

+



Pr uk ∈ D′





uk ∈U

     Pr uk ∈ D′ . Pr v¯ i ∈ D′ +

v¯ i ∈V¯ 1

uk ∈U

Note that a vertex v¯ i will be added in D′ only if all its neighbours that were in the a priori solution D are all absent from V ′ . Let mi be the number of clauses containing literal xi . Then, vertex v¯ i has mi +1 neighbours. In other words, Pr[¯vi ∈ D′ ] = (1/2)mi +1 . On the other hand, since any scenario G′ = G[V ′ ] has probability (1/2)n , it holds that Pr[uk ∈ D′ ] = (1/2)n · NG′ (uk ), where NG′ (uk ) is the number of scenarios where vertex uk belongs to D′ . Observe that uk can be added in D′ only if vi and vj are both absent, i.e., when the assignment induced by this scenario does not satisfy φ . Furthermore, any time M adds a vertex uk in D′ , no other vertex of U will be added, since a single vertex (uk ) is sufficient for dominating all the other vertices of U (that form a clique). Hence, for any unsatisfied formula φ , only one vertex of U will be added in D′ . Consequently:



Pr uk ∈ D′ =





uk ∈U

 n  n  1 1 · NG′ (uk ) = · Nφ¯ 2

2

uk ∈U

where Nφ¯ denotes the number of assignments that do not satisfy φ . Therefore: E (G, D, M) =

n 2

+

  1 mi +1 v¯ i ∈V¯ 1

2

 n +

1

2

· Nφ .

So, if E (G, D, M) could be calculated in polynomial time, then Nφ could be so, contradicting the #P-completeness of # 2sat. 

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Algorithm 2 A simpler modification strategy S. Input: A graph G = (V , E ), an a priori dominating set D and a set V ′ ⊆ V Output: A dominating set D′ for G[V ′ ] D′ := ∅; for vi ∈ V ′ do if vi ∈ D then D′ := D′ ∪ {vi }; else if Γ (vi ) ∩ (D ∩ V ′ ) = ∅ then D′ := D′ ∪ {vi }; return D′ .

Fig. 4. The subgraph G[V ′ ] and the solution computed by S in Example 1.

3.3. A simpler modification strategy under which probabilistic min dominating set belongs to NP Let us now specify a slightly simpler recourse model. Consider the modification strategy S specified in Algorithm 2. There, S first takes D ∩ V ′ in D′ and then completes it with all the non-dominated vertices of V ′ . It is easy to see that the complexity of S is polynomial, since it is bounded by the sum of the degrees of the vertices of V ′ \ D that is at most O(|E |). With the modification strategy S, the approach used to complete the dominating set in the subgraph is no more depending on the order of nodes are selected for the completion of the dominating set. This allows to compute the induced functional faster. Example 1. Consider the graph of Fig. 1 and a dominating set D = {v1 , v5 , v7 } and its subgraph induced by the set V ′ = {v1 , v2 , v3 , v6 , v7 , v9 } (Fig. 4). For these vertices two cases are possible: 1. either a vertex is in D and it will be part of D′ also (this is the case of v1 and v7 ); 2. or a vertex does not belong to D and in this case it will be added to D′ if all its neighbours that were in D are not in V ′ (this is the case of v3 , v6 and v9 ). The set D′ is shown in Fig. 4 by the bold-circled vertices.



Let us now study the functional induced by S. Reasoning as in (2), if vi ̸∈ D but vi ∈ V ′ , it will be included in D′ only if all itsneighbours in G that belonged to D are not in V ′ ; in this case, its contribution to E(G, D, S) in (1) will be equal to α × pi vj ∈Γ (vi )∩D (1 − pj ). Based upon these remarks, starting from (1) we get:

E(G, D, S) =



pi + α ·



vi ∈D

vi ̸∈D V ′ ⊆V



pi + α ·



=

vi ∈D

vi ̸∈D

pi ·

Pr V ′ · 1(vi ∩V ′ =vi ) · 1Γ (vi )∩(D∩V ′ )=∅

 



1 − pj .





(3)

vj ∈Γ (vi )∩D

It is easy to see that, from (3), E(G, D, S) can be computed in polynomial time. Also, setting pi = p, i.e., considering that vertices of G have identical probabilities (this is quite natural if we assume identical sensors), one gets:

E(G, D, S) = p|D| + α



p(1 − p)|Γ (vi )∩D| .

(4)

vi ̸∈D

For the rest of the paper we study this (restricted) version of probabilistic min dominating set, denoted by probabilistic min dominating set(S). Unfortunately, even if the functional is written in an analytical closed form, this form does not derive a compact combinatorial characterization for the optimal a priori solution of probabilistic min dominating set(S). In particular, the form of the functional does not imply a solution, for instance, of some well-defined weighted version of the (deterministic) min dominating set. This is due to the second term of (3). There, the ‘‘costs’’ assigned to the vertices depend on the structure of the a priori solution chosen and of the present subgraph of G. The following results hold for probabilistic min dominating set(S).

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(a) Elementary patterns.

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(b) Examples of partitions. Fig. 5. Partitioning nodes with elementary patterns.

Proposition 1. probabilistic min dominating set(S) is NP-hard, even in bipartite graphs and inapproximable in polynomial time within ratio O(log n). Proof. The inclusion of probabilistic min dominating set(S) in NP is immediately deduced from (3) that is computable in polynomial time. Also, if in (4) we set p = 1, then E(G, D, S) = |D|, i.e., the a priori solution minimizing E is a minimum dominating set in G. So, probabilistic min dominating set(S), having as particular case min dominating set, inherits all its hardness results either in exact computation [19,25], or in polynomial approximation [18]. Note the O(log n) lower bound of [18] mentioned in the statement of the proposition is originally addressed to min set cover problem. But, by a classical approximability preserving reduction (that preserves both constant ratios and ratios depending on the sizes of the instances [34]), this bound also applies to min dominating set.  The main results in Section 4 imply that probabilistic min dominating set(S) is polynomial in trees with degrees bounded by O(log n) and in general trees assuming identical probabilities. It remains however open if it is polynomial in general trees with distinct vertex-probabilities, which seems to be a difficult problem. 4. Probabilistic dominating set in paths, cycles and trees We handle in this section probabilistic min dominating set(S) in paths, cycles and trees. Let us recall that min dominating set in these graphs is polynomial. We prove that probabilistic min dominating set(S) in paths and cycles remains polynomial for any vertex-probability, while probabilistic min dominating set(S) in trees is polynomial either when the maximum degree of the input tree is bounded, or when the vertex-probabilities are all equal. Recall that the contribution of a node refers to its weight and to the probability for this node to be present in D′ for a given a priori solution D and for the modification strategy S adopted in Section 3.3. Let us recall that the contribution of a node vi  that belongs to D is pi , and for a node that does not belong to D, its contribution amounts to α · pi vj ∈Γ (vi )∩D (1 − pj ). This notion will be extended to a set: the contribution C (V ′ ) of a node-set V ′ is the contribution of the nodes of V ′ in D′ . In other words, C (V ′ ) is the sum of the contributions of all the nodes in V ′ . 4.1. Paths and cycles Given a path we consider its nodes labelled in the following way: the leftmost endpoint of the path is labelled by v1 (which might be referred as the first, or left end node), while the rightmost endpoint will be labelled by vn (last or right end node). Of course, all nodes in the path between v1 and vn will be labelled in increasing order from left to right. We show how probabilistic min dominating set(S) in paths can be solved by dynamic programming. First, let us make a preliminary remark that will help us to build the final algorithm. Remark 1. For a given node vi in a dominating set, the ‘‘next’’ dominating node (if any) will be vi+1 , vi+2 or vi+3 . Indeed, if none of them is in the dominating set, then vi+2 would not be dominated.  The general idea of the dynamic programming algorithm goes as follows. Given some dominating set D, we build a unique partition (V1 , V2 , . . . , V|D| ) of V according to the following rules:

