Maximum Stable Set Problem Research Articles

The r-coloring and maximum stable set problem in hypergraphs with bounded matching number and edge size

Motivated by the analogous questions in graphs, we study the complexity of coloring and stable set problems in hypergraphs with forbidden substructures and bounded edge size. Letting ν(G) denote the maximum size of a matching in H, we obtain complete dichotomies for the complexity of the following problems parametrized by fixed r,k,s∈N:•r-Coloring in hypergraphs G with edge size at most k and ν(G)≤s;•r-Precoloring Extension in k-uniform hypergraphs G with ν(G)≤s;•r-Precoloring Extension in hypergraphs G with edge size at most k and ν(G)≤s;•Maximum Stable Set in k-uniform hypergraphs G with ν(G)≤s;•Maximum Weight Stable Set in k-uniform hypergraphs with ν(G)≤s; as well as partial results for r-Coloring in k-uniform hypergraphs ν(G)≤s. We then turn our attention to 2-Coloring in 3-uniform hypergraphs with forbidden induced subhypergraphs, and give a polynomial-time algorithm when restricting the input to hypergraphs excluding a fixed one-edge hypergraph. Finally, we consider linear 3-uniform hypergraphs (in which every two edges share at most one vertex), and show that excluding an induced matching in G implies that ν(G) is bounded by a constant; and that 3-coloring linear 3-uniform hypergraphs G with ν(G)≤532 is NP-hard.

Jordan symmetry reduction for conic optimization over the doubly nonnegative cone: theory and software

A common computational approach for polynomial optimization problems (POPs) is to use (hierarchies of) semidefinite programming (SDP) relaxations. When the variables in the POP are required to be nonnegative – as is the case for combinatorial optimization problems, for example – these SDP problems typically involve nonnegative matrices, i.e. they are conic optimization problems over the doubly nonnegative cone. The Jordan reduction, a symmetry reduction method for conic optimization, was recently introduced for symmetric cones by Parrilo and Permenter [Mathematical Programming 181(1), 2020]. We extend this method to the doubly nonnegative cone, and investigate its application to known relaxations of the quadratic assignment and maximum stable set problems. We also introduce new Julia software where the symmetry reduction is implemented.

Optimizing over the Closure of Rank Inequalities with a Small Right-Hand Side for the Maximum Stable Set Problem via Bilevel Programming

In the context of the maximum stable set problem, rank inequalities impose that the cardinality of any set of vertices contained in a stable set be, at most, as large as the stability number of the subgraph induced by such a set. Rank inequalities are very general, as they subsume many classical inequalities such as clique, hole, antihole, web, and antiweb inequalities. In spite of their generality, the exact separation of rank inequalities has never been addressed without the introduction of topological restrictions on the induced subgraph and the tightness of their closure has never been investigated systematically. In this work, we propose a methodology for optimizing over the closure of all rank inequalities with a right-hand side no larger than a small constant without imposing any restrictions on the topology of the induced subgraph. Our method relies on the exact separation of a relaxation of rank inequalities, which we call relaxed k-rank inequalities, whose closure is as tight. We investigate the corresponding separation problem, a bilevel programming problem asking for a subgraph of maximum weight with a bound on its stability number, whose study could be of independent interest. We first prove that the problem is [Formula: see text]-hard and provide some insights on its polyhedral structure. We then propose two exact methods for its solution: a branch-and-cut algorithm (which relies on a family of faced-defining inequalities which we introduce in this paper) and a purely combinatorial branch-and-bound algorithm. Our computational results show that the closure of rank inequalities with a right-hand side no larger than a small constant can yield a bound that is stronger, in some cases, than Lovász’s Theta function, and substantially stronger than bounds obtained with standard inequalities that are valid for the stable set problem, including odd-cycle inequalities and wheel inequalities. Summary of Contribution: This paper proposes two original methods for solving a challenging cut-separation problem (of bilevel type) for a large class of inequalities valid for one of the key operations research problems, namely, the max stable set problem. An extensive set of experimental results validates the proposed methods. All the source code and data sets are available online on GitHub.

