Efficient Implementation of Color Coding Algorithm for Subgraph Isomorphism Problem

Abstract

We consider the subgraph isomorphism problem where, given two graphs G (source graph) and F (pattern graph), one is to decide whether there is a (not necessarily induced) subgraph of G isomorphic to F . While many practical heuristic algorithms have been developed for the problem, as pointed out by McCreesh et al. [JAIR 2018], for each of them there are rather small instances which they cannot cope. Therefore, developing an alternative approach that could possibly cope with these hard instances would be of interest.A seminal paper by Alon, Yuster and Zwick [J. ACM 1995] introduced the color coding approach to solve the problem, where the main part is a dynamic programming over color subsets and partial mappings. As with many exponential-time dynamic programming algorithms, the memory requirements constitute the main limiting factor for its usage. Because these requirements grow exponentially with the treewidth of the pattern graph, all existing implementations based on the color coding principle restrict themselves to specific pattern graphs, e.g., paths or trees. In contrast, we provide an efficient implementation of the algorithm significantly reducing its memory requirements so that it can be used for pattern graphs of larger treewidth. Moreover, our implementation not only decides the existence of an isomorphic subgraph, but it also enumerates all such subgraphs (or given number of them).We provide an extensive experimental comparison of our implementation to other available solvers for the problem.

Current Algorithms for Detecting Subgraphs of Bounded Treewidth are Probably Optimal

“…Most notably, the classic Color-Coding algorithm by Alon, Yuster and Zwick [12] solves the problem by a Las Vegas algorithm in expected time O(n tw(H)+1 g(k)), or by a deterministic algorithm in time O(n tw(H)+1 g(k)), where g is a computable function (and O(•) is used to suppress factors that are polylogarithmic in the input size). In other words, if the pattern graph H has treewidth bounded by some constant, the problem is fixed-parameter tractable when parameterized by k. The Color-Coding algorithm is also relevant for practical purposes: Recently, it has received an efficient implementation, which tested well against state-of-the-art programs for Subgraph Isomorphism [53].…”