• set V1 := {v1 } and k := 1; • we visit vertices vi from v1 to vn−1 and: – if vi ∈ D then k := k + 1 and Vk := ∅; – else, Vk := Vk ∪ {vi+1 }. By Remark 1, any Vk is necessarily of the form D1 , or D2 , or D3 . In other words, any dominating set D on a path (except possibly the very last node of this path) can be partitioned into elementary patterns of shapes D1 , D2 or D3 , which are illustrated in Fig. 5.

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Algorithm 3 An optimal dynamic programming algorithm for probabilistic min dominating set(S) in paths. Input: A path P = (v1 , . . . , vn ) Output: An optimal a priori dominating set D∗ for P for i := 1 to n do for j := 1 to 3 do compute C (Dj (i)) as given in (5a) to (5d); C (W0 ) := 0; C (W1 ) := p1 ; C (W2 ) := min{C (W1 ) + C (D1 (2)), C (W0 ) + C (D2 (2))}; C (W3 ) := min{C (W2 ) + C (D1 (3)), C (W1 ) + C (D2 (3))}; for i := 4 to n do C (Wi ) := minj=1,2,3 {C (Wi−j ) + C (Dj (i))}; if C (Wn ) < C (Wn−1 ) + α pn (1 − pn−1 ) then D⋆ := Wn ; elseD⋆ := Wn−1 ; return D∗ .

Note that the contribution of a given pattern can be determined, regardless of the rest of the sequence, but depending on its position in the path. To be more precise, denoting by Dj (i) the pattern Dj with its dominating node being vi : C (D1 (i)) = pi C (D2 (i)) = pi + α pi−1 (1 − pi )

∀i i=2

(5b)

C (D2 (i)) = pi + α pi−1 (1 − pi ) (1 − pi−2 )

i>3

(5c)

C (D3 (i)) = pi + α pi−1 (1 − pi ) + α pi−2 (1 − pi−3 )

i > 4.

(5d)

(5a)

Indeed, a pattern D1 has clearly a contribution of pi , namely, the probability of its only node, which is a dominating one. The contribution of D2 (i) depends on its position: either i = 2, then its last node is the second one of the path. Its first node v1 (which is also the first node of the path) is only dominated by v2 ; this amounts to (5b). For any other i, it is clear that vi−1 is dominated by both vi and vi−2 , given the way a sequence is built: any sequence ends with a dominating node, so that vi−2 has to be a dominating node. This leads to (5c). For the same reason, in a sequence D3 (i) (which can only appear with i > 4), vi−1 is only dominated by vi , and vi−2 is only dominated by vi−3 , which leads to (5d). This will be very useful for the method. Indeed, if a pattern has no impact in terms of contribution neither on the preceding sequence, nor on the following one, then it is quite easy to give a recursive definition of Wi which is the sequence up to node vi of minimum contribution, where vi is a dominating node:

C (Wi ) =

 0    p 1

     min C Wi−j + C Dj (i) j=1,2          min C Wi−j + C Dj (i) j=1,2,3

i=0 i=1 i = 2, 3

(6)

4 6 i 6 n.

We are ready now to specify a dynamic programming algorithm (Algorithm 3), optimally solving probabilistic min dominating set in paths in polynomial time. It receives as input a path (v1 , . . . , vn ) and a probability pi for each vi ∈ V and outputs an optimal a priori dominating set D⋆ . Proposition 2. probabilistic min dominating set(S) in paths can be solved in polynomial time. Proof. We denote by CV (D) the contribution of a dominating set D on the path V = (v1 , . . . , vn ); we are going to prove that D∗ computed by Algorithm 3 is such that CV (D⋆ ) = minD {CV (D)}. The functional of a dominating set D on a path V = V1 ∪ V2 , is the sum of the contributions of D on the sub-paths V1 and V2 . In particular, for V1 = (v1 , . . . , vn−1 ) and V2 = (vn ) we have: CV (D) = CV1 (D) + CV2 (D). Concerning vn , only two cases are possible: 1. either vn is in D (so, vn is the last dominating node), 2. or vn is not in D (consequently, the last dominating node is vn−1 ). As vn must be dominated, the last dominating node cannot be other than vn−1 , or vn . So, for D = D⋆ we must have: CV (D⋆ ) = min{C (Wn ), C (Wn−1 ) + α pn (1 − pn−1 )}, where Wi has been defined just above. So, in order to compute CV (D⋆ ), we need to determine C (Wn ) and C (Wn−1 ). For that, we use the recursive definition of Wi given in (6). Indeed, a sequence Wi always ends with a dominating node. So, leaving aside particular cases when i 6 3, it ends necessarily with one of the three patterns defined earlier. If it ends with D1 (resp., D2 or D3 ) then it should be completed with a minimal sequence up to node vi−1 (resp., vi−2 or vi−3 ), namely Wi−1 (resp., Wi−2 or Wi−3 ). Among these three possibilities, the minimal contribution is chosen for C (Wi ). In all, 3n contributions C (Dj (i)) have to be computed and, in order to compute a Wi , one basically compares 3 possible values, which amounts to 6n operations to get an optimal a priori solution for probabilistic min dominating set(S) in paths. This concludes the proof. 

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Fig. 6. 6 cases regarding positions of the first and last dominating nodes.

