Maximum Stable Set Problem Research Articles

An Error Analysis for Polynomial Optimization over the Simplex Based on the Multivariate Hypergeometric Distribution

We study the minimization of fixed-degree polynomials over the simplex. This problem is well-known to be NP-hard, as it contains the maximum stable set problem in graph theory as a special case. In this paper, we consider a rational approximation by taking the minimum over the regular grid, which consists of rational points with denominator $r$ (for given $r$). We show that the associated convergence rate is $O(1/r^2)$ for quadratic polynomials. For general polynomials, if there exists a rational global minimizer over the simplex, we show that the convergence rate is also of the order $O(1/r^2)$. Our results answer a question posed by De Klerk, Laurent, and Sun [Math. Program., 151 (2015), pp. 433--457]. and improves on previously known $O(1/r)$ bounds in the quadratic case.

Sorting common operations to minimize the number of tardy jobs

We study an operation scheduling problem where a finite set of jobs with due dates must be completed by one machine: each job is completed as soon as a specific subset of unit operations is done. Distinct jobs may share operations, and when an operation is done, it is done for all the jobs that share it. The goal is to schedule operations so that the (weighted) number of tardy jobs is minimized. We reformulate the problem as maximum stable set problem on a special graph and study its structure. Valid inequalities and optimality cuts are derived, separated, and tested in a computational experience that identifies some features of hard instances and the potential contribution of the addition, at root, of various cut classes. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 64(4), 306–320 2014

A Refined Error Analysis for Fixed-Degree Polynomial Optimization over the Simplex

We consider the problem of minimizing a fixed-degree polynomial over the standard simplex. This problem is well known to be NP-hard, since it contains the maximum stable set problem in combinatorial optimization as a special case. In this paper, we revisit a known upper bound obtained by taking the minimum value on a regular grid, and a known lower bound based on Polya’s representation theorem. More precisely, we consider the difference between these two bounds and we provide upper bounds for this difference in terms of the range of function values. Our results refine the known upper bounds in the quadratic and cubic cases, and they asymptotically refine the known upper bound in the general case.

Handelman’s hierarchy for the maximum stable set problem

The maximum stable set problem is a well-known NP-hard problem in combinatorial optimization, which can be formulated as the maximization of a quadratic square-free polynomial over the (Boolean) hypercube. We investigate a hierarchy of linear programming relaxations for this problem, based on a result of Handelman showing that a positive polynomial over a polytope with non-empty interior can be represented as conic combination of products of the linear constraints defining the polytope. We relate the rank of Handelman's hierarchy with structural properties of graphs. In particular we show a relation to fractional clique covers which we use to upper bound the Handelman rank for perfect graphs and determine its exact value in the vertex-transitive case. Moreover we show two upper bounds on the Handelman rank in terms of the (fractional) stability number of the graph and compute the Handelman rank for several classes of graphs including odd cycles and wheels and their complements. We also point out links to several other linear and semidefinite programming hierarchies.

A note on the Cornaz–Jost transformation to solve the graph coloring problem

In this note, we use a reduction by Cornaz and Jost from the graph (max-)coloring problem to the maximum (weighted) stable set problem in order to characterize new graph classes where the graph coloring problem and the more general max-coloring problem can be solved in polynomial time.

A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems

Using the semi-tensor product of matrices, this paper investigates the maximum (weight) stable set and vertex coloring problems of graphs with application to the group consensus of multi-agent systems, and presents a number of new results and algorithms. Firstly, by defining a characteristic logical vector and using the matrix expression of logical functions, an algebraic description is obtained for the internally stable set problem, based on which a new algorithm to find all the internally stable sets is established for any graph. Secondly, the maximum (weight) stable set problem is considered, and a necessary and sufficient condition is presented, by which an algorithm to find all the maximum (weight) stable sets is obtained. Thirdly, the vertex coloring problem is studied by using the semi-tensor product method, and two necessary and sufficient conditions are proposed for the colorability, based on which a new algorithm to find all the k-coloring schemes and minimum coloring partitions is put forward. Finally, the obtained results are applied to multi-agent systems, and a new protocol design procedure is presented for the group consensus problem. The study of illustrative examples shows that the results/algorithms presented in this paper are very effective.

How tight is the corner relaxation? Insights gained from the stable set problem

The corner relaxation of a mixed-integer linear program is a central concept in cutting plane theory. In a recent paper Fischetti and Monaci provide an empirical assessment of the strength of the corner and other related relaxations on benchmark problems. In this paper we give a precise characterization of the bounds given by these relaxations for the edge formulation of the maximum stable set problem in a graph.

On the accuracy of uniform polyhedral approximations of the copositive cone

We consider linear optimization problems over the cone of copositive matrices. Such conic optimization problems, called copositive programs, arise from the reformulation of a wide variety of difficult optimization problems. We propose a hierarchy of increasingly better outer polyhedral approximations to the copositive cone. We establish that the sequence of approximations is exact in the limit. By combining our outer polyhedral approximations with the inner polyhedral approximations due to de Klerk and Pasechnik [SIAM J. Optim. 12 (2002), pp. 875–892], we obtain a sequence of increasingly sharper lower and upper bounds on the optimal value of a copositive program. Under primal and dual regularity assumptions, we establish that both sequences converge to the optimal value. For standard quadratic optimization problems, we derive tight bounds on the gap between the upper and lower bounds. We provide closed-form expressions of the bounds for the maximum stable set problem. Our computational results shed light on the quality of the bounds on randomly generated instances.

