Given a graph G = (V , E), the independent set problem is that of finding a maximum-cardinality subset S of V such that no two vertices in S are adjacent. We introduce two fast local search routines for this problem. The first can determine in linear time whether a maximal solution can be improved by replacing a single vertex with two others. The second routine can determine in O(m ) time (where is the highest degree in the graph) whether there are two solution vertices than can be replaced by a set of three. We also present a more elaborate heuristic that successfully applies local search to find near-optimum solutions to a wide variety of instances. We test our algorithms on instances from the literature as well as on new ones proposed in this paper.

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This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arborescence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms. Powerful new cuts expressed over a very large set of variables could be added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very significant improvements over previous algorithms. Several open instances could be solved to optimality.

We say that a graph G has the CIS-property and call G a CIS-graph if each maximal clique and each maximal stable set of G intersect. By definition, G is a CIS-graph if and only if the complementary graphḠ is a CIS-graph too. In this paper we give some necessary and some sufficient conditions for the CIS-property to hold. In general, problems of efficient characterization and recognition of CIS-graphs remain open. Given an integer k ≥ 2, a comb (or k-comb) S k is a graph with 2k vertices k of which, v 1 ,. .. , v k , form a clique C, while others, v ′ 1 ,. .. , v ′ k , form a stable set S, and (v i , v ′ i) is an edge for all i = 1,. .. , k, and there are no other edges. The complementary graphS k is called an anti-comb (or k-anti-comb). Clearly, S and C switch in the complementary graphs. Obviously, the combs and anti-combs are not CIS-graphs, since C ∩ S = ∅. Hence, if a CIS-graph G contains an induced comb (respectively, anti-comb) then it must be settled, that is, G must contain a vertex v connected to all vertices of C and to no vertex of S. However, these conditions are only necessary but not sufficient for the CIS-property to hold. Our main result is the following theorem: G is a CIS-graph whenever G contains no induced 3-combs and 3-anti-combs, and every induced 2-comb is settled in G. We also generalize the concept of CIS-graph as follows. Given integer d ≥ 2 and a complete graph whose edges are colored by d colors G = (V ; E 1 ,. .. , E d), we say that G is a CIS-dgraph (has the CIS-d-property) if d i=1 C i = ∅ whenever C i is a maximal color i-free subset of V , that is, (v, v ′) ∈ E i for no v, v ′ ∈ C i. Clearly, in case d = 2 we return to the concept of CIS-graphs. (More accurately, CIS-2-graph is a pair of two complementary CIS-graphs.) We conjecture that each CIS-d-graph is a Gallai graph, that is, it contains no triangle colored by 3 distinct colors. We obtain results supporting this conjecture and also show that if it holds then characterization and recognition of CIS-d-graphs is easily reduced to characterization and recognition of CIS-graphs.

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