By adapting the method, one can extend the result of Proposition 2 to cycles. It suffices to choose an arbitrary node as starting point v1 (whose neighbours will be v2 and vn ), and to notice that there are three possible positions regarding the ‘‘first’’ (i.e. the lowest-labelled) dominating node, namely v1 itself, v2 or v3 . Depending on the position of the first dominating node, there are different possibilities for the position of the ‘‘last’’ (e.g. highest labelled) one: 1. the first is v1 and the last vn , or vn−1 , or vn−2 ; 2. the first is v2 , the last vn or vn−1 ; 3. the first is v3 , the last is necessarily vn (or else v1 is not dominated). So, we have a total of 6 cases for the position of the ‘‘first’’ and ‘‘last’’ dominating nodes. These 6 cases are illustrated in Fig. 6, and will be compared by the algorithm. In order to take all these cases into account, we will have to define three kinds of partial sequences Wi , Wi′ and Wi′′ which represent, respectively, the three cases defined earlier. More precisely, Wi represents the optimal sequence ending with node vi and starting with pattern D1 , Wi′ corresponds to ending with node vi but starts with D2 and Wi′′ does the same but starts with D3 . The contributions of these three sequences are as follows:

 p1    p 1 + p 2       C (Wi ) = min C Wi−j + C Dj (i) j = 1 , 2           min C Wi−j + C Dj (i) j=1,2,3

i=1 i=2 i=3 i > 4.

This definition differs from that defined in (6) only on case i = 2, to ensure that v1 will remain a dominating node anyway. For the same reason, we define C (Wi′ ) and C (Wi′′ ) as follows:

α p (1 − p ) + p 1 2 2     α p 1 − p + p ( ) 1 2 2 + p3         C Wi′ = min C Wi′−j + C Dj (i)  j = 1 , 2     min C W ′  + C Dj (i) j=1,2,3

i −j

i=2 i=3

i>5

α p (1 − p ) + α p (1 − p ) + p n 2 3 3 1     α p 1 − p + α p 1 − p + p ( ) ( ) 2 3 3 + p4  ′′   1   n    ′′ C Wi = min C Wi−j + C Dj (i)  j = 1 ,2     min C W ′′  + C Dj (i) j=1,2,3

i−j

(7)

i=4

i=3 i=4 i=5

(8)

i > 6.

The dynamic programming Algorithm 4, optimally solves probabilistic min dominating set(S) in cycles in polynomial time. It receives as input a cycle (v1 , . . . , vn , v1 ) and a probability pi for each vi ∈ V and outputs an optimal a priori dominating set D⋆ . Proposition 3. probabilistic dominating set(S) in cycles can be solved in polynomial time. Proof. Using these sequences, in order to build an optimal dominating set, one defines the optimal solution D∗ as the minimal dominating set over the 6 cases defined earlier, whose structure is shown in Fig. 7 and whose contributions amount

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Algorithm 4 An optimal dynamic programming algorithm for probabilistic min dominating set(S) in cycles. Input: A cycle Q = (v1 , . . . , vn , v1 ) Output: An optimal a priori dominating set D∗ for Q for i := 1 to n do for j := 1 to 3 do compute C (Dj (i)) as given in (5a) to (5d); C (W1 ) := p1 ; C (W2 ) := p1 + p2 ; C (W3 ) := minj=1,2 {C (W3−j ) + C (Dj (3))}; for i := 4 to n do C (Wi ) := minj=1,2,3 {C (Wi−j ) + C (Dj (i))}; C (W2′ ) := α p1 (1 − p2 ) + p2 ; C (W3′ ) := α p1 (1 − p2 ) + p2 + p3 ;      C (W4′ ) := minj=1,2 C W4′ −j + C Dj (4) ; for i := 5 to n do      C (Wi′ ) := minj=1,2,3 C Wi′−j + C Dj (i) ; C (W3′′ ) := α p1 (1 − pn ) + α p2 (1 − p3 ) + p3 ; C (W4′′ ) := α p1 (1 −pn ) + α p2 (1 − p3 ) +p3 + p4 ; C (W5′′ ) := minj=1,2 C W5′′−j + C Dj (5) ; for i := 6 to n do      C (Wi′′ ) := minj=1,2,3 C Wi′′−j + C Dj (i) ; compute C (D∗ ) as given in (9); return D∗ .

(a) Case 1a.

(b) Case 1b.

(d) Case 2a.

(c) Case 1c.

(e) Case 2b.

(f) Case 3.

Fig. 7. Overall structures on the 6 cases.

to:

 Wn     W + α p 1 − pn−1 ) (1 − p1 ) (  n −1 n   W  ∗ n−2 + α pn−1 (1 − pn−2 ) + α pn (1 − p1 ) C D = min  Wn′ − α pn .p1 (1 − p2 )     Wn′ −1 + α pn (1 − pn−1 )    ′′ Wn

(Case 1a) 

   (Case 1b)    (Case 1c)  (Case 2a)  

  (Case 2b)    (Case 3)

.

(9)

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Fig. 8. Partitioning nodes in three subsets.

In (7), we consider that v1 has a contribution of α p1 (1 − p2 ). Actually, this represents its contribution in the partial sequence but, in the end, its contribution might be decreased in a dominating set that includes vn (Case 2a). Precisely, the contribution would decrease from α p1 (1 − p2 ) down to α p1 (1 − p2 )(1 − pn ). Thus a decrease of α pn · p1 (1 − p2 ). In (8), we consider that v1 has a contribution of α p1 (1 − pn ). Indeed, it is not dominated in any partial sequence Wi′′ , but will have to be dominated by vn in our final solution, so that its contribution has to be α p1 (1 − pn ) anyway. Here, since three types of sequences are built, the number of operations required to compute an optimal solution is 12n which concludes the proof.  4.2. Trees With a similar, yet more complex method, one can generalize the result of Proposition 2 from paths (trees of maximum degree bounded by 2) to trees with bounded maximum degrees. In what follows, v1 will denote the root of the input tree T , Ti will denote the subtree rooted at node vi (so, T1 = T ), Fi will denote the set of children of given node vi , and vfi the father of vi . Consider a tree T and a dominating set D on T . For any subtree Ti of T , there are only three possible configurations regarding its root vi : 1. vi ∈ D; 2. vi ∈ ̸ D, and vfi ∈ ̸ D; in this case, the root has to be dominated in the subtree Ti (i.e., dominated by at least one of its children); 3. vi ̸∈ D and vfi ∈ D; in this case, the root might be non-dominated in the subtree Ti (i.e., not dominated by any of its children). Considering Cases 1–3, for a given dominating set D in a tree T , it is always possible to partition nodes of T into three sets: D, S and N, where D is the dominating set itself, S is the set of nodes that are dominated, but not by their fathers, and N is the set of nodes dominated by their fathers (and also possibly by some of their children). Fig. 8 gives an example of such a partition. Now, let us analyse what possible cases can occur for nodes of Fi with respect to the status of vi . The following cases can occur:

