A novel evolutionary formulation of the maximum independent set problem

We introduce a novel evolutionary formulation of the problem of finding a maximum independent set of a graph. The new formulation is based on the relationship that exists between a graph's independence number and its acyclic orientations. It views such orientations as individuals and evolves them with the aid of evolutionary operators that are very heavily based on the structure of the graph and its acyclic orientations. The resulting heuristic has been tested on some of the Second DIMACS Implementation Challenge benchmark graphs, and has been found to be competitive when compared to several of the other heuristics that have also been tested on those graphs

A Heuristic for the Maximum Independent Set Problem Based on Optimization of a Quadratic over a Sphere

For a given graph the maximum independent set problem is to find a maximum subset of vertices no two of which are adjacent. We propose a heuristic for the maximum independent set problem which utilizes classical results for the problem of optimization of a quadratic function over a sphere. The e#ciency of the approach is confirmed by results of numerical experiments on DIMACS benchmarks

Algorithms for the Maximum Independent Set Problem

This thesis focuses mainly on the Maximum Independent Set (MIS) problem. Some related graph theoretical combinatorial problems are also considered. As these problems are generally NP-hard, we study their complexity in hereditary graph classes, i.e. graph classes defined by a set F of forbidden induced subgraphs. We revise the literature about the issue, for example complexity results, applications, and techniques tackling the problem. Through considering some general approach, we exhibit several cases where the problem admits a polynomial-time solution. More specifically, we present polynomial-time algorithms for the MIS problem in: + some subclasses of $S_{2;j;k}$-free graphs (thus generalizing the classical result for $S_{1;2;k}$-free graphs); + some subclasses of $tree_{k}$-free graphs (thus generalizing the classical results for subclasses of P5-free graphs); + some subclasses of $P_{7}$-free graphs and $S_{2;2;2}$-free graphs; and various subclasses of graphs of bounded maximum degree, for example subcubic graphs. Our algorithms are based on various approaches. In particular, we characterize augmenting graphs in a subclass of $S_{2;k;k}$-free graphs and a subclass of $S_{2;2;5}$-free graphs. These characterizations are partly based on extensions of the concept of redundant set [125]. We also propose methods finding augmenting chains, an extension of the method in [99], and finding augmenting trees, an extension of the methods in [125]. We apply the augmenting vertex technique, originally used for $P_{5}$-free graphs or banner-free graphs, for some more general graph classes. We consider a general graph theoretical combinatorial problem, the so-called Maximum -Set problem. Two special cases of this problem, the so-called Maximum F-(Strongly) Independent Subgraph and Maximum F-Induced Subgraph, where F is a connected graph set, are considered. The complexity of the Maximum F-(Strongly) Independent Subgraph problem is revised and the NP-hardness of the Maximum F-Induced Subgraph problem is proved. We also extend the augmenting approach to apply it for the general Maximum Π -Set problem. We revise on classical graph transformations and give two unified views based on pseudo-boolean functions and αff-redundant vertex. We also make extensive uses of α-redundant vertices, originally mainly used for $P_{5}$-free graphs, to give polynomial solutions for some subclasses of $S_{2;2;2}$-free graphs and $tree_{k}$-free graphs. We consider some classical sequential greedy heuristic methods. We also combine classical algorithms with αff-redundant vertices to have new strategies of choosing the next vertex in greedy methods. Some aspects of the algorithms, for example forbidden induced subgraph sets and worst case results, are also considered. Finally, we restrict our attention on graphs of bounded maximum degree and subcubic graphs. Then by using some techniques, for example ff-redundant vertex, clique separator, and arguments based on distance, we general these results for some subclasses of $S_{i;j;k}$-free subcubic graphs

Optimization Methods for Cluster Analysis in Network-based Data Mining

This dissertation focuses on two optimization problems that arise in network-based data mining, concerning identification of basic community structures (clusters) in graphs: the maximum edge weight clique and maximum induced cluster subgraph problems. We propose a continuous quadratic formulation for the maximum edge weight clique problem, and establish the correspondence between its local optima and maximal cliques in the graph. Subsequently, we present a combinatorial branch-and-bound algorithm for this problem that takes advantage of a polynomial-time solvable nonconvex relaxation of the proposed formulation. We also introduce a linear-time-computable analytic upper bound on the clique number of a graph, as well as a new method of upper-bounding the maximum edge weight clique problem, which leads to another exact algorithm for this problem. For the maximum induced cluster subgraph problem, we present the results of a comprehensive polyhedral analysis. We derive several families of facet-defining valid inequalities for the IUC polytope associated with a graph. We also provide a complete description of this polytope for some special classes of graphs. We establish computational complexity of the separation problems for most of the considered families of valid inequalities, and explore the effectiveness of employing the corresponding cutting planes in an integer (linear) programming framework for the maximum induced cluster subgraph problem