An Incremental Search Heuristic for Coloring Vertices of a Graph

Abstract

Graph coloring is one of the fundamentally known NP-complete problem. Several heuristics have been developed to solve this problem, among which greedy coloring is the most naturally used one. In greedy coloring, vertices are traversed following an order and hence performance of it highly depends on finding a good order. In this paper, we propose an incremental search heuristic (ISH) which considers some ρ1 random orders and for each of them it calls a selective search (SS) procedure with parameter ρ2. Given an order, SS considers only those orders which produces equal or less number of distinct colors than a given order. We showed that those orders can be partitioned into disjoint subsets of equivalent orders. To make effective search, SS selects and evaluates only one order from such a subset. Analytically we have shown that ISH can solve the graph coloring instances on sparse graph in expected polynomial time. Through simulations, we have evaluated ISH on 86 challenging benchmarks and compare results with state of the art existing algorithms. We observed that ISH significantly outperforms existing algorithms specially for sparse graphs and also produces reasonably good results for others.