• vi ∈ D; then children of vi (if any) might be in D, or in N; • vi ∈ S; then children of vi might be in D or in S, and at least one of them should be in D; • vi ∈ N; in this case, children of vi might be in D or in S. It is interesting to notice that the situation of a given node impacts only its own contribution if the status of its children is known, so that, once more, it is possible to run a dynamic programming method. It will be based upon three partial solutions for each subtree Ti , namely, Wi , Wi′ and Wi′′ , where Wi (resp., Wi′ , Wi′′ ) is the partial solution of minimum contribution for subtree Ti rooted at vi when vi ∈ D (resp., vi ∈ S, vi ∈ N). Proposition 4. probabilistic min dominating set(S) in trees of maximum degree bounded by k + 1 can be solved in O∗ (2k ), where O∗ (·) notation ignores the terms that are polynomial in n. Proof. Partial solution Wi can be computed on a tree of any depth. Children of vi can belong either to D, or to N, so that there are 2|Fi | possible combinations to evaluate (bounded by 2k ). The combination minimizing the overall contribution leads to

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Algorithm 5 An optimal dynamic programming algorithm for probabilistic min dominating set(S) in trees. Input: A tree T rooted at node v1 Output: An optimal a priori dominating set D∗ for T for i := 1 to n do   if Fi == ∅; then C (Wi ) := pi ; C (Wi′ ) := M; C (Wi′′ ) := α pi 1 − pfi ; vi is labelled; while v1 is not labelled do choose a vertex vi not labelled such that ∀vj ∈ Fi , vj is labelled; compute C (Wi ) as given in (10); compute C (Wi′ ) as given in (12); compute C (Wi′′ ) as given in (11); vi is labelled;   C (D∗ ) = min C (W1 ), C (W1′ ) ; ∗ return D . the structure we are trying to define, namely Wi . Setting Di = D ∩ Fi (the subset of children of vi that are dominating nodes), C (Wi ) is defined by: C (Wi ) = min D i ⊆F i

 





vj ∈Di

pi +







C Wj +

vj ∈Fi \Di

   C Wj′′ .  

(10)

This value is quite easy to initialize with leaves where C (Wi ) = pi . Now, let us tackle Wi′′ , which can also be initialized on leaves (unlike Wi′ ). Indeed, if a leaf is dominated, then its father has to be dominating, or else it would not be dominated, so that a leaf can only be in D or in N. As we said earlier, children of vi in a partial solution Wi′′ can belong either to D or to S. Thus, the following holds:



 ′′

C Wi

 

= min α pi 1 − pfi Di ⊆Fi  





1 − pj +





C Wj +

vj ∈Di

vj ∈Di





vj ∈Fi \Di

   C Wj′ .  

(11)

Note there are also at most 2k combinations to examine in this case. Finally, let us specify Wi′ . In this case, children of vi should be in S or in D and at least one of them should be in D. Of course, by definition, none of them can be in N. Once more, each combination of sons in S or in D (at most 2k combinations) leads to a specific contribution for the subtree Ti . The partial solution Wi′ is the one minimizing this contribution. Thus, the following holds:



 ′

C Wi =

min

Di ⊆Fi ,|Di |>1

  

α pi





1 − pj +

vj ∈Di

 vj ∈Di





C Wj +

 vj ∈Fi \Di

   C Wj′ .  

(12)

Note that such a value can be computed for any subtree Ti of depth at least 1 (not on leaves). This might be a problem when computing a value for Wi′ or for Wi′′ on a tree Ti of depth 1 since, according to (11) and (12), in order to compute C (Wi′ ) and C (Wi′′ ), one needs values C (Wj′ ) for all the children vj of vi but these values do not exist for leaves. To keep all formulævalid and still ensure that a leaf will never be in S, we will initialize C (Wj′ ) to an arbitrarily large value M for any vj that is a leaf. Thus, when applying (11) and (12), all leaf-children of vi will be forced to be in Di . Note also, that definitions of C (Wi′ ) and C (Wi′′ ) differ only by a factor (1 − pfi ) in the contribution of vi . We are ready now to give a dynamic programming algorithm (Algorithm 5), optimally solving probabilistic min dominating set(S) in trees in polynomial time. It receives as input a tree T rooted at node v1 and a probability pi for each vi ∈ V and outputs an optimal a priori dominating set D⋆ . The dynamic programming Algorithm 5 runs in a ‘‘bottom up’’ way. It is initialized with leaves where values C (Wi ), C (Wi′ ) and C (Wi′′ ) are equal to pi , M and α pi (1 − pfi ), respectively. Then, each subtree Ti is associated with structures Wi , Wi′ and Wi′′ (apart from leaves that are associated only with Wi and Wi′′ , and the overall tree which will be associated only with Wi and Wi′ for obvious reasons), whose computation relies on the same structures on subtrees induced by children of the root vi . In other words, in order to compute, for instance, a value C (Wi ), one needs to have already computed values C (Wj ) and C (Wj′′ ) for all children vj of vi . Since the method is easily initialized with leaves, all values can be computed for all subtrees, starting from leaves and ending with the whole tree. Finally, the optimum D∗ is given by D∗ = argminW =W1 ,W ′ (C (W )). In all, O(2k n) = O∗ (2k ) operations are necessary to 1 compute D∗ .  Corollary 1. probabilistic min dominating set(S) in trees with maximum degree bounded by O(log n) can be solved in polynomial time.

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Corollary 2. probabilistic min dominating set(S) in trees is fixed parameter tractable with respect to parameter ‘‘maximum degree’’. As one can see from the proof of Proposition 4, the bound 2k for evaluating C (Wi )’s, C (Wi′ )’s and C (Wi′′ )’s was somewhat greedy and includes the real complexity of this evaluation that depends on which of the children of vi are in D, the quantity of them depending on their probabilities that affect C (vi ). So, the real complexity of the method presented is exponential to the number of distinct vertex probabilities of the children of vi which can be as large as k. In what follows, we conclude this section by restricting ourselves to the natural case where all the vertices of the tree have the same probability. Proposition 5. probabilistic min dominating set(S) is polynomial in general trees with equiprobable nodes. Proof. We will use the same kind of recursion as in Proposition 4, but we will adapt slightly its definition to take advantage of the equiprobability property. What is interesting and useful with this property, is that the contribution of a given dominated node does not depend on which of its neighbours are dominating ones, but only on how many of them are dominating. Indeed, if Fi ∪ {vfi } represents the set of neighbours of a dominated node vi (i ̸= 1), then: C (vi ) = α p(1 − p)

       Fi ∪ vfi ∩D

Now, let us specify the dynamic programming method. The idea is basically the same as in Proposition 4, but given the equiprobability assumption, (10)–(12) can be computed in O(k log k) instead of O(2k ), which enables us to apply the method to trees of any structure, and any degree. If one considers thata given node has h children in D (0 6 h 6 k), then one does not have to check all the possible  configurations (at most

k h

) for the h children in D. Indeed, the contribution of vi does not depend on which children are

in D, but only on how many of them, namely h. Thus the following holds:

 p h C (vi ) = α p(1 − p)  h +1 α p(1 − p)

vi ∈ D vi ∈ S , (h > 1) vi ∈ N .

