Speeding Up Graph Algorithms via Switching Classes

TL;DR

This paper introduces a novel approach using switching classes to significantly accelerate various fundamental graph algorithms, surpassing traditional methods in simplicity and efficiency.

Contribution

The work demonstrates that leveraging switching classes can lead to super-polylogarithmic speedups in key graph algorithms, offering a simpler alternative to existing complex techniques.

Findings

Achieved super-polylogarithmic speedups for diameter, transitive closure, and maximum matching algorithms.

Provided a simpler method compared to existing sophisticated pre-processing and data-structure techniques.

Enhanced bounds for several fundamental graph problems using switching class properties.

Abstract

Given a graph $G$, a vertex switch of $v \in V(G)$ results in a new graph where neighbors of $v$ become nonneighbors and vice versa. This operation gives rise to an equivalence relation over the set of labeled digraphs on $n$ vertices. The equivalence class of $G$ with respect to the switching operation is commonly referred to as $G$'s switching class. The algebraic and combinatorial properties of switching classes have been studied in depth; however, they have not been studied as thoroughly from an algorithmic point of view. The intent of this work is to further investigate the algorithmic properties of switching classes. In particular, we show that switching classes can be used to asymptotically speed up several super-linear unweighted graph algorithms. The current techniques for speeding up graph algorithms are all somewhat involved insofar that they employ sophisticated pre-processing, data-structures, or use "word tricks" on the RAM model to achieve at most a $O(\log(n))$ speed up for sufficiently dense graphs. Our methods are much simpler and can result in super-polylogarithmic speedups. In particular, we achieve better bounds for diameter, transitive closure, bipartite maximum matching, and general maximum matching.