On the phase transitions of graph coloring and independent sets

We study combinatorial indicators related to the characteristic phase transitions associated with coloring a graph optimally and finding a maximum independent set. In particular, we investigate the role of the acyclic orientations of the graph in the hardness of finding the graph's chromatic number and independence number. We provide empirical evidence that, along a sequence of increasingly denser random graphs, the fraction of acyclic orientations that are `shortest' peaks when the chromatic number increases, and that such maxima tend to coincide with locally easiest instances of the problem. Similar evidence is provided concerning the `widest' acyclic orientations and the independence number

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

Приближенный алгоритм решения задачи нахождения максимального независимого множества вершин графа

Предложен приближенный алгоритм решения задачи нахождения максимального независимого множества вершин графа. С помощью этого алгоритма улучшено известное рекордное значение мощности максимального независимого множества для одного из графов.Пропонується наближений алгоритм розв'язання задачі знаходження максимальної незалежної множини вершин графа. За допомогою цього алгоритму покращено відоме рекордне значення потужності максимальної незалежної множини для одного з графів.In the paper, an approximate algorithm for solving the problem of finding a maximum independent set in a graph is proposed. With the help of this algorithm the known record value of cardinality of the maximum independent set is improved for one of the graphs

Analysis of an exhaustive search algorithm in random graphs and the n^{c\log n} -asymptotics

We analyze the cost used by a naive exhaustive search algorithm for finding a maximum independent set in random graphs under the usual G_{n,p} -model where each possible edge appears independently with the same probability p. The expected cost turns out to be of the less common asymptotic order n^{c\log n}, which we explore from several different perspectives. Also we collect many instances where such an order appears, from algorithmics to analysis, from probability to algebra. The limiting distribution of the cost required by the algorithm under a purely idealized random model is proved to be normal. The approach we develop is of some generality and is amenable for other graph algorithms.Comment: 35 page

Finding a maximum set of independent chords in a circle

Abstract Chang, R.C. and H.S. Lee, Finding a maximum set of independent chords in a circle, Information Processing Letters 41 (1992) 99-102. In this note we propose an O(nmI algorithm for finding a maximum independent set of m chords which are incident to n vertices on a circle. This result can be applied to improving the time complexity of the algorithm for partitioning simple polygons into a minimum number of uniformly monotone polygons

From matchings to independent sets

In 1965, Jack Edmonds proposed his celebrated polynomial-time algorithm to find a maximum matching in a graph. It is well-known that finding a maximum matching in G is equivalent to finding a maximum independent set in the line graph of G. For general graphs, the maximum independent set problem is NP-hard. What makes this problem easy in the class of line graphs and what other restrictions can lead to an efficient solution of the problem? In the present paper, we analyze these and related questions. We also review various techniques that may lead to efficient algorithms for the maximum independent set problem in restricted graph families, with a focus given to augmenting graphs and graph transformations. Both techniques have been used in the solution of Edmonds to the maximum matching problem, i.e. in the solution to the maximum independent set problem in the class of line graphs. We survey various results that exploit these techniques beyond the line graphs