(13)

Now, we will use this, to simplify computations of C (Wi ), C (Wi′ ) and C (Wi′′ ). We will detail the simplification for C (Wi′ ) only, and simply present the final results for the other two, but the ideas and methods are exactly the same as in the proof of Proposition 4. Let us denote by Wi′,h the partial solution Wi′ where the number |Di | of children of vi in D is equal to h. Of course, if one computes C (Wi′,h ) for each possible h between 1 and |Fi |, it is quite easy to compute C (Wi′ ) since C (Wi′ ) = min16h6|Fi | {C (Wi′,h )}. We will show that it is possible to generate C (Wi′,h ) for each possible h in time k log k. Combining (12) and (13), one gets: C Wi′,h =





min

Di ⊂Fi ,|Di |=h

 

α p(1 − p)h +



min

 

Di ⊂Fi ,|Di |=h

v ∈D





C Wj +

vj ∈Di



= α p(1 − p)h +



C (Wj′ )

vj ∈Fi \Di







C Wj +

C (Wj′ )

    

 vj ∈Fi \Di             = α p(1 − p)h + C Wj′ + min C Wj − C Wj′  Di ⊂Fi ,|Di |=h  v ∈F v ∈D j

= α p(1 − p)h +

j

i

i

j

 

 

 ′

C Wj +

vj ∈Fi

min

Di ⊂Fi ,|Di |=h

v ∈D j

i

i

∆′j

 

(14)



where ∆′j = C (Wj ) − C (Wj′ ) for all j. This value represents the ‘‘additional cost’’ for a given subtree Tj to have its root vj as dominating node instead of dominated one. Indeed, if the root of the subtree is dominating then it must have the structure Wj and weight C (Wj ), whereas if the root is dominated then it must have the structure Wj′ and weight C (Wj′ ). Obviously, this value can be negative if the partial solution Wj (with the node vj considered as dominating) has a better expected value than the partial solution Wj′ (with vj considered dominated). Understanding this, one also understands that the min operator can be computed in time O(k log k) for every h. Indeed, it suffices to compute the values ∆′j for all j, and to sort them in increasing order (which takes O(k log k) operations). Then, in order to compute the min for a given h, simply pick as subset D the h nodes vj corresponding to the h first values ∆′j in the sorted list. Furthermore, one needs to sort ∆′j ’s only once to get any Wi′,h for all possible h. Indeed, once sorted, one

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can use the sorted list to compute all Wi′,h : just pick the lowest ∆′j (thus the first element of the sorted list) to compute Wi′,1 , the two lowest (first and second elements of this same sorted list) to compute Wi′,2 , and so on, for each h. This method returns the proper min as well as its associated structure. Indeed, if one wants to minimize the sum of h values in a set of k values, one only needs to pick the h lowest values. In this case, the h subtrees with lowest ‘‘additional costs’’ (linked to setting the root as dominating instead of as dominated) will be the h that actually have their respective roots as dominating nodes, the |Fi | − h other subtrees having their roots set as dominated nodes. Obviously, given the recursive structure of the method, one knows the optimal partial solutions associated with all these subtrees. By the same method, and setting ∆′′j = C (Wj ) − C (Wj′′ ), one gets:





C Wi,h = p +

 

 ′′

C Wj

+

vj ∈Fi

C Wi′′,h = α p(1 − p)h+1 +



min

Di ⊂Fi ,|Di |=h

 



 

 ′

C Wj +

vj ∈Fi

∆′′j

v ∈D j

i

min

 

(15)



Di ⊂Fi ,|Di |=h

  v ∈D j

i

∆′j

 

.

(16)



Each of these values can be computed for all h in O(k log k) in exactly the same way as described previously. Using these values, one directly defines Wi and Wi′′ as follows: C (Wi ) = min

 

C Wi′′ = min

 

06h6|Fi |





06h6|Fi |

C Wi,h



C Wi′′,h



.

The rest of the algorithm is the same as that described in Proposition 4. In all, each value C (Wi ), C (Wi′ ) or (Wi′′ ) takes O(k log k) operations to be computed, and the method computes 3n of them. So, the overall running time of the method is O(n2 log n), which concludes the proof.  In order to make clear how C (Wi ), C (Wi′ ) and (Wi′′ ) are computed, in the Appendix we run the algorithm on a small instance, and detail some interesting steps. 5. Polynomial approximation of probabilistic min dominating set(S) 5.1. Graphs with identical vertex-probabilities We consider in this section that the vertex-probabilities of the input graph are identical. In other words, referring to our wireless sensors problem, the probability that a sensor fails is the same for all of them. As we have already mentioned, min dominating set is approximate equivalent to min set cover, in the sense that an approximation algorithm for one of them can be transformed in polynomial time into an approximation algorithm for the other one achieving the same approximation ratio. Recall also that the natural greedy algorithm for min set cover achieves approximation ratio either 1 + ln |Smax | where |Smax | is the cardinality of the largest set Smax in the set-system describing the instance of min set cover [24,26], either O(log n) where n is the cardinality of the ground set describing this instance [38]. In the transformation of min dominating set into min set cover, any set S of the set-system becomes a vertex with degree |S | + 1. So, in the derived instance of min dominating set, ∆ = |Smax | + 1, where ∆ denotes the maximum degree of the derived graph. For facility and because of the form of the functional given in (4), we will use in what follows the former of the above ratios. The following easy lemma will be used later. Lemma 1 ([6]). For any instance of min dominating set of size n and maximum degree ∆, any minimal (for inclusion) solution (hence, the minimum-size one also) has size bounded below by n/(∆+1). Based upon Lemma 1, one can easily prove that the simplest algorithm consisting of taking all the vertices of the network in an a priori solution, achieves approximation ratio ∆ + 1. Indeed, if D = V , no recourse penalty one has to pay for any sensor’s failure since the surviving ones already dominate ˆ an optimal solution for probabilistic min dominating themselves. So, (4) becomes E(G, V , S) = pn. Denote by D∗ and D set(S) and for min dominating set in G, respectively. Remark that since D∗ is a feasible solution for min dominating set, we have:

 ∗    D  > Dˆ  .

(17)

  ˆ |D∗ | > Dˆ  > n/(∆+1). A simple inspection of (4) suffices Also, by Lemma 1 (that, as pointed out above, holds also for D), to see that E(G, D∗ , S) > p|D∗ | > p · n/(∆+1). Then putting together E(G, V , S) and E(G, D∗ , S) the following result holds immediately.

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Proposition 6. probabilistic min dominating set(S) is approximable within ratio ∆ + 1, in graphs with maximum degree ∆, when vertices have identical probabilities. Let us now consider a particular case of the recourse model considered until now, where we assume that recourse has no additional cost, i.e., α = 1. We will show that in this case, the ratio of Proposition 6 can be improved slightly. For this, let us first make the following remark. Remark 2 ([33]). Let D be a minimal dominating set in a graph G = (V , E ). Then, V \ D is also a dominating set and, moreover, the smaller of them is smaller than n/2.  Recall now (4) and observe that the following framing can be easily derived for E(G, D, S) for the case where α = 1:

E(G, D, S) 6 p|D| + (n − |D|)p(1 − p) = p2 |D| + pn(1 − p) ∆

E(G, D, S) > p|D| + p(n − |D|)(1 − p) = p 1 − (1 − p)



∆



(18) ∆

|D| + pn(1 − p) .

