Finding minimal unsolvable structures for constructive generation of graph 3-colorability instances

Abstract

We have developed the method for systematically generating very hard graph 3-colorability (3COL) instances to clarify phase transition mechanism of combinatorial search problems. In our method, 3COL instances can be generated by repeatedly embedding minimal unsolvable graphs (MUGs). In this paper, we find larger-scale MUGs, which are necessary for our generation method, using genetic algorithms (GAs). We also experimentally demonstrate that the computational cost to solve our 3COL instances generated by using MUGs found by GAs is of an exponential order of the number of vertices and our instances are extraordinarily harder than randomly generated instances.