Solving The FeedbackVertex Set Problem On Undirected Graphs Lorenzo Brunetta, Francesco Maoli and Marco Trubian Dipartimento di Elettronica e Informazione Politecnico di Milano Piazza Leonardo da Vinci 32, 20133 Milano, ITALY Abstract Feedback problems consist of removing a minimal number of vertices of a directed or undirected graph in order to make it acyclic. The problem is known to b e NP - complete. In this pap er we consider the variant on undirected graphs. The p olyhedral structure of the FeedbackVertex Set p olytop e is studied. We prove that this p olytop e is full dimensional and show that some inequalities are facet de ning. We describ e a new large class of valid constraints, the subset inequalities. A branch-and-cut algorithm for the exact solution of the problem is then outlined, and separation algorithms for the inequalities studied in the pap er are prop osed. A Lo cal Search heuristic is describ ed next. Finally we create a library of 1400 random generated instances with the geometric structure suggested by the applications, and we computationally compare the two algorithmic approaches on our library. Key words: feedbackvertex set, Branch-and-cut, lo cal search heuristic, tabu search. 1 Intro duction We consider undirected graphs G =V; E, where E is the edge set and V is the vertex set, with vertex weights c 2 IR, v 2 V .We denote by uv the edge of E having u and v as v end{p oints. Twovertices u and v are adjacent if there exists an edge uv 2 E connecting them. A sequence of vertices v ;v ; :::; v of G is called a path if v v 2 E , for i =2; :::; k . 1 2 k i1 i Vertex v is the origin and vertex v is the end of the path. If v = v a path is said to 1 k 1 k 1 be a cycle . A graph G is acyclic if it do es not contain cycles. A chord is an edge v v 2 E i j connecting two non consecutivevertices in a cycle. The vertex degree of vertex v in G is the numb er of the edges uv 2 E having v as end{p oint, and it is denoted by dv . P V For y 2 IR and S V ,we indicate with y S the sum y . v v 2S The undirected feedback vertex set UFVS problem consists of removing a vertex P subset F of minimum weight cF = c F V in an undirected graph in order v v2F to make it acyclic. We assume that a graph made of a single vertex v do es not contain cycles, and its feedbackvertex set is the empty set. The problem is known to b e NP -complete see [13], since the vertex cover problem can b e reduced to it; there exist p olynomial time algorithms for particular top ologies [6, 15 and 17]. The relevance of this problem arises in several areas. For example, it mo dels the problem of removing dead locks in a system of pro cessors, see [20]. Applications of the FeedbackVertex Set Problem to constraint satisfaction and Bayesian inference are re- p orted in [2] and [7]. In telecommunications it is helpful in nding the minimum number of vertices of control for monitoring a network. Another application of the problem with vertex weights equal to 1 cardinality case is in the context of op erating systems for the removal of dead locks created by cyclical pro cesses' requests of already allo cated resources. Finally it is relevant in the study of \monop olies" in synchronous distributed systems, as intro duced in [18, 19], where connection networks are undirected graphs of b ounded degree, namely, grids and toroidal grids. Several approximate and heuristic approaches have app eared in the literature on the problem on directed and undirected graphs, see for example [1], [2], [3], [4], [12], [16] and [20]. A p olyhedral approach to the FVS problem on directed graphs is presented in [10]. In this pap er we describ e a system based on a branch-and-cut algorithm for nding a UFVS of minimum weight in an undirected graph and a lo cal search heuristic. The comp onents of our system are: a set of exact and heuristic pro cedures for the separation of violated inequalities b elonging to a partial description of the UFVS p olytop e, an enumeration pro cedure that combines branching with cutting-planes techniques, and an eXploring Tabu SearchXTS pro cedure see [8] to nd a go o d upp er b ound on the ob jective function. We describ e these comp onents in the following sections. We create a library of 1400 random generated instances with the geometric structure suggested by the applications. Finally,we rep ort on our computational exp erience in Section 5. 2 2 Polyhedral results 0 A subgraph of G is the graph G =S; F , where S V and F E . The subgraph 0 G =S; E S is said to b e induced by the vertex set S V if E S is the set of edges having b oth end vertices in S . V We denote byIR the real vector space whose comp onents are indexed by the elements V U U of V . With every subset U V we asso ciate a vector x 2 IR where a comp onent x is v equal to 1 if v 2 U , and to 0 otherwise. B A For a given vertex set A and B A the incidencevector 2f0;1g of B is de ned B B by setting =1 if a2 B and =0 if a62 B . a a The feedback vertex set polytope QG of the graph G is the convex hull of incidence vectors of all feedbackvertex sets of G. Therefore the minimum weight UFVS of G can b e found, in principle, by solving the following linear program: min cx s.t. x 2 QG: 1 Recall that, givenavertex set W of G, the set W E of the edges b etween W and V n W is called a cut. A graph G is q {connected if j W j q for all ;W V. This is equivalenttosaying that G contains q edge-disjoint paths b etween any pair of vertices by de nition, a graph made of a single vertex is not q {connected for all q . A bridge of G is a cut of size 1. The ane hull of QGischaracterized in the following theorem. Theorem 2.1 If dv 2 for al l v 2 V of G, then QG is ful l dimensional, i.e., dimQG = jV j. Proof. If there exists a subset of the vertex set U = fv 2 V : j v j < 2g,by de nition of 2 feedbackvertex set we can reduce the size of the vertex set V by removing the set U : let 2 ~ V b e the resulting graph. We can rep eat this reduction, since the vertices of degree lower than 2, and the edges having at least one end vertex in U do not b elong to any cycle of 2 V . This fact reduces the dimension of QGbyjU j. 2 V nfv g If G is 2-connected, we can construct a feasible solution for all v 2 V , since is V a UFVS, thus obtaining jV j solutions. The other p oint is given by that is, obviously, a UFVS. Supp ose that G is not 2-connected, then G can b e decomp osed in 2-connected comp o- nents linked by bridges. Assume, w.l.o.g., that there exists only one bridge and two 3 shores. The shores of the bridge are the 2-connected graphs G =V;EV and 1 1 1 G =V ;EV , with V = V [ V and V \ V = ;, then QG and QG are full 2 2 2 1 2 1 2 1 2 dimensional, and dimQG = dimQG + dimQG is full dimensional. 1 2 The hyp othesis of dv 2 for all v 2 V seems not to b e restrictive considering that the applications describ ed in the literature are usually on 2-connected graphs. The optimal solution to 1 is the incidence vector of an optimal UFVS. Let C b e the set of all cycles C in G. An LP relaxation of QGisgiven by the following system of inequalities, xC 1; for all cycles C 2C 2 0 x 1: 3 ~ Consequently, the inequalities 2-3 de ne an LP relaxation QGofQG and pro- vide an integer programming formulation for the UFVS problem on G. The inequalities 2 are called cycle inequalities : their p olyhedral analysis is in theorem 2.4. We rst examine the trivial inequalities 0 x 1 for every v 2 V . v Theorem 2.2 For every vertex v 2 V , the inequality x 0 is facet de ning. v V nfv g Proof. Let v be a vertex of V , then is a UFVS that satis es x =0. We can v V nfw;vg construct a feasible solution for all w 2 V nfvg, since is a UFVS that satis es x =0. v Theorem 2.3 For every vertex v 2 V , the inequality x 1 is facet de ning. v V Proof. One p oint is given by that is a UFVS that satis es x = 1. Let v b e a given v Vnfw g vertex of V , then for all w 2 V nfvg, is a UFVS that satis es x =1.