A Composition Algorithm for Very Hard Graph 3-Colorability Instances

Graph colorability (COL) is a constraint satisfaction problem, which has been studied in the context of computational complexity and combinatorial search algorithms. It is also interesting as subjects of heuristics[2]. Many research reports discuss the complexity of COL [2,3,4,8,9,10]. Examples of possible candidates of order parameters that explain the mechanism making COLs very hard include the 3-paths[10], the minimal unsolvable subproblems[8], and the frozen developments[4]. Instead of generate-and-test approaches, we propose a constructive approach producing 3-colorablity problems (3COLs) that are exceptionally hard for usual backtracking algorithms adopting Brelaz heuristics and for Smallk coloring program[1]. Instances generated by our procedure (1) are 4-critical, (2) include no near-4-cliques(n4c's; 4-cliques with 1 edge removed) as subgraphs, and (3) have the degree of every node limited to 3 or 4: quasi-regular. This research was supported in part by the Ministry of Education, Culture, Sports, Science and Technology of Japan, Grant-in-Aid for Scientific Research (B)(2), No. 14380134, 2002-2005.