Counting Subgraphs in Somewhere Dense Graphs
Year: 2024
Type: article
Abstract: . We study the problems of counting copies and induced copies of a small pattern graph [Formula: see text] in a large host graph [Formula: see text]. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns [Formula: see text]. In this work, we address the more challenging task of analyzing the complexity for restricted patterns and restricted hosts. Specifically, we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time [Formula: see text] for some computable function [Formula: see text]. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes [Formula: see text] as our central objects of study and establish the following crisp dichotomies as consequences of the exponential time hypothesis: (1) Counting [Formula: see text]-matchings in a graph [Formula: see text] is fixed-parameter tractable if and only if [Formula: see text] is nowhere dense. (2) Counting [Formula: see text]-independent sets in a graph [Formula: see text] is fixed-parameter tractable if and only if [Formula: see text] is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if [Formula: see text] is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting [Formula: see text]-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in [Formula: see text]-colorable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting [Formula: see text]-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19). At the same time, our proofs are much simpler: using structural characterizations of somewhere dense graphs, we show that a colorful version of a recent breakthrough technique for analyzing pattern counting problems (Curticapean, Dell, Marx; STOC 17) applies to any subgraph-closed somewhere dense class of graphs, yielding a unified view of our current understanding of the complexity of subgraph counting.
Source: SIAM Journal on Computing
DOI: 10.1137/22m1535668
Authors: Marco Bressan, Leslie Ann Goldberg, Kitty Meeks, Marc Roth
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4402930157