The maximum common subgraph problem is an N P-hard problem which is very difficult to solve with exact approaches. To speed up the solution process, we may decompose it into independent subproblems which are solved in parallel. We describe a new decomposition method which exploits the structure of the problem to decompose it. We compare this structural decomposition with domain-based decompositions, which basically split variable domains. Experimental results show us that the structural decomposition leads to better speedups on two classes of instances, and to worse speedups on one class of instances.

The Sum Colouring Problem is an N P-hard problem derived from the well-known graph colouring problem. It consists in finding a proper colouring which minimizes the sum of the assigned colours rather than the number of those colours. This problem often arises in scheduling and resource allocation. In this paper, we conduct an in-depth evaluation of ILP and CP's capabilities to solve this problem, with several improvements. Moreover, we propose to combine ILP and CP in a tree decomposition with a bounded height. Finally, those methods are combined in a portfolio approach to take advantage from their complementarity. 2 The sum colouring problem An undirected graph G = (V, E) is defined by a set V of nodes and a set E ⊆ V × V of edges. Each edge of G is an undirected pair of nodes. We note