QoS routing in cluster OLSR by using the artificial intelligence model MSSP in the big data environnment

<p><span id="docs-internal-guid-75ba661c-7fff-7cfb-9afe-b7c901c3fe82"><span>The most complex problems, in data science and more specifically in artificial intelligence, can be modeled as cases of the maximum stable set problem (MSSP). this article describes a new approach to solve the MSSP problem by proposing the continuous hopfield network (CHN) to build optimized link state protocol routing (OLSR) protocol cluster. our approach consists in proposing in two stages: the first acts at the level of the choice of the OLSR master cluster in order to quickly make a local minimum using the CHN, by modeling the MSSP problem. As for the second step, the objective is the improvement of the precision making a solution of efficient at the first rank of neighborhood as a linear constraint, and at the end, to find the resolution of the model using the CHN. We will show that this model determines a good solution of the MSSP problem. To test the theoretical results, we propose a comparison with a classic OLSR.</span></span></p>

Polyhedral approximations of the semidefinite cone and their application

We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

A New Hybrid Routing Protocol Using a Modified K-Means Clustering Algorithm and Continuous Hopfield Network for VANET

Vehicular Ad-hoc Networks (VANET) offer several user applications for passengers and drivers, as well as security and internet access applications. To ensure efficient data transmission between vehicles, a reliable routing protocol is considered a significant challenge. This paper suggests a new clustering-based routing protocol combining a modified K-Means algorithm with Continuous Hopfield Network and Maximum Stable Set Problem (KMRP) for VANET. In this way, the basic input parameters of the K-Means algorithm, such as the number of clusters and the initial cluster heads, will not be selected randomly, but using Maximum Stable Set Problem and Continuous Hopfield Network. Then the assignment of vehicles to clusters will be carried out according to Link Reliability Model as a metric that replaces the distance parameter in the K-Means algorithm. Finally, the cluster head is selected by weight function according to the amount of free buffer space, the speed, and the node degree. The simulation results have proved that the designed protocol performs better in a highway vehicular environment, compared to the most recent schemes designed for the same objective. In fact, KMRP reduces traffic congestion, and thus provides a significant increase in Throughput. In addition, KMRP decreases the transmission delay and guarantees the stability of the clusters in high density and mobility, which acts better in terms of the Packet Delivery Ratio.

A smaller extended formulation for the odd cycle inequalities of the stable set polytope

For sparse graphs, the odd cycle polytope can be used to compute useful bounds for the maximum stable set problem quickly. Yannakakis (1991) introduced an extended formulation for the odd cycle inequalities of the stable set polytope, which provides a direct way to optimize over the odd cycle polytope in polynomial time, although there can exist exponentially many odd cycles in given graphs in general. We present another extended formulation for the odd cycle polytope that uses less variables and inequalities than Yannakakis’ formulation. Moreover, we compare the running time of both formulations as relaxations of the maximum stable set problem in a computational study.

BDD-based optimization for the quadratic stable set problem

The quadratic stable set problem (QSSP) is a natural extension of the well-known maximum stable set problem. The QSSP is NP-hard and can be formulated as a binary quadratic program, which makes it an interesting case study to be tackled from different optimization paradigms. In this paper, we propose a novel representation for the QSSP through binary decision diagrams (BDDs) and adapt a hybrid optimization approach which integrates BDDs and mixed-integer programming (MIP) for solving the QSSP. The exact framework highlights the modeling flexibility offered through decision diagrams to handle nonlinear problems. In addition, the hybrid approach leverages two different representations by exploring, in a complementary way, the solution space with BDD and MIP technologies. Machine learning then becomes a valuable component within the method to guide the search mechanisms. In the numerical experiments, the hybrid approach shows to be superior, by at least one order of magnitude, than two leading commercial MIP solvers with quadratic programming capabilities and a semidefinite-based branch-and-bound solver.

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.

