Counting Subgraphs in Somewhere Dense Graphs

Bressan, M., Goldberg, L. A., Meeks, K. and Roth, M. (2023) Counting Subgraphs in Somewhere Dense Graphs. In: 14th Innovations in Theoretical Computer Science Conference (ITCS 2023), Cambridge, Massachusetts, USA, 10-13 January 2023, ISBN 9783959772631

Abstract

We study the problems of counting copies and induced copies of a small pattern graph H in a large host graph G. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns H. In this work, we address the more challenging task of analysing 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 f(H)·|G|O(1) for some computable function f. 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 G as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis: Counting k-matchings in a graph G ∈ G is fixed-parameter tractable if and only if G is nowhere dense. Counting k-independent sets in a graph G ∈ G is fixed-parameter tractable if and only if G is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if G 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 k-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in F-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting k-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19). At the same time our proofs are much simpler: using structural characterisations of somewhere dense graphs, we show that a colourful version of a recent breakthrough technique for analysing 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.

• Item Type: Conference Proceedings
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Meeks, Dr Kitty
• Authors: Bressan, M., Goldberg, L. A., Meeks, K., and Roth, M.
• College/School: College of Science and Engineering > School of Computing Science
• ISSN: 1868-8969
• ISBN: 9783959772631
• Copyright Holders: Copyright © Marco Bressan, Leslie Ann Goldberg, Kitty Meeks, and Marc Roth
• First Published: First published in LIPIcs, Volume 251
• Publisher Policy: Reproduced under a Creative Commons license

Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
312197		Beyond One Solution in Combinatorial Optimisation	Kitty Meeks	Engineering and Physical Sciences Research Council (EPSRC)	EP/V032305/1	Computing Science