Maximum Weight Stable Set Problem Research Articles

Combining decomposition approaches for the Maximum Weight Stable Set problem

The Maximum Weight Stable Set Problem (MWS) is a well-known NP-hard problem. A popular way to study MWS is to detect graph classes for which MWS can be solved in polynomial time. In this context some decomposition approaches have been introduced in the literature, in particular decomposition by homogeneous sets and decomposition by clique separators, which had several applications in the literature. Brandstädt and Hoàng (2007) [6] showed how to combine such two decomposition approaches and stated the following result: If the MWS problem can be solved in polynomial time for every subgraph of a graph G which has no homogeneous set and has no clique separators then so can the problem for G. This result, namely Corollary 9 of [6], was used in various papers.Unfortunately, it was stated in a successive paper by Brandstädt and Giakoumakis (2015) [5] that “Actually, [6] does not give a proof for this, and it seems that Corollary 9 of [6] promises too much; later attempts for proving it failed, and thus, it is unproven and its use has to be avoided.”This manuscript introduces a proof of Corollary 9 of [6]. Furthermore a third decomposition approach, namely decomposition by anti-neighborhoods of vertices, is combined together with such two decomposition approaches. Then the various papers in which Corollary 9 of [6] was used would be fixed.

Combining Decomposition Approaches for the Maximum Weight Stable Set Problem

Combining Decomposition Approaches for the Maximum Weight Stable Set Problem

An $${\mathcal {O}}(n^2 \log {n})$$ algorithm for the weighted stable set problem in claw-free graphs

A graph G(V, E) is claw-free if no vertex has three pairwise non-adjacent neighbours. The maximum weight stable set (MWSS) problem in a claw-free graph is a natural generalization of the matching problem and has been shown to be polynomially solvable by Minty and Sbihi in 1980. In a remarkable paper, Faenza, Oriolo and Stauffer have shown that, in a two-step procedure, a claw-free graph can be first turned into a quasi-line graph by removing strips containing all the irregular nodes and then decomposed into {claw, net}-free strips and strips with stability number at most three. Through this decomposition, the MWSS problem can be solved in $$\mathcal{O}(|V|(|V| \log |V| + |E|))$$ time. In this paper, we describe a direct decomposition of a claw-free graph into {claw, net}-free strips and strips with stability number at most three which can be performed in $$\mathcal{O}(|V|^2)$$ time. In two companion papers we showed that the MWSS problem can be solved in $$\mathcal{O}(|E| \log |V|)$$ time in claw-free graphs with $$\alpha (G) \le 3$$ and in $$\mathcal{O}(|V| \sqrt{|E|})$$ time in {claw, net}-free graphs with $$\alpha (G) \ge 4$$ . These results prove that the MWSS problem in a claw-free graph can be solved in $$\mathcal{O}(|V|^2 \log |V|)$$ time, the same complexity of the best and long standing algorithm for the MWSS problem in line graphs.

The class of (P7,C4,C5)‐free graphs: Decomposition, algorithms, and χ‐boundedness

AbstractAs usual, () denotes the path on vertices, and () denotes the cycle on vertices. For a family of graphs, we say that a graph is ‐free if no induced subgraph of is isomorphic to any graph in . We present a decomposition theorem for the class of ‐free graphs; in fact, we give a complete structural characterization of ‐free graphs that do not admit a clique‐cutset. We use this decomposition theorem to show that the class of ‐free graphs is ‐bounded by a linear function (more precisely, every ‐free graph satisfies ). We also use the decomposition theorem to construct an algorithm for the minimum coloring problem, an algorithm for the maximum weight stable set problem, and an algorithm for the maximum weight clique problem for this class, where denotes the number of vertices and the number of edges of the input graph.

An $$\mathcal{O}(m\log n)$$ O ( m log n ) algorithm for the weighted stable set problem in claw-free graphs with $$\alpha ({G}) \le 3$$ α ( G ) ≤ 3

In this paper we show how to solve the Maximum Weight Stable Set Problem in a claw-free graph G(V, E) with $$\alpha (G) \le 3$$ź(G)≤3 in time $$\mathcal{O}(|E|\log |V|)$$O(|E|log|V|). More precisely, in time $$\mathcal{O}(|E|)$$O(|E|) we check whether $$\alpha (G) \le 3$$ź(G)≤3 or produce a stable set with cardinality at least 4; moreover, if $$\alpha (G) \le 3$$ź(G)≤3 we produce in time $$\mathcal{O}(|E|\log |V|)$$O(|E|log|V|) a maximum weight stable set of G. This improves the bound of $$\mathcal{O}(|E||V|)$$O(|E||V|) due to Faenza, Oriolo and Stauffer.

