We exhibit links between pseudo-Boolean optimization, graph theory and logic. We show the equivalence of maximizing a pseudo-Boolean function and finding a maximum weight stable set; symmetrically minimizing a pseudo-Boolean function is shown to be equivalent to solving a weighted satisfiability proble

de Werra, D.

Hammer, P.

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Ann Oper Res (2007) 149:67–73DOI 10.1007/s10479-006-0101-0Weighted stability number of graphs and weightedsatisfiability: The two facets of pseudo-BooleanoptimizationD. de Werra · P. L. HammerPublished online: 2 December 2006C© Springer Science + Business Media, LLC 2007ForewordIn the break following the first hour of a Boolean seminar held in the late ’70s at the EcolePolytechnique Fe´de´rale de Lausanne, one of the participants approached us with a fascinatingidea. He told us that he had noticed that by associating to a graph a certain pseudo-Booleanfunction, simplifying afterwards a particular posiform of that function, and finally associatinga graph to the resulting expression, the stability number of the original graph would be reducedby exactly one unit in this sequence of transformations.The name of the seminar participant who made this remark was Christian Ebeneg-ger, an economist by training, a member of the Faculty of Architecture, and an enthu-siastic believer in operations research. Following Christian’s weekly persistent pointersto more and more convincing examples of graphs whose stability number he had calcu-lated by repeated applications of the above transformations, it was finally realized thathis proposed transformations were not only valid, but did in fact always yield the desiredresult.Christian left us suddenly and prematurely in 2002. It is to his memory that we dedicatethis note.D. de Werra ()IMA - EPFL, Lausanne, Suissee-mail: dewerra.ima@epfl.chP. L. HammerRUTCOR, Rutgers University, 640 Bartholomew Rd, Piscataway NJ 088254-8003, USAe-mail: hammer@rutcor.rutgers.eduSpringer68 Ann Oper Res (2007) 149:67–731 An introductionAmong the integer programming problems that attracted the attention of many researchers,the maximum weight stable set problem (MWSP) has generated a variety of contributionsbased on very different techniques.Given a graph G = (V, E) where each vertex i has a (positive) weight w(i) one wants tofind a stable set S (i.e., a subset S ⊆ V of vertices such that no two vertices in S are linkedby an edge (Berge, 1973) with maximum weight w(S) = ∑(w(i) | i ∈ S). It is formulatedas follows:Max z =n∑i=1w(i)xi ≡ αw(G)s.t. xi + x j ≤ 1 for all [i, j] ∈ E (MWSP)xi ∈ {0, 1} for all i ∈ VWhen all weights w(i) are one αw(G) will be written α(G); it is the stability number of G.2 A formulation in pseudo-Boolean termsAs observed by Hamor (1980), this problem can also be viewed as the maximization of apseudo-Boolean function, i.e., a real-valued function of Boolean variables. It is obtainedas follows: we take a family of complete bipartite graphs (not necessarily induced) G j =(A∗j ; B∗j ; E j ) ( j = 1, . . . , q) that cover the edge set E , we associate to each Gj a 0-1 variablex j , and we set:f (x1, . . . , xn) =∑i∈Vw(i)TiwhereTi =∏j∈Aix j∏k∈Bixk (1)andAi = { j : i ∈ A∗j }, Bi = { j : i ∈ B∗j }Notice that this representation of f is a posiform (there is no constant term and allcoefficients w(i) are positive). Clearly the maximum of f will give the maximum weightof a stable set in G: each Ti with value 1 will correspond to a vertex i of G which will beincluded in the set S; it follows from the construction of f that no two adjacent vertices ofG can both be in S.In fact given any pseudo-Boolean function g, we can express it as g = K + f where Kis a constant and f a posiform, by using the equality x j = 1 − x j .Any given pseudo-Boolean function g can generally be written as a constant plus a posi-form in many different ways.SpringerAnn Oper Res (2007) 149:67–73 693 The conflict graph and its useMaximizing g amounts to maximizing an associated posiform f ; it was also observed thatconversely maximizing a posiform f can be reduced to finding a stable set with maximumweight in a graph G f = (V, E) called the conflict graph of f , and defined in the followingway:If f = ∑ni=1 w(i) Ti where each Ti is defined by (1) where Ai , Bi ⊆ {1, . . . , n}Ai ∩ Bi =∅, then to each i we associate a vertex i of G f with weight w(i), and define the edge set byE = {[i, j] : (Ai ∩ B j ) ∪ (A j ∩ Bi ) = ∅}. This means that in G f there is an edge betweenvertices i and j if the terms Ti and Tj are in conflict (there is a k such that xk occurs in oneterm and xk in the other term).It is clear that max {w(S) : S stable set in G f } = max f (x1, . . . , xn).So the problem of finding a stable set of maximum weight in a graph G is equivalent