A tutorial on branch and cut algorithms for the maximum stable set problem

AbstractThis tutorial provides an overview of various characteristics of effective branch and cut type algorithms for the maximum stable set problem. We discuss several facet‐defining inequalities for the stable set polytope along with their separation routines. In particular, we review implementation tweaks for the separation routines and reference empirical studies, illustrating the performance of these cutting planes for benchmark graphs. In addition to the polyhedral study, we present basic preprocessing, discuss heuristic methods particularly suited within a branch and cut framework, and examine a branching rule.

On $3$-Colorable $P_5$-Free Graphs

A graph is $P_5$-free when it does not contain a $P_5$ (that is, a path with five vertices) as an induced subgraph. The class of $P_5$-free graphs is of particular interest, especially with respect to the still unknown complexity status of the maximum stable set problem in that class. We investigate the class of $3$-colorable $P_5$-free graphs. We give a complete description of the structure of those graphs and derive a linear-time algorithm that tests membership in this class. Moreover, the algorithm is able to find a maximum weight stable set of a $3$-colorable $P_5$-free graph in linear time.

On Polar, Trivially Perfect Graphs

<p>During the last decades, different types of decompositions have been processed in the field of graph theory. In various problems, for example in the construction of recognition algorithms, frequently appears the so-called weakly decomposition of graphs.<br />Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. Recognizing a polar graph is known to be NP-complete. For this class of graphs, polynomial algorithms for the maximum stable set problem are unknown and algorithms for the dominating set problem are also NP-complete.<br />In this paper we characterize the polar graphs using the weakly decomposition, give a polynomial time algorithm for recognizing graphs that are both trivially perfect and polar, and directly calculate the domination number. For the stability number and clique number, we give polynomial time algorithms. </p>

Alternating direction augmented Lagrangian methods for semidefinite programming

We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration, our basic algorithm minimizes the augmented Lagrangian function for the dual SDP problem sequentially, first with respect to the dual variables corresponding to the linear constraints, and then with respect to the dual slack variables, while in each minimization keeping the other variables fixed, and then finally it updates the Lagrange multipliers (i.e., primal variables). Convergence is proved by using a fixed-point argument. For SDPs with inequality constraints and positivity constraints, our algorithm is extended to separately minimize the dual augmented Lagrangian function over four sets of variables. Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that our algorithms are robust and very efficient due to their ability or exploit special structures, such as sparsity and constraint orthogonality in these problems.

A Combined Parallel Lagrangian Decomposition and Cutting-Plane Generation for Maximum Stable Set Problems

We propose an integer programming formulation for the problem of finding the maximum k -partite induced sub-graph of a graph G based on representatives of stable sets. We investigate upper bounds provided by the solution, via a parallel sub-gradient algorithm, of a Lagrangian decomposition that breaks up this formulation into maximum weighted stable set problems for sub-graphs of G . Some computational experiments were carried out with an effective multi-threaded parallel implementation in a multi-core system, and their results are presented.

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.

A Branch and Cut solver for the maximum stable set problem

This paper deals with the cutting-plane approach to the maximum stable set problem. We provide theoretical results regarding the facet-defining property of inequalities obtained by a known project-and-lift-style separation method called edge-projection, and its variants. An implementation of a Branch and Cut algorithm is described, which uses edge-projection and two other separation tools which have been discussed for other problems: local cuts (pioneered by Applegate, Bixby, Chvatal and Cook) and mod-k cuts. We compare the performance of this approach to another one by Rossi and Smiriglio (Oper. Res. Lett. 28:63---74, 2001) and discuss the value of the tools we have tested.

Stability preserving transformations of graphs

Graph transformations proved useful for many algorithmic problems. In the present paper, we study this tool with respect to the maximum stable set problem. We first review available results on this topic and then propose an approach to uniformly describe and systematically develop graph transformations that do not change the size of a maximum stable set in the graph. The approach is illustrated by a number of new transformations.

The 0–1 inverse maximum stable set problem

In this paper we study the 0–1 inverse maximum stable set problem, denoted by I S { 0 , 1 } . Given a graph and a fixed stable set, it is to delete the minimum number of vertices to make this stable set maximum in the new graph. We also consider I S { 0 , 1 } against a specific algorithm such as G r e e d y and 2 o p t , aiming to delete the minimum number of vertices so that the algorithm selects the given stable set in the new graph; we denote them by I S { 0 , 1 } , g r e e d y and I S { 0 , 1 } , 2 o p t , respectively. Firstly, we show that they are NP-hard, even if the fixed stable set contains only one vertex. Secondly, we achieve an approximation ratio of 2 − Θ ( 1 l o g Δ ) for I S { 0 , 1 } , 2 o p t . Thirdly, we study the tractability of I S { 0 , 1 } for some classes of perfect graphs such as comparability, co-comparability and chordal graphs. Finally, we compare the hardness of I S { 0 , 1 } and I S { 0 , 1 } , 2 o p t for some other classes of graphs.

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.

The stable set polytope of quasi-line graphs

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today. Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. The Ben Rebea conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that the Ben Rebea conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph. In this paper, we give a proof of the Ben Rebea conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.

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.