An [formula omitted] algorithm for the weighted stable set problem in {claw, net}-free graphs with [formula omitted

In this paper we show that a connected {claw, net}-free graph G(V,E) with α(G)≥4 is the union of a strongly bisimplicial clique Q and at most two clique-strips. A clique is strongly bisimplicial if its neighborhood is partitioned into two cliques which are mutually non-adjacent and a clique-strip is a sequence of cliques {H0,…,Hp} with the property that Hi is adjacent only to Hi−1 and Hi+1. By exploiting such a structure we show how to solve the Maximum Weight Stable Set Problem in such a graph in time O(|V||E|), improving the previous complexity bound of O(|V||E|).

Some results on stable sets for k-colorable P₆-free graphs and generalizations

Graphs and Algorithms This article deals with the Maximum Weight Stable Set (MWS) problem (and some other related NP-hard problems) and the class of P-6-free graphs. The complexity status of MWS is open for P-6-free graphs and is open even for P-5-free graphs (as a long standing open problem). Several results are known for MWS on subclasses of P-5-free: in particular, MWS can be solved for k-colorable P-5-free graphs in polynomial time for every k (depending on k) and more generally for (P-5, K-p)-free graphs (depending on p), which is a useful result since for every graph G one can easily compute a k-coloring of G, with k not necessarily minimum. This article studies the MWS problem for k-colorable P-6-free graphs and more generally for (P-6, K-p)-free graphs. Though we were not able to define a polynomial time algorithm for this problem for every k, this article introduces: (i) some structure properties of P-6-free graphs with respect to stable sets, (ii) two reductions for MWS on (P-6; K-p)-free graphs for every p, (iii) three polynomial time algorithms to solve MWS respectively for 3-colorable P-6-free, for 4-colorable P-6-free, and for (P-6, K-4)-free graphs (the latter allows one to state, together with other known results, that MWS can be solved for (P-6, F)-free graphs in polynomial time where F is any four vertex graph).

Complexity results for equistable graphs and related classes

The class of equistable graphs is defined by the existence of a cost structure on the vertices such that the maximal stable sets are characterized by their costs. This graph class, not contained in any nontrivial hereditary class, has so far been studied mostly from a structural point of view; characterizations and polynomial time recognition algorithms have been obtained for special cases.We focus on complexity issues for equistable graphs and related classes. We describe a simple pseudo-polynomial-time dynamic programming algorithm to solve the maximum weight stable set problem along with the weighted independent domination problem in some classes of graphs, including equistable graphs. Our results are obtained within the wider context of Boolean optimization; corresponding hardness results are also provided. More specifically, we show that the above problems are APX-hard for equistable graphs and that it is co-NP-complete to determine whether a given cost function on the vertices of a graph defines an equistable cost structure of that graph.

Some observations on maximum weight stable sets in certain P5-free graphs

The maximum weight stable set problem (MWS) is the weighted version of the maximum stable set problem (MS), which is NP-hard. The class of P 5-free graphs – i.e., graphs with no induced path of five vertices – is the unique minimal class, defined by forbidding a single connected subgraph, for which the computational complexity of MS is an open question. At the same time, it is known that MS can be efficiently solved for ( P 5 , F ) -free graphs, where F is any graph of five vertices different to a C 5. In this paper we introduce some observations on P 5-free graphs, and apply them to introduce certain subclasses of such graphs for which one can efficiently solve MWS. That extends or improves some known results, and implies – together with other known results – that MWS can be efficiently solved for ( P 5 , F ) -free graphs where F is any graph of five vertices different to a C 5.

On clique separators, nearly chordal graphs, and the Maximum Weight Stable Set Problem

Clique separators in graphs are a helpful tool used by Tarjan as a divide-and-conquer approach for solving various graph problems such as the Maximum Weight Stable Set (MWS) Problem, Maximum Clique, Graph Coloring and Minimum Fill-in, but few examples of graph classes having clique separators are known. We use this method to solve MWS in polynomial time for two classes where the unweighted Maximum Stable Set (MS) Problem is solvable in polynomial time by augmenting techniques but the complexity of the MWS problem was open. Another example, namely a result by Alekseev for the MWS problem on a subclass of P 5 -free graphs obtained by clique separators, can be improved by our techniques. We also combine clique separators with decomposition by homogeneous sets in graphs and use the following notion: A graph is nearly Π if for each of its vertices, the subgraph induced by the set of its nonneighbors has property Π . We deal with the cases Π ∈ { chordal,perfect } . This also simplifies a result obtained by a method called struction.

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.

