A safeness condition for minimal separators based on vertex connectivity

Abstract

The treewidth is a measure that quantifies how tree-like a graph is. Its interest stems from the fact that many problems on graphs that are NP-complete for arbitrary graphs become solvable in polynomial time when restricted to graphs with bounded treewidth. Unfortunately, computing the treewidth of a graph is itself an NP-complete problem. A preprocessing technique that has proven to be highly effective at reducing the size of a graph for which the treewidth is to be computed is splitting the graph into multiple smaller subgraphs using separators that are safe for treewidth, allowing the treewidth of the complete graph to be computed as the maximum treewidth over these subgraphs. This paper introduces a new class of safe separators that largely generalises the existing classes of safe separators and opens up the possibility of designing even more powerful preprocessing techniques. An experimental evaluation on a set of synthetic graphs with known safe separators demonstrates that decomposing the graphs using their safe separators has the potential to reduce the execution time of algorithms by up to 10 times in many cases, with a maximum of more than 100 times in the most favourable cases. A heuristic decomposition algorithm is also presented that uses the community structure of a graph to decompose it into subgraphs. The evaluation of this heuristic algorithm on both the synthetic graphs and the graphs from the PACE challenge of 2017 using the same algorithms shows that the potential reduction in execution time is largely maintained while having only a slight impact on the quality as a result of the heuristic-based decomposition.