An exact algorithm for the maximum stable set problem

Abstract

We describe a new branch-and-bound algorithm for the exact solution of the maximum cardinality stable set problem. The bounding phase is based on a variation of the standard greedy algorithm for finding a colouring of a graph. Two different node-fixing heuristics are also described. Computational tests on random and structured graphs and very large graphs corresponding to 'real-life' problems show that the algorithm is competitive with the fastest algorithms known so far. © 1994 Kluwer Academic Publishers

Multi-neighborhood tabu search for the maximum weight clique problem

“…In order to show the relative effectiveness of our MN/TS for the MWCP, we first compare MN/TS with two state of the art algorithms from the literature [22,20]. The main comparison criterion is the quality of the solutions found.…”

“…To augment the above comparison, Table 4 contrasts the results of our MN/TS with those of the AugSearch algorithm reported in [20]. The authors of the AugSearch algorithm used a subset of 36 DIMACS graphs with weighting function in which the weight w i of vertex i is set equal to (i mod 10) + 1.…”

“…Along with the proposed algorithm, a set of MWCP benchmark instances using the DIMACS graphs are also introduced (we use these instances in our experiments). In [20], an augmentation algorithm is described which is based on edge projections for the equivalent maximum weight stable set problem. Other representative studies include a deterministic iterated greedy construction algorithm using a nonlinear programming formulation [7], and a distributed computational network algorithm [6].…”

Stability preserving transformations of graphs

“…It is based on a transformation which, given a graph G and an induced subgraph H of G, substitutes H together with its neighbourhood by a new graph in such a way that the stability number of the resulting graph is exactly α(H) less that that of G. This generalization includes several particular graph transformations studied in the literature, such as crown reduction (was used to prove fixed-parameter tractability of the minimum vertex cover problem [12]), cycle shrinking (a key tool to solve the maximum matching problem, which is equivalent to the maximum stable set problem in the class of line graphs [29]), clique reduction (was used to derive polynomial time algorithms for the maximum stable set problem in several particular classes of graphs [13,26,23]. Also, the edge projection, which is a specialization of the clique reduction when restricted to edges, has been used in [32,33] to develop some heuristics for the stable set problem).…”

CABOB: A Fast Optimal Algorithm for Winner Determination in Combinatorial Auctions

“…Three fundamentally different approaches have been taken: (1) designing algorithms that provably find an optimal solution but are slow on some problem instances (Sandholm 2002a;Fujishima et al 1999;Sandholm and Suri 2003;Andersson et al 2000;de Vries and Vohra 2003;Gonen and Lehmann 2000;Lehmann and Gonen 2001;Leyton-Brown et al 2000b;van Hoesel and Müller 2001;Balas and Yu 1986;Babel and Tinhofer 1990;Babel 1991;Xue 1991, 1996;Nemhauser and Sigismondi 1992;Mannino and Sassano 1994;Pardalos and Desai 1991;Loukakis and Tsouros 1983), (2) designing algorithms that are provably fast but fail to find an optimal (or even close to optimal) solution to some problem instances (Lehmann et al 2002, Hoos and Boutilier 2000, Zurel and Nisan 2001, Anandalingam et al 2002, and (3) restricting the bundles on which bids can be submitted so severely that the remaining problem can be solved optimally and provably fast (Rothkopf et al 1998, Tennenholtz 2000, Penn and Tennenholtz 2000, van Hoesel and Müller 2001, Lehmann et al 2005.…”