Equivalences and differences in conic relaxations of combinatorial quadratic optimization problems

Various conic relaxations of quadratic optimization problems in nonnegative variables for combinatorial optimization problems, such as the binary integer quadratic problem, quadratic assignment problem (QAP), and maximum stable set problem have been proposed over the years. The binary and complementarity conditions of the combinatorial optimization problems can be expressed in several ways, each of which results in different conic relaxations. For the completely positive, doubly nonnegative and semidefinite relaxations of the combinatorial optimization problems, we discuss the equivalences and differences among the relaxations by investigating the feasible regions obtained from different representations of the combinatorial condition which we propose as a generalization of the binary and complementarity condition. We also study theoretically the issue of the primal and dual nondegeneracy, the existence of an interior solution and the size of the relaxations, as a result of different representations of the combinatorial condition. These characteristics of the conic relaxations affect the numerical efficiency and stability of the solver used to solve them. We illustrate the theoretical results with numerical experiments on QAP instances solved by SDPT3, SDPNAL+ and the bisection and projection method.

Determining the Number of Clusters using Neural Network and Max Stable Set Problem

One of the most difficult problems, in cluster analysis is the determination of the number of clusters in a data set. Solving this problem consists in detecting and finding the best number of clusters, which is an input parameter for the clustering problems. In this paper, we propose a new approach using the Maximum Stable Set Problem (MSSP) combined by Continuous Hopfield Network (CHN) to determine the number of clusters, which is a basic input parameter for K-Means method. By testing the theoretical results, the proposed approach was validated on a real application for the text mining. Some numerical examples and computational experiments assess the effectiveness of this approach as demonstrated in this paper.

Solving vertex coloring problems as maximum weight stable set problems

In Vertex Coloring Problems, one is required to assign a color to each vertex of an undirected graph in such a way that adjacent vertices receive different colors, and the objective is to minimize the cost of the used colors. In this work we solve four different coloring problems formulated as Maximum Weight Stable Set Problems on an associated graph. We exploit the transformation proposed by Cornaz and Jost (2008), where given a graph G, an auxiliary graph Gˆ is constructed, such that the family of all stable sets of Gˆ is in one-to-one correspondence with the family of all feasible colorings of G. The transformation in Cornaz and Jost (2008) was originally proposed for the classical Vertex Coloring and the Max-Coloring problems; we extend it to the Equitable Coloring Problem and the Bin Packing Problem with Conflicts. We discuss the relation between the Maximum Weight Stable formulation and a polynomial-size formulation for the VCP, proposed by Campêlo et al. (2008) and called the Representative formulation. We report extensive computational experiments on benchmark instances of the four problems, and compare the solution method with the state-of-the-art algorithms. By exploiting the proposed method, we largely outperform the state-of-the-art algorithm for the Max-coloring Problem, and we are able to solve, for the first time to proven optimality, 14 Max-coloring and 2 Equitable Coloring instances.

On the computational complexity of the virtual network embedding problem

Given a graph representing a substrate (or physical) network with node and edge capacities and a set of virtual networks with node capacity demands and node-to-node traffic demands, the Virtual Network Embedding problem (VNE) calls for an embedding of (a subset of) the virtual networks onto the substrate network which maximizes the total profit while respecting the physical node and edge capacities. In this work, we investigate the computational complexity of VNE. In particular, we present a polynomial-time reduction from the maximum stable set problem which implies strong NP-hardness for VNE even for very special subclasses of graphs and yields a strong inapproximability result for general graphs. We also consider the special cases obtained when fixing one of the dimensions of the problem to one. We show that VNE is still strongly NP-hard when a single virtual network request is present or when each virtual network request consists of a single virtual node and that it is weakly NP-hard for the case with a single physical node.

Average case polyhedral complexity of the maximum stable set problem

We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via linear programs (LPs), in two distinct models. In the uniform model, the constraints of the LP are not allowed to depend on the input graph, which should be encoded solely in the objective function. There we prove a $2^{\Omega(n/ \log n)}$ lower bound with probability at least $1 - 2^{-2^n}$ for every LP that is exact for a randomly selected set of instances; each graph on at most n vertices being selected independently with probability $p \geq 2^{-\binom{n/4}{2}+n}$. In the non-uniform model, the constraints of the LP may depend on the input graph, but we allow weights on the vertices. The input graph is sampled according to the G(n, p) model. There we obtain upper and lower bounds holding with high probability for various ranges of p. We obtain a super-polynomial lower bound all the way from $p = \Omega(\log^{6+\varepsilon} / n)$ to $p = o (1 / \log n)$. Our upper bound is close to this as there is only an essentially quadratic gap in the exponent, which currently also exists in the worst-case model. Finally, we state a conjecture that would close this gap, both in the average-case and worst-case models.

