Finding and counting vertex-colored subtrees

TL;DR

This paper introduces new fixed-parameter tractable algorithms for finding and counting vertex-colored subtrees in graphs, improving efficiency and providing experimental validation on real datasets.

Contribution

It develops algebraic FPT algorithms for the Graph Motif problem and its variants, with new results on counting complexity and practical performance evaluation.

Findings

FPT algorithms with improved running times for Graph Motif.

Counting problem is FPT when M is a set, W[1]-hard when M has two colors.

Experimental results show competitive performance on real datasets.

Abstract

The problems studied in this article originate from the Graph Motif problem introduced by Lacroix et al. in the context of biological networks. The problem is to decide if a vertex-colored graph has a connected subgraph whose colors equal a given multiset of colors $M$. It is a graph pattern-matching problem variant, where the structure of the occurrence of the pattern is not of interest but the only requirement is the connectedness. Using an algebraic framework recently introduced by Koutis et al., we obtain new FPT algorithms for Graph Motif and variants, with improved running times. We also obtain results on the counting versions of this problem, proving that the counting problem is FPT if M is a set, but becomes W[1]-hard if M is a multiset with two colors. Finally, we present an experimental evaluation of this approach on real datasets, showing that its performance compares favorably with existing software.