Finding and Counting Patterns in Sparse Graphs

TL;DR

This paper introduces new parameters, matched treedepth and matched treewidth, enabling faster algorithms for finding and counting small patterns in sparse graphs, improving over previous methods especially for cycles and paths.

Contribution

The paper defines matched treedepth and matched treewidth and demonstrates their effectiveness in designing faster algorithms for pattern detection in sparse graphs.

Findings

Faster algorithms for cycles of length up to 11.

Efficient algorithms for paths of length up to 10.

Introduction of matched treedepth and matched treewidth parameters.

Abstract

We consider algorithms for finding and counting small, fixed graphs in sparse host graphs. In the non-sparse setting, the parameters treedepth and treewidth play a crucial role in fast, constant-space and polynomial-space algorithms respectively. We discover two new parameters that we call matched treedepth and matched treewidth. We show that finding and counting patterns with low matched treedepth and low matched treewidth can be done asymptotically faster than the existing algorithms when the host graphs are sparse for many patterns. As an application to finding and counting fixed-size patterns, we discover $\otilde(m^3)$-time \footnote{$\otilde$ hides factors that are logarithmic in the input size.}, constant-space algorithms for cycles of length at most $11$ and $\otilde(m^2)$-time, polynomial-space algorithms for paths of length at most $10$.