New applications of clique separator decomposition for the Maximum Weight Stable Set problem

Graph decompositions such as decomposition by clique separators and modular decomposition are of crucial importance for designing efficient graph algorithms. Clique separators in graphs were used by Tarjan as a divide-and-conquer approach for solving various problems such as the Maximum Weight Stable Set (MWS) problem, Colouring and Minimum Fill-in. The basic tool is a decomposition tree of the graph whose leaves have no clique separator (so-called atoms), and the problem can be solved efficiently on the graph if it is efficiently solvable on its atoms. We give new examples where the clique separator decomposition works well for the MWS problem; our results improve and extend various recently published results. In particular, we describe the atom structure for some new classes of graphs whose atoms are P 5 -free (the P 5 is the induced path with five vertices) and obtain new polynomial time results for the MWS problem. The complexity of this problem on the class of P 5 -free graphs is still unknown.

P6- and triangle-free graphs revisited: structure and bounded clique-width

The Maximum Weight Stable Set (MWS) Problem is one of the fundamental problems on graphs. It is well-known to be NP-complete for triangle-free graphs, and Mosca has shown that it is solvable in polynomial time when restricted to P6- and triangle-free graphs. We give a complete structure analysis of (nonbipartite) P6- and triangle-free graphs which are prime in the sense of modular decomposition. It turns out that the structure of these graphs is extremely simple implying bounded clique-width and thus, efficient algorithms exist for all problems expressible in terms of Monadic Second Order Logic with quantification only over vertex predicates. The problems Vertex Cover, MWS, Maximum Clique, Minimum Dominating Set, Steiner Tree, and Maximum Induced Matching are among them. Our results improve the previous one on the MWS problem by Mosca with respect to structure and time bound but also extends a previous result by Fouquet, Giakoumakis and Vanherpe which have shown that bipartite P6-free graphs have bounded clique-width. Moreover, it covers a result by Randerath, Schiermeyer and Tewes on polynomial time 3-colorability of P6- and triangle-free graphs.

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.

Efficient robust algorithms for the Maximum Weight Stable Set Problem in chair-free graph classes

Modular decomposition of graphs is a powerful tool for designing efficient algorithms for problems on graphs such as Maximum Weight Stable Set (MWS) and Maximum Weight Clique. Using this tool we obtain O( n· m) time algorithms for MWS on chair- and xbull-free graphs which considerably extend an earlier result on bull- and chair-free graphs by De Simone and Sassano (the chair is the graph with vertices a, b, c, d, e and edges ab, bc, cd, be, and the xbull is the graph with vertices a, b, c, d, e, f and edges ab, bc, cd, de, bf, cf). Moreover, our algorithm is robust in the sense that we do not have to check in advance whether the input graphs are indeed chair- and xbull-free.

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.

( P5,diamond)-free graphs revisited: structure and linear time optimization

Using the concept of prime graphs and modular decomposition of graphs, we give a complete structure description of ( P 5,diamond)-free graphs implying that these graphs have bounded clique width (the P 5 is the induced path with five vertices a, b, c, d, e and four edges ab, bc, cd, de, and the diamond consists of four vertices a, b, c, d such that a, b, c form an induced path with edges ab, bc, and vertex d is adjacent to a, b and c). The structure and bounded clique width of this graph class allows to solve several algorithmic problems on this class in linear time, among them the problems Maximum Weight Stable Set (MWS), Maximum Weight Clique, Domination, Steiner Tree and in general every algorithmic problem which is, roughly speaking, expressible in a certain kind of Monadic Second-Order Logic using quantification only over vertex but not over edge set predicates. This improves previous results on ( P 5,diamond)-free graphs in several ways: We give a complete structure description of prime ( P 5,diamond)-free graphs, we do not only solve the MWS problem on this class, we achieve linear time algorithms (instead of a recent time bound O ( n m ) ), and we can do all this on a larger graph class containing ( P 5,diamond)-free graphs which admits linear time recognition.

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.

Maximum Weight Stable Set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time

Minty has shown that the Maximum Weight Stable Set (MWS) Problem can be solved in polynomial time when restricted to claw-free graphs. We show that the structure of graphs being both claw-free and co-claw-free is very simple which implies bounded clique width for this graph class. It is known that for graph classes of bounded clique width (assuming that k-expressions can be determined in linear time), there are linear time algorithms for all problems expressible in Monadic Second Order Logic with quantification only over vertex sets. The problems MWS, Maximum Clique, Domination and Steiner Tree, for example, are expressible in this way. Moreover, we describe the structure of prime graphs being both H- and H -free for any four-vertex graph H and obtain bounded clique width for these graph classes except the 2K 2 - and C 4 -free graphs.

A LINEAR TIME ALGORITHM FOR THE GENERALIZED STABLE SET PROBLEM ON TRIANGULATED BIDIRECTED GRAPHS

The generalized stable set problem is an extension of the maximum weight stable set problem for undirected graphs to bidirected graphs. This paper shows that the problem on triangulated bidirected graphs is solvable in linear time. We also propose an exact branch and bound algorithm for the general problem by applying the linear time algorithm.