(19)

Take D to be the smallest between the (1 + ln ∆)-approximate solution computed by the greedy algorithm in G and the complement of it with respect to V as a priori solution for min dominating set. Remark that, so computed, the size of D guarantees the ratio 1 + ln ∆ and, simultaneously, according to Remark 2, it is smaller than n/2. Applying the rightmost bound of (18) to D and the rightmost bound of (19) to D∗ , we immediately get:

   

ˆ  + p(1 − p)n E(G, D, S) 6 p2 |D| + pn(1 − p) 6 p2 (1 + ln ∆) D     E G, D , S > p 1 − (1 − p)∆ D∗  + pn(1 − p)∆ . 

∗

(20) (21)

Combining (20) and (21), we have the following for the approximation ratio of D, first taking into account that it achieves approximation ratio 1 + ln ∆, in (23), then taking into account (17) (in the first inequality of (24)), then that n > |D∗ | (second inequality of (24)), and finally using Lemma 1 (in the first inequality of (25)):

E(G, D, S) E (G, D∗ , S)

p|D| + n(1 − p)  6  1 − (1 − p)∆ |D∗ | + n(1 − p)∆

(22)

   

ˆ  + n(1 − p) p(1 + ln ∆) D

 6  1 − (1 − p)∆ |D∗ | + n(1 − p)∆

(23)

p(1 + ln ∆) |D∗ | + n(1 − p) p(1 + ln ∆) |D∗ | + n(1 − p)  6 6  |D∗ | 1 − (1 − p)∆ |D∗ | + n(1 − p)∆

(24)

6 p(1 + ln ∆) + (1 − p)(∆ + 1) = ∆ + 1 − p(∆ − ln ∆).

(25)

∗

Recall (22) and use Remark 2 for D and Lemma 1 for D . Then, some very easy algebra gives:

  (∆ + 1) 1 − 2p ∆+1 6 6 . ∗ ∆ E (G, D , S) 1 + ∆(1 − p) 1 + ∆(1 − p)∆ E(G, D, S)

(26)

Ratio in (25) is bounded above by ∆ − ln ∆ for p > (ln ∆+1)/(∆−ln ∆) > ln(e∆)/(∆+1−ln ∆). On the other hand, for arbitrarily large ∆’s, if p 6 ln(e∆)/(∆+1−ln ∆), expression ∆(1 − p)∆ in the denominator of (26) is very close to e−1 = 0.37 and the ratio given by (26) becomes (∆+1)/1.37. So, the following holds for probabilistic min dominating set(S) under the simplified recourse model. Proposition 7. The set D selected to be the smallest between the (1 + ln ∆)-approximate solution computed by the greedy algorithm in G and the complement of it with respect to V achieves for probabilistic min dominating set(S), under the simplified recourse model, approximation ratio bounded above by ∆ − ln ∆, where ∆ is the maximum degree of G, when sensors have identical failure probabilities. The mathematical analysis leading to the result of Proposition 7 seems to be quite tight. Indeed, a numerical study of the ratio in the first inequality of (26) with respect to p derives that function f (p) = (∆+1)(1−(p/2))/(1+∆(1−p)∆ ) attains its maximum value for p = 2 ln ∆/∆. In this case, after an easy algebra, the maximum is ∆ − ln ∆. 5.2. Distinct vertex-probabilities Let us now suppose that the sensors of the network are heterogeneous so that each of them has its own failure probability that can be different from that of another sensor in the network. This, in our model becomes to assume that the vertices of the input graph have distinct probabilities. We will show that, in this case, any probabilistic min dominating set(S)-solution 2 achieves approximation ratio bounded above by O(∆ /log ∆).

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Recall (3) and observe that the dominating set D, consisting of the whole vertex-set of the input-graph satisfies the trivial inequality:

E(G, D, S) 6



pi .

(27)

vi ∈V

Fix an optimal a priori dominating set D∗ and, for a probability p′ that will be fixed later, partition the vertices of G into four subsets: D∗1 : the set of vertices of D∗ whose probabilities are at least p′ ; furthermore, set |D∗1 | = κ ; D∗2 : the rest of vertices of D∗ , i.e., D∗2 = D∗ \ D∗1 ;

¯ ∗1 : the set ΓD¯ ∗ (D∗1 ) of neighbours of D∗1 in D¯ ∗ = V \ D∗ , i.e., D¯ ∗1 = Γ (D∗1 ) ∩ D¯ ∗ ; D

¯ ∗2 : the set ΓD¯ ∗ (D∗2 ) \ D¯ ∗1 , i.e., the vertices of D¯ ∗ that have no neighbours in D∗1 , i.e., D¯ ∗2 = D¯ ∗ \ ΓD¯ ∗ (D∗1 ). D Denote, for simplicity, by p, the largest vertex-probability. Now, taking into account the partition above, (3) can be rewritten as:



E G, D∗ , S =







pi +

vi ∈D∗1



pi + α

vi ∈D∗2

> κ p′ +



vi ∈D∗2



>

pi + α 1 − p′





∆ 

 vi ∈D∗2

vi ∈D∗1

pi +

1 − pj



vj ∈Γ (vi )∩D∗2

pi

(28)



>

pi + 1 − p′



∆ 

vi ∈D∗2

pi .

(29)

vi ∈D¯ ∗2

pi

vi ∈V ′ ∆

6

(1 − p )

6

pi

pi



pi

vi ∈D∗2 ∪D¯ ∗2

x

(1 − p′ )∆

.

(30)

x

vi ∈D∗2 ∪D¯ ∗2

 pi >



pi

v ∈V p′ ) ∆ i



vi ∈D¯ ∗2

2





pi



(1 −



pi + α 1 − p

vi ∈D¯ ∗ vi ∈V

6

vi ∈D¯ ∗2

  ′ ∆



pi

vi ∈V pi x

pi + (1 − p′ )∆



Suppose now that

pi



/ for some x to be also defined later. Then, using (27) and (29), it holds that: 



pi >

vi ∈D∗2 ∪D¯ ∗2

vi ∈V

6

E (G, D∗ , S)





vi ∈D¯ ∗2

 E(G, D, S)

1 − pj + α

vi ∈D¯ ∗1

vi ∈D∗2

Let us first suppose that



vj ∈Γ (vi )∩D∗

vi ∈D¯ ∗1

pi + α(1 − p)∆





pi

pi 6

x−1

vi ∈V pi x.



x

vi ∈D¯ ∗1

/ Then, for





vi ∈D∗1

pi +



vi ∈D¯ ∗1

pi , it holds that:

pi .