tofinding the maximum of a pseudo-Boolean function (and in particular of a posiform).4 The structionNow if we have a graph G = (V, E) with positive weights on its vertices, there are manydifferent ways of covering its edges by complete bipartite graphs; as a consequence there aremany different posiforms that could be maximized to find αw(G).Since there is usually a choice, one may wonder whether there are some “good” choicesto be made. This is precisely what Christian Ebenegger noticed and this idea was exploited inEbenegger, Hammer, and de Werra (1984) for defining the struction (from STability numberRedUCTION).For any vertex v of a graph, let N (v) be the set of its neighbors. Consider now a graph G =(V, E) and choose a vertex ao in V ; Assume N (ao) = {a1, . . . , ap} and let R = V − N [ao] ={ap+1, . . . , a|V |−1} where N [ao] = N (ao) ∪ {ao}. We construct the following covering of theedge set E by |V | − 1 stars:(a) For each i = 1, . . . , p , we take the star with vertex set{ai , ao} ∪ {a j ∈ N (ai ) ∩ N (ao) : j > i} ∪ (N (ai ) ∩ R)and with center ai .(b) For each i = p + 1, . . . , |V | − 1, we take the star with vertex set{ai } ∪ {a j ∈ N (ai ) ∩ R : j > i}and with center ai .One obtains the posiformf =|V |−1∑i=ow(ai ) T (ai )whereT (ao) =p∏i=1xiSpringer70 Ann Oper Res (2007) 149:67–73a5x5x¯1x¯3 a4x4x¯3a6 x6x¯3x¯4a3 x3a0x¯1x¯2x¯3x¯4x¯5a2 x2x¯1a1 x1Fig. 1 A graph G with the posiform f.T (ai ) =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩xi∏a j ∈N (ai )1≤ j<i≤px j (1 ≤ i ≤ p),xi∏a j ∈N (ai )∩N (a0)x j∏a j ∈N (ai )∩Rp< j<i<|V |x jFor simplicity purposes we shall assume that all weights are one. An example is given inFig. 1. For this example, after the transformation of f , we getf = 1 + x1x3 + x1x2x3 + x3x1x4 + x1x3x2x4 + x1x3x2x5 + x1x2x3x4x5In general, this covering is always such that∑|V |−1i=o T (ai ) can be rewritten as:1 +∑aq ∈N (ar )1≤q<r≤pxq xr∏1≤s<qxs∏at ∈N (ar )q<t<rx t .which is of the form 1 + g, where g is a posiform. So in case all weights w(ai ) are one, wehave f = 1 + g. Since g is a posiform, we can now consider its conflict graph, which wecall G(ao); for the example of Fig. 1, we get the graph given in Fig. 2.Since α(G) = max f = 1 + max g, for the graph G(ao), we have α(G(ao)) = α(G) − 1.5 A direct reductionIn fact G(ao) could have been obtained directly from G by the following transformation:(1) remove vertices ao, a1, . . . , ap from G(2) add to the rest R of the graph a set of new verticesW = {vi j : 1 ≤ i < j ≤ p and [vi , v j ] /∈ E}SpringerAnn Oper Res (2007) 149:67–73 712,42,31,31,42,54,5 a6G(a0)Fig. 2 The graph G(ao) for thegraph G of Fig. 1(3) join two new vertices vi, j and vk,l by an edge whenever i = k or if [v j , v] ∈ E ; join anew vertex vi, j to a vertex u ∈ R by an edge whenever [vi , u] ∈ E or [v j , u] ∈ E .One can then show that α(G(ao)) = α(G) − 1; but it is worth insisting on the fact that thisconstruction was inspired by pseudo-Boolean manipulations and the choice of an appropriatecovering of edges of E by complete bipartite graphs was crucial for deriving this reductionof the stability number α(G).6 Using the struction in an algorithmWe may now think of computing α(G) by repeatedly applying the struction until one gets agraph Gk−1 with α(G(k−1)) = 1; G(k−1) is a clique and we get α(G(o)) = α(G) = k.This does clearly not provide a polynomial algorithm for a general graph G; the numberof vertices in the consecutive graphs G() may increase exponentially.However in special classes of graphs, by making systematically good choices of the centerao of the struction, one may avoid this inconvenience; this was exploited for some subclassesof claw-free graphs for instance (see Hammer, Mahadev, and de Werra, 1985).For a weighted graph G, the basic struction operation can be extended to reduce αw(G)by a fixed amount (for instance, the minimum weight of the nodes of G), but we shall notdiscuss this here.The example of the struction is an interesting illustration of a situation in which pseudo-Boolean techniques have led to the discovery of some graph transformation (reducing thestability number in our case) which could afterwards be explained and justified directly. Inthat sense pseudo-Boolean algebra has played a role of catalyst in the procedure of designinga graph transformation.There are other situations where the same phenomenon occurred (see for instance (Alexeet al., 2004; Hertz, 1995, 1997) for variations and extensions of such transformations).7 Another reduction of pseudo-Boolean inspirationIt is worth mentioning also the case of “magnets” that generalize a simple neighborhoodreduction. A magnet in a graph G = (V, E) is a pair (a, b) of adjacent vertices such thatevery vertex in N (a) − N (b) is adjacent to every vertex in N (b) − N (a). If (a, b) is a magnetSpringer72 Ann Oper Res (2007) 149:67–73dx¯1x¯3 e x¯1x3ax1x2g x¯2c x¯2b x¯1x2f x1Gd egcbfGFig. 