Maximum Stable Set Problem Research Articles

Resolution Branch and Bound and an Application: The Maximum Weighted Stable Set Problem

We propose a new resolution algorithm, called resolution branch and bound (RBB), where a branch-and-bound scheme is empowered by exploiting the information contained in a family of closed subproblems, collected by a full resolution phase. In particular, we use this information to define a new branching rule that seems able to reduce the risk of incurring inappropriate branchings. We apply RBB and the proposed branching rule to the maximum weighted stable set problem, as its features allow us to speed up a time-consuming step in the full resolution phase. To compute upper bounds, we generalize to the weighted case the polynomial time procedure provided by Mannino and Sassano [Mannino, C., A. Sassano. 1994. An exact algorithm for the maximum stable set problem. Computational Optim. Appl. 3 243–258] for the unweighted case. Computational results validate the effectiveness of the provided branching rule and the good performance of RBB on many DIMACS benchmarks.

Stable sets of maximum weight in ( [formula omitted], banner)-free graphs

We prove that on ( P 7 , banner)-free graphs the maximum weight stable set problem is solvable in polynomial time.

A linear programming reformulation of the standard quadratic optimization problem

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note, we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LP's whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.

Augmenting chains in graphs without a skew star

The augmenting chain technique has been applied to solve the maximum stable set problem in the class of line graphs (which coincides with the maximum matching problem) and then has been extended to the class of claw-free graphs. In the present paper, we propose a further generalization of this approach. Specifically, we show how to find an augmenting chain in graphs containing no skew star, i.e. a tree with exactly three vertices of degree 1 of distances 1, 2, 3 from the only vertex of degree 3. As a corollary, we prove that the maximum stable set problem is polynomially solvable in a class that strictly contains claw-free graphs, improving several existing results.

On Extracting Maximum Stable Sets in Perfect Graphs Using Lovász's Theta Function

We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lovasz's theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its induced subgraphs. We develop an efficient, polynomial-time algorithm to extract a maximum stable set in a perfect graph using the theta function. Our algorithm iteratively transforms an approximate solution of the semidefinite formulation of the theta function into an approximate solution of another formulation, which is then used to identify a vertex that belongs to a maximum stable set. The subgraph induced by that vertex and its neighbors is removed and the same procedure is repeated on successively smaller graphs. We establish that solving the theta problem up to an adaptively chosen, fairly rough accuracy suffices in order for the algorithm to work properly. Furthermore, our algorithm successfully employs a warm-start strategy to recompute the theta function on smaller subgraphs. Computational results demonstrate that our algorithm can efficiently extract maximum stable sets in comparable time it takes to solve the theta problem on the original graph to optimality.

D.C. Versus Copositive Bounds for Standard QP

The standard quadratic program (QPS) is minx??xTQx, where $$\Delta\subset\Re^n$$ is the simplex ? = {x ? 0 ? ?i=1n xi = 1}. QPS can be used to formulate combinatorial problems such as the maximum stable set problem, and also arises in global optimization algorithms for general quadratic programming when the search space is partitioned using simplices. One class of `d.c.' (for `difference between convex') bounds for QPS is based on writing Q=S?T, where S and T are both positive semidefinite, and bounding xT Sx (convex on ?) and ?xTx (concave on ?) separately. We show that the maximum possible such bound can be obtained by solving a semidefinite programming (SDP) problem. The dual of this SDP problem corresponds to adding a simple constraint to the well-known Shor relaxation of QPS. We show that the max d.c. bound is dominated by another known bound based on a copositive relaxation of QPS, also obtainable via SDP at comparable computational expense. We also discuss extensions of the d.c. bound to more general quadratic programming problems. For the application of QPS to bounding the stability number of a graph, we use a novel formulation of the Lovasz ? number to compare ?, Schrijver's ??, and the max d.c. bound.

A Semidefinite Programming Relaxation for the Generalized Stable Set Problem

In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grotschel, Lovasz and Schrijver to the generalized stable set problem. We define a convex set which serves as a relaxation problem, and show that optimizing a linear function over the set can be done in polynomial time. This implies that the generalized stable set problem for perfect bidirected graphs is polynomial time solvable. Moreover, we prove that the convex set is a polytope if and only if the corresponding bidirected graph is perfect. The definition of the convex set is based on a semidefinite programming relaxation of Lovasz and Schrijver for the maximum weight stable set problem, and the equivalent representation using infinitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof.

A Linear Programming Reformulation of the Standard Quadratic Optimization Problem

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NPhard, and contains the maximum stable set problem in graphs as a special case. In this note we show that the SQO problem may be reformulated as an (exponentially sized) linear program.

Separating multi-oddity constrained shortest circuits over the polytope of stable multisets

The maximum stable set problem is NP -hard. Koster and Zymolka introduced as a generalization the stable multiset problem by allowing vertices multiple times subject to vertex- and edge capacities and introduced cycle inequalities. We derive an efficient separation algorithm for them.

P5-free augmenting graphs and the maximum stable set problem

The complexity status of the maximum stable set problem in the class of P 5-free graphs is unknown. In this paper, we first propose a characterization of all connected P 5-free augmenting graphs. We then use this characterization to detect families of subclasses of P 5-free graphs where the maximum stable set problem has a polynomial time solution. These families extend several previously studied classes.