Discrete Optimization with Decision Diagrams

We propose a general branch-and-bound algorithm for discrete optimization in which binary decision diagrams (BDDs) play the role of the traditional linear programming relaxation. In particular, relaxed BDD representations of the problem provide bounds and guidance for branching, and restricted BDDs supply a primal heuristic. Each problem is given a dynamic programming model that allows one to exploit recursive structure, even though the problem is not solved by dynamic programming. A novel search scheme branches within relaxed BDDs rather than on values of variables. Preliminary testing shows that a rudimentary BDD-based solver is competitive with or superior to a leading commercial integer programming solver for the maximum stable set problem, the maximum cut problem on a graph, and the maximum 2-satisfiability problem. Specific to the maximum cut problem, we tested the BDD-based solver on a classical benchmark set and identified tighter relaxation bounds than have ever been found by any technique, nearly closing the entire optimality gap on four large-scale instances.

A Strengthened General Cut-Generating Procedure for the Stable Set Polytope

In a previous work we have presented a procedure for generating rank and non-rank valid inequalities for the stable set polytope based on clique projection and lifting operations. In this work we propose to apply another lifting operation and give some sufficient conditions for this new procedure to generate facet defining inequalities. Computational experience shows that the proposed approach allows to obtain tighter upper bounds for the maximum stable set problem.

A Polynomial Turing-Kernel for Weighted Independent Set in Bull-Free Graphs

The maximum stable set problem is NP-hard, even when restricted to triangle-free graphs. In particular, one cannot expect a polynomial time algorithm deciding if a bull-free graph has a stable set of size $k$, when $k$ is part of the instance. Our main result in this paper is to show the existence of an FPT algorithm when we parameterize the problem by the solution size $k$. A polynomial kernel is unlikely to exist for this problem. We show however that our problem has a polynomial size Turing-kernel. More precisely, the hard cases are instances of size $O(k^5)$. As a byproduct, if we forbid odd holes in addition to the bull, we show the existence of a polynomial time algorithm for the stable set problem. We also prove that the chromatic number of a bull-free graph is bounded by a function of its clique number and the maximum chromatic number of its triangle-free induced subgraphs. All our results rely on a decomposition theorem of bull-free graphs due to Chudnovsky which is modified here, allowing us to provide extreme decompositions, adapted to our computational purpose.

A polyhedral study of the maximum stable set problem with weights on vertex-subsets

Given a graph G=(V,E), a family of nonempty vertex-subsets S⊆2V, and a weight w:S→R+, the maximum stable set problem with weights on vertex-subsets consists in finding a stable set I of G maximizing the sum of the weights of the sets in S that intersect I. This problem arises within a natural column generation approach for the vertex coloring problem. In this work we perform an initial polyhedral study of this problem, by introducing a natural integer programming formulation and studying the associated polytope. We address general facts on this polytope including some lifting results, we provide connections with the stable set polytope, and we present three families of facet-inducing inequalities.

A Lagrangian–DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems

We propose an efficient computational method for linearly constrained quadratic optimization problems (QOPs) with complementarity constraints based on their Lagrangian and doubly nonnegative (DNN) relaxation and first-order algorithms. The simplified Lagrangian---completely positive programming (CPP) relaxation of such QOPs proposed by Arima, Kim, and Kojima in 2012 takes one of the simplest forms, an unconstrained conic linear optimization problem with a single Lagrangian parameter in a CPP matrix variable with its upper-left element fixed to 1. Replacing the CPP matrix variable by a DNN matrix variable, we derive the Lagrangian---DNN relaxation, and establish the equivalence between the optimal value of the DNN relaxation of the original QOP and that of the Lagrangian---DNN relaxation. We then propose an efficient numerical method for the Lagrangian---DNN relaxation using a bisection method combined with the proximal alternating direction multiplier and the accelerated proximal gradient methods. Numerical results on binary QOPs, quadratic multiple knapsack problems, maximum stable set problems, and quadratic assignment problems illustrate the superior performance of the proposed method for attaining tight lower bounds in shorter computational time.