Proximity Search For Maximal Subgraph Enumeration

TL;DR

This paper introduces proximity search, a novel technique for maximal subgraph enumeration that creates more efficient solution graphs with polynomial out-degree, enabling polynomial-delay enumeration of various complex subgraph problems.

Contribution

It formalizes proximity search and canonical reconstruction to build solution graphs with smaller out-degree, enabling output-polynomial algorithms for previously unsolved enumeration problems.

Findings

Output-polynomial algorithms for Maximal Bipartite Subgraphs

Output-polynomial algorithms for Maximal k-Degenerate Subgraphs

Output-polynomial algorithms for Maximal Induced Chordal Subgraphs

Abstract

This paper proposes a new general technique for maximal subgraph enumeration which we call proximity search, whose aim is to design efficient enumeration algorithms for problems that could not be solved by existing frameworks. To support this claim and illustrate the technique we include output-polynomial algorithms for several problems for which output-polynomial algorithms were not known, including the enumeration of Maximal Bipartite Subgraphs, Maximal k-Degenerate Subgraphs (for bounded k), Maximal Induced Chordal Subgraphs, and Maximal Induced Trees. Using known techniques, such as reverse search, the space of all maximal solutions induces an implicit directed graph called "solution graph" or "supergraph", and solutions are enumerated by traversing it; however, nodes in this graph can have exponential out-degree, thus requiring exponential time to be spent on each solution. The novelty of proximity search is a formalization that allows us to define a better solution graph, and a technique, which we call canonical reconstruction, by which we can exploit the properties of given problems to build such graphs. This results in solution graphs whose nodes have significantly smaller (i.e., polynomial) out-degree with respect to existing approaches, but that remain strongly connected, so that all solutions can be enumerated in polynomial delay by a traversal. A drawback of this approach is the space required to keep track of visited solutions, which can be exponential: we further propose a technique to induce a parent-child relationship among solutions and achieve polynomial space when suitable conditions are met.