3 The magnet transformationin G, then the edges adjacent to a or b can be covered by the following two complete bipartitesubgraphs G1 = (N (b) − N (a), N (a) − N (b), E1) and G2 = ({a, b}, N (a) ∩ N (b), E2) ofG. If we cover the edges of E − (E1 ∪ E2) with complete bipartite subgraphs G3, . . . , Gq ,then the associated posiform f = ∑v∈V T (v) is such that Ta = x1 · x2, Tb = x1 · x2, so wehave Ta + Tb = (x1 + x1)x2 = x2 (assuming all weights equal to one). Hence f has the samemaximum as g = ∑v∈V −{a,b} T (v) + x2.The conflict graph G ′ = (V ′, E ′) associated with g satisfies |V ′| = |V | − 1 and α(G ′) =α(G). This construction is illustrated in Fig. 3.Again, one could have constructed directly the graph G ′ by deleting in G vertex a togetherwith all edges [v, b] with v ∈ N (b) − N (a). Notice that G ′ is not an induced subgraph ofG and also that the simple neighborhood reduction corresponds to the case where N (b) −N (a) = ∅.8 Minimizing posiformsHaving observed the equivalence of the weighted maximum stable set problem and themaximization of a posiform, we may wonder whether the minimization of a posiform alsohas some interpretation. In fact, there is one which is illustrated by the following example:Let f = 3 x1x2x3x4 + 4 x1x3x5 + x4x5 + 2 x2x5We can associate with each term Ti =∏j∈Ai x j∏k∈Bi xk (with Ai ∩ Bi = ∅) a clauseCi =(∨j∈Aix j)∨(∨k∈Bixk)with weight w(i).Here we would have:C1 = x1 ∨ x2 ∨ x3 ∨ x4 with w(1) = 3C2 = x1 ∨ x3 ∨ x5 with w(2) = 4C3 = x4 ∨ x5 with w(3) = 1C4 = x2 ∨ x5 with w(4) = 2SpringerAnn Oper Res (2007) 149:67–73 73Assume we want to find an assignment of values true or false (1 or 0) to the variablesx1, . . . , x4 such that the set of clauses that are satisfied has a maximum total weight.Obviously, this is the so called weighted satisfiability problem (WSP)(Garey and John-son, 1979). One sees that a clause Ci will be violated if and only if the corresponding termTi is one.So minimizing f will minimize the total weight of the set of violated clauses; this amountsto maximizing the weight of the set of clauses that are satisfied. This is precisely the WSP.Obviously to every WSP one can associate a posiform f such that its minimum will solvethe WSP. The two problems are thus equivalent.Notice that the WSP may be interpreted as a weighted covering problem of edges byvertices in a hypergraph.However a simple interpretation in terms of stable sets in a graph does not seem to be athand for the minimization of a posiform.9 A brief conclusionWe have seen here how some pseudo-Boolean methods may contribute to discovering purelygraph theoretical operations that have proven to be essential in several procedures related tothe stability number of graphs.Besides this we have observed that minimizing a posiform is equivalent to the weightedMAX SAT problem, which is another famous integer programming problem.This may again suggest new approaches and may lead to discovering other polynomiallysolvable cases as well.It is likely that there are still many situations in graph theory and in combinatorial op-timization in general where such pseudo-Boolean methods will suggest original ways ofcomputing exact or approximate solutions.ReferencesAlexe, A., P.L. Hammer, V.V. Lozin, and D. de Werra. (2004). “Struction Revisited.” Discrete Applied Math-ematics, 132, 27–46.Berge, C. (1973). Graphs and Hypergraphs. North-Holland: Amsterdam.Ebenegger, C., P.L. Hammer, and D. de Werra. (1984). “Pseudoboolean Functions and Stability of Graphs.”Annals of Discrete Mathematics, 19, 83–98.Hamor, A. (1980). “Stories of the One-Zero-One Nights: Abou Boul in Graphistan.” In P. Hansen and D.de Werra (Eds.), Regards sur la the´orie des graphes. Lausanne, Presses Polytechniques Romandes, pp.29–38.Hertz, A. (1995). “Polynomially Solvable Cases for the Maximum Stable Set Problem.” Discrete AppliedMathematics, 60, 195–210.Hertz, A. (1985). “On the Use of Boolean Methods for the Computation of the Stability Number.” DiscreteApplied Mathematics, 76, 183–203.Garey, M.R. and D.S. Johnson. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. New York, Freeman.Hammer, P.L., N.V.R. Mahadev, and D. de Werra. (1985). “Stability in CAN-Free Graphs.” J. Combin. Theory,B38, 23–30.Hammer, P.L., N.V.R. Mahadev, and D. de Werra. (1985). “The Struction of a Graph: Applications to CN-FreeGraphs.” Combinatorica, 5, 141–147.Springer

Weighted stability number of graphs and weighted satisfiability: The two facets of pseudo-Boolean optimization

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Weighted stability number of graphs and weighted satisfiability: The two facets of pseudo-Boolean optimization

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