On the structure and stability number of P5- and co-chair-free graphs

We give a O(nm) time algorithm for the maximum weight stable set (MWS) problem on P 5- and co-chair-free graphs without recognizing whether the (arbitrary) input graph is P 5- and co-chair-free. This algorithm is based on the fact that prime P 5- and co-chair-free graphs containing 2 K 2 are matched co-bipartite graphs and thus have very simple structure, and for 2 K 2-free graphs, there is a polynomial time algorithm for the MWS problem due to a result of Farber saying that 2 K 2-free graphs contain at most O(n 2) maximal stable sets. A similar argument can be used for ( P 5,co-P)-free graphs; their prime graphs are 2 K 2-free. Moreover, we give a complete classification of ( P 5,co-chair, H)-free graphs with respect to their clique width when H is a one-vertex P 4 extension; this extends the characterization of ( P 5, P 5 ,co-chair)-free graphs called semi- P 4- sparse by Fouquet and Giakoumakis. For H being a house, P, bull or gem, the class of ( P 5,co-chair, H)-free graphs has bounded clique width and very simple structure whereas for the other four cases, namely H being a co-gem, chair, co- P or C 5, the class has unbounded clique width due to a result of Makowsky and Rotics. Bounded clique width implies linear time algorithms for all algorithmic problems expressible in LinEMSOL—a variant of Monadic Second Order Logic including the MWS Problem.

Stable sets in two subclasses of banner-free graphs

The maximum stable set problem is NP-hard, even when restricted to banner-free graphs. In this paper, we use the augmenting graph approach to attack the problem in two subclasses of banner-free graphs. We first provide both classes with the complete characterization of minimal augmenting graphs. Based on the obtained characterization, we prove polynomial solvability of the problem in the class of ( banner, P 8)-free graphs, improving several existing results.

Some results on maximum stable sets in certain P5-free graphs

We introduce some large classes of P 5-free graphs for which the maximum stable set problem can be efficiently solved. The solution algorithms are based on the search of augmenting vertices. Then a method for extending some of them to the weighted case is provided.

Finding augmenting chains in extensions of claw-free graphs

Finding augmenting chains is in the heart of the maximum matching problem, which is equivalent to the maximum stable set problem in the class of line graphs. Due to the celebrated result of Edmonds, augmenting chains can be found in line graphs in polynomial time. Minty and Sbihi generalized this result to claw-free graphs. In this paper we extend it to larger classes. As a particular consequence, a new polynomially solvable case for the maximum stable set problem has been detected.

Stability number of bull- and chair-free graphs revisited

De Simone showed that prime bull- and chair-free graphs containing a co-diamond are either bipartite or an induced cycle of odd length at least five. Based on this result, we give a complete structural characterization of prime (bull,chair)-free graphs having stability number at least four as well as of (bull,chair,co-chair)-free graphs. This implies constant-bounded clique width for these graph classes which leads to linear time algorithms for some algorithmic problems. Moreover, we obtain a robust O ( n m ) time algorithm for the maximum weight stable set problem on bull- and chair-free graphs without testing whether the (arbitrary) input graph is bull- and chair-free. This improves previous results with respect to structural insight, robustness and time bounds.

A Derivation of Lovász' Theta via Augmented Lagrange Duality

A recently introduced dualization technique for binary linear programs with equality constraints, essentially due to Poljak et al. [13], and further developed in Lemarechal and Oustry [9], leads to simple alternative derivations of well-known, important relaxations to two well-known problems of discrete optimization: the maximum stable set problem and the maximum vertex cover problem. The resulting relaxation is easily transformed to the well-known Lovasz θ number.

Computational Experience with Stable Set Relaxations

We investigate relaxations for the maximum stable set problem based on the Lovasz number $\vartheta(G)$ as an initial upper bound. We strengthen this relaxation by adding two classes of cutting planes, odd circuit and triangle inequalities. We present computational results using this tighter model on many classes of graphs.

A new Greedoid: the family of local maximum stable sets of a forest

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set if it is a maximum stable of the subgraph of G spanned by S∪ N( S), where N( S) is the neighborhood of S. One theorem of Nemhauser and Trotter Jr. (Math. Programming 8 (1975) 232–248), working as a useful sufficient local optimality condition for the weighted maximum stable set problem, ensures that any local maximum stable set of G can be enlarged to a maximum stable set of G. In this paper we demonstrate that an inverse assertion is true for forests. Namely, we show that for any non-empty local maximum stable set S of a forest T there exists a local maximum stable set S 1 of T, such that S 1⊂ S and | S 1|=| S|−1. Moreover, as a further strengthening of both the theorem of Nemhauser and Trotter Jr. and its inverse, we prove that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set.

A branch-and-cut algorithm for the maximum cardinality stable set problem

We propose a branch-and-cut algorithm for the Maximum Cardinality Stable Set problem. Rank constraints of general structure are generated by executing clique separation algorithms on a modified graph obtained with edge projections. A branching scheme exploiting the available inequalities is also introduced. A computational experience on the DIMACS benchmark graphs validates the effectiveness of the approach.

Stability in P5- and banner-free graphs

We prove polynomial solvability of the maximum stable set problem in P 5-free banner-free (( P 5, B)-free) graphs, where P 5 is the simple path on five vertices and a banner is the graph with vertices a, b, c, d, e and edges ( a, b),( b, c),( b, d),( c, e),( d, e).