New polynomial delay bounds for maximal subgraph enumeration by proximity search

Abstract

In this paper we propose polynomial delay algorithms for several maximal subgraph listing problems, by means of a seemingly novel technique which we call proximity search. Our result involves modeling the space of solutions as an implicit directed graph called “solution graph”, a method common to other enumeration paradigms such as reverse search. Such methods, however, can become inefficient due to this graph having vertices with high (potentially exponential) degree. The novelty of our algorithm consists in providing a technique for generating better solution graphs, reducing the out-degree of its vertices with respect to existing approaches, and proving that it remains strongly connected. Applying this technique, we obtain polynomial delay listing algorithms for several problems for which output-sensitive results were, to the best of our knowledge, not known. These include Maximal Bipartite Subgraphs, Maximal k-Degenerate Subgraphs (for bounded k), Maximal Induced Chordal Subgraphs, and Maximal Induced Trees. We present these algorithms, and give insight on how this general technique can be applied to other problems.