Planar Subgraph Isomorphism Revisited

TL;DR

This paper presents a significant improvement in the algorithmic complexity for planar subgraph isomorphism, achieving a single exponential time bound while maintaining linear dependence on the host graph size.

Contribution

It introduces an embedded dynamic programming technique that reduces the pattern dependency to single exponential, solving an open problem and enhancing efficiency.

Findings

Achieves 2^O(k) n running time for fixed pattern size k

Provides enumeration of all subgraph solutions within the same asymptotic time

Establishes an exponential bound on partial solutions based on pattern size

Abstract

The problem of Subgraph Isomorphism is defined as follows: Given a pattern H and a host graph G on n vertices, does G contain a subgraph that is isomorphic to H? Eppstein [SODA 95, J'GAA 99] gives the first linear time algorithm for subgraph isomorphism for a fixed-size pattern, say of order k, and arbitrary planar host graph, improving upon the O(n^\sqrt{k})-time algorithm when using the ``Color-coding'' technique of Alon et al [J'ACM 95]. Eppstein's algorithm runs in time k^O(k) n, that is, the dependency on k is superexponential. We solve an open problem posed in Eppstein's paper and improve the running time to 2^O(k) n, that is, single exponential in k while keeping the term in n linear. Next to deciding subgraph isomorphism, we can construct a solution and enumerate all solutions in the same asymptotic running time. We may list w subgraphs with an additive term O(w k) in the running time of our algorithm. We introduce the technique of "embedded dynamic programming" on a suitably structured graph decomposition, which exploits the topology of the underlying embeddings of the subgraph pattern (rather than of the host graph). To achieve our results, we give an upper bound on the number of partial solutions in each dynamic programming step as a function of pattern size--as it turns out, for the planar subgraph isomorphism problem, that function is single exponential in the number of vertices in the pattern.