(31)

vi ∈V

¯ ∗1 | 6 ∆κ and, using (31), the following holds: Remark, furthermore, that, since |D∗1 | = κ , |D κ 1p + κ p >



pi +

vi ∈D∗1



pi >

vi ∈D¯ ∗1

x−1  x

vi ∈V

pi =⇒

 vi ∈V

pi 6

x x−1

κ p(∆ + 1).

(32)

Remark finally that, from (27) and (28), one can immediately derive another easy expression for the approximation ratio,  namely, E(G,D,S)/E(G,D∗ ,S) 6 vi ∈V pi/κ p′ , since E(G, D∗ , S) > κ p′ . This, using (32), becomes:

E(G, D, S) E (G, D∗ , S)

6

xp(∆ + 1)

(x −

1)p′

6

x(∆ + 1)

(x − 1)p′

.

If we fix p′ = ln ∆/∆ and x = ∆/ln ∆, then both ratios given in (30) and (33) become ≈ the section and the following holds.

(33) ∆2/ln ∆

as claimed in the beginning of

Proposition 8. probabilistic min dominating set(S), under the recourse model is approximable within ratio O(∆ /ln ∆) in graphs with distinct vertex probabilities. 2

Corollary 3. probabilistic min dominating set(S) is approximable within constant ratio in bounded-degree graphs under the recourse model considered and for any probability-system.

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6. Conclusion Probabilistic combinatorial optimization seems to be an interesting and appropriate mathematical framework for handling numerous risk management natural problems where, faced with some disaster that has destroyed a part of the solution, whatever this solution represents, the objective of a decision maker is to restore it, or to efficiently compute one having several good properties, for example, having a good quality (under a predefined criterion) minimizing, or maximizing the reconstruction costs. In this paper we have studied emergency management for a wireless sensor network problem modelled as a probabilistic version of min dominating set. We have shown that, even under simple (and ‘‘generous’’) recourse models, the problem handled is very hard. We then proposed a very simple model (that just enters in the final solution all the non-dominated vertices of the final instance) under which probabilistic min dominating set(S) can be proved to be in NP. But even in this case, this has led to a problem that is quite more difficult to handle than its original deterministic version, in particular when trying to approximately solve it. We have proposed algorithms for paths, cycles and trees (cases where min dominating set is polynomial), as well as we have tried a first study of approximability of probabilistic min dominating set(S) in general graphs. What is missing with respect to the latter issue is to propose algorithms that are proper to the probabilistic nature of this problem. This seems to be a hard issue since no compact characterization of the optimal a priori solution can be derived from (3) but work is in progress. Let us note that the proof of Theorem 1 encompasses more problems than min dominating set. For instance, recall [30] where the probabilistic max independent set is studied. There, two simple greedy strategies for the second stage completion of the a priori solution studied, are similar to Algorithm 1. Since in the proof of Theorem 1, one can immediately remark that both the a priori solution D considered there and the final solution D′ are maximal (for inclusion) independent sets, the following corollary can be immediately derived. Corollary 4. Calculations of the functionals of problems probabilistic max independent set 2 and 3 in [30] are both #Pcomplete. Basic assumption in this paper was that the second stage solution D′ (recourse) has to be computed very quickly. If this requirement is somewhat relaxed, for example, if the objective is to determine an optimal second-stage solution in the finally present subgraph G[V ′ ] (always subject to the constraint that the vertices of the a priori solution that survive in G[V ′ ] must be included in D′ ), it is possible to model this problem’s version as an integer linear program. A scenario s in S corresponds to a subset of vertices that will be present; so, |S | = 2n . We associate with every vi ∈ V a variable x0i with: x0i =



1 0

if vi belongs to the a priori solution otherwise

and with any scenario s we add a variable: ysi =



1 0

if vi is added during the second stage in the solution for s otherwise.

Then the integer linear program is the following:

(P )

   min        s.t.   

n 



 pi x0i

+α



× Pr(s)

s∈S

i=1

 

x0i + ysi +

x0j + ysj > 1

vj ∈Γ (vi )

           

ysi

ysi = 0 ysi 6 1 − x0i x0i ∈ {0, 1} ysi ∈ {0, 1}



∀s ∈ S , ∀vi ∈ s

(1)

∀s ∈ S , ∀vi ̸∈ s ∀s ∈ S , ∀vi ∈ s i = 1, . . . , n ∀s ∈ S , ∀vi ∈ V

(2) (3) (4) (5)

where Pr(s) is the probability of scenario s, and Γ (vi ) is the open neighbourhood of vi and:

• the constraint block (1) guarantees that any vertex of scenario s is dominated; • the constraint block (2) expresses the fact that if vi is absent from scenario s, then it cannot be added in any solution for this scenario;

• the constraint block (3) expresses the fact that a vertex of the a priori solution surviving in s cannot be added again (indeed this constraint could be omitted because we deal with a minimization problem and so it will not be taken into account). If the recourse strategy is fixed, quantities ysi are no more real variables because the strategy will induce their value for all the vertices present in s. This, with respect to program (P ) amounts to add more constraints on the variables ysi deteriorating so the value of its optimum. In this case, variables ysi only serve for calculating the functional of the a priori solution and if their values are not unknown, then the number of binary variables of (P ) becomes polynomial. But in this case also the number of constraints remains exponential since one needs to calculate the value of the solution in any of the possible sub-graphs.

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Fig. 9. A tree T.

Acknowledgements The very pertinent comments and suggestions of two anonymous Reviewers are gratefully acknowledged. Appendix An example of the algorithm of Proposition 5 Suppose p = 0.2, α = 1 and consider the tree of Fig. 9. The algorithm is initialized with leaves: C (W3 ) = C (W5 ) = C (W7 ) = C (W8 ) = C (W9 )= 0.2 ′ ′ C W3′  = C W5′ = C W7′ = C W  8 ′′ = C W  9 ′′=  M ′′ ′′ ′′ C W3 = C W5 = C W7 = C W8 = C W9 = 0.8 · 0.2 = 0.16 ∆′3 = ∆′5 = ∆′7 = ∆′8 = ∆′9 = 0.2 − M ∆′′3 = ∆′′5 = ∆′′7 = ∆′′8 = ∆′′9 = 0.2 − 0.16 = 0.04. With all leaves initialized, it is possible to find substructures Wi , Wi′ and Wi′′ on subtrees rooted at v6 and v4 . Let us start with v6 . Here it is senseless to sort children vj of v6 in increasing values of ∆′j or ∆′′j since they all have the same values ∆′j and ∆′′j . This means that if, in a given substructure, only one child vj of v6 must be dominating, then it is indifferent to choose v8 or v9 as such a node (in (34a) and (35a), we will choose arbitrarily v8 ). Applying (15), we get the following:









C W6,0 = p +



C Wi′′ = 0.2 + 2 × 0.16 = 0.52





i=8,9

C W6,1 = p +



C Wi′′ + ∆′′8 = 0.2 + 2 × 0.16 + 0.04 = 0.56





(34a)

i=8,9





C W6,2 = p +



C Wi′′ +





i=8,9



∆′′i = 0.2 + 2 × 0.16 + 2 × 0.04 = 0.6

(34b)

i=8,9

and thus, C (W6 ) = minh=0,1,2 {C (W6,h )} = C (W6,0 ) = 0.52. We now use (14) to get C (W6′ ): C W6′ ,1 = p(1 − p) +













(35a)

i=8,9

C W6′ ,2 = p(1 − p)2 +



C Wi′ + ∆′8 = 0.16 + 2M + 0.2 − M = 0.36 + M

 i=8,9

C Wi′ +







∆′i

i=8,9

= 0.128 + 2M + 2(0.2 − M ) = 0.528. This example shows clearly the role of M in forcing all leaves to be either dominating nodes, or dominated by their fathers. In any structure with one leaf that is neither dominating, nor dominated by its father (as it is the case in W6′ ,1 ), the overall weight will be in O(M ). This is the case in (35a). Thus, for an arbitrarily large M, W6′ ,1 cannot be picked as optimal substructure, and C (W6′ ) = minh=1,2 {C (W6′ ,h )} = C (W6,2 ) = 0.528. Similarly, applying (16), one gets C (W6′′ ) = minh=1,2 {C (W6′′,h )} = C (W6′′,2 ) = 0.5024. All these results and corresponding substructures are illustrated in Fig. 10. Substructures W4 , W4′ and W4′′ are computed in the same way and are illustrated in Fig. 11.

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Fig. 10. Substructures W6 , W6′ and W6′′ .

Fig. 11. Substructures W4 , W4′ and W4′′ .

Now, all substructures rooted at all children of v2 have been computed; we continue by computing substructures rooted at v2 . First, let us order children vi of v2 in increasing order of ∆′i and ∆′′i . Recall that ∆′5 = 0.2 − M and ∆′′5 = 0.04 and notice that ∆′6 = −0.08 and ∆′′6 = 0.0176. Thus, the two sorted lists are: (∆′5 , ∆′6 ) and (∆′′6 , ∆′′5 ). From the first sorted list, we can assert that if v2 is dominated and if it has only one dominating child, then this child must be v5 , since ∆′5 appears first in the list. From the second sorted list, we can assert that if v2 is dominating and if it has only one dominating child, then this child must be v6 , since ∆′′6 appears first in the list. We now compute substructure for v2 :















C W2,0 = p +

C Wi′′ = 0.8624





i=5,6



C W2,1 = p +

C Wi′′ + ∆′′6 = 0.88





i=5,6



C W2,2 = p +

C Wi′′ +





i=5,6



∆′′i = 0.92

i=5,6

and C (W2 ) = C (W2,0 ) = 0.8624. Then: C W2′ ,1 = p(1 − p) +











i=5,6

C W2′ ,2 = p(1 − p)2 +



C Wi′ + ∆′5 = 0.888





C Wi′ +





i=5,6



∆′i = 0.776

i=5,6

and C (W2 ) = C (W2,2 ) = 0.776. Finally: ′

′

C W2′′,0 = p(1 − p) +



C





W2′′,1





= p(1 − p) +





C Wi′ + ∆′5 = 0.856





i=5,6



and C (W2 ) = ′′



i=5,6

2

C W2′′,2 = p(1 − p)3 +



C Wi′ = 0.688 + M

C (W2′′,2 )



C Wi′ +

i=5,6







∆′i = 0.7504

i=5,6

= 0.7504. All results and structures are represented in Fig. 12.

Finally, let us compute structures W1 and W1′ . Note that we do not compute W1′′ , since v1 has no father, and a fortiori it cannot be dominated by such a node. As before, we first build sorted lists with ∆′i ’s and ∆′′i ’s for all children vi of v1 . These sorted lists are (∆′3 , ∆′4 , ∆′2 ) and (∆′′4 , ∆′′3 , ∆′′2 ) since:

∆′2 = 0.0864 ∆′′2 = 0.112

∆′3 = 0.2 − M ∆′′3 = 0.04

∆′4 = 0 ∆′′4 = 0.032.

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Fig. 12. Substructures W2 , W2′ and W2′′ .

Fig. 13. Substructures W1 and W1′ .

Suppose that v1 is dominated. The first sorted list indicates us that:

• if v1 has 1 dominating child, this must be v3 ; this leads to (36a); • if v1 has 2 dominating children, these must be v3 and v4 ; this leads to (36b). Similarly, suppose that v1 is dominating. According to the second sorted list:

• if v1 has 1 dominating child, this must be v4 , leading to (37a); • if v1 has 2 dominating children, these must be v4 and v3 , leading to (37b). Thus:









C W1,0 = p +



C Wi′′ = 1.4384





i=2,3,4

C W1,1 = p +



C Wi′′ + ∆′′4 = 1.4704





(36a)

i=2,3,4





C W1,2 = p +







i=2,3,4





C W1,3 = p +





C Wi′′ +

∆′′i = 1.5104

(36b)

i=3,4



C Wi′′ +





i=2,3,4

∆′′i = 1.6224

i=2,3,4

and C (W1 ) = C (W1,0 ) = 1.4384. Also: C W1′ ,1 = p (1 − p) +









C W1′ ,3 = p + (1 − p)3









(37a)

i=2,3,4

C W1′ ,2 = p(1 − p)2 +



C Wi′ + ∆′′3 = 1.496



C Wi′ +





i=2,3,4

 i=2,3,4



∆′′i = 1.464

i=3,4

C Wi′ +





 i=2,3,4

∆′′i = 1.5504

(37b)

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and C (W1′ ) = C (W1,2 ) = 1.464. Corresponding structures are shown in Fig. 13. The optimum on this instance is given by W1 , since C (W1 ) < C (W1′ ). References [1] J. Akbari Torkestani, M.R. Meybodi, Approximating the minimum connected dominating set in stochastic graphs based on learning automata, in: Proc. International Conference on Information Management and Engineering, IEEE, 2009, pp. 672–676. [2] I. Averbakh, O. Berman, D. Simchi-Levi, Probabilistic a priori routing-location problems, Naval Res. Logist. 41 (1994) 973–989. [3] P. Balaprakash, M. Birattari, T. Stützle, M. Dorigo, Estimation-based metaheuristics for the probabilistic traveling salesman problem, Comput. Oper. Res. 37 (11) (2010) 1939–1951. [4] J. Barta, V. Leggieri, R. Montemanni, P. Nobili, C. Triki, Some valid inequalities for the probabilistic minimum power multicasting problem, Electron. Notes Discrete Math. 36 (2010) 463–470. [5] J. Barta, R. 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