Counting Small Induced Subgraphs with Hereditary Properties

Focke, Jacob and Roth, Marc
(2022) Counting Small Induced Subgraphs with Hereditary Properties.
In: STOC 2022.
Conference: STOC ACM Symposium on Theory of Computing

This is the latest version of this item.

[img]
Preview
Text
2111.02277.pdf

Download (487kB) | Preview
Official URL: https://doi.org/10.1145/3519935.3520008

Abstract

We study the computational complexity of the problem #IndSub(\Phi) of counting k-vertex induced subgraphs of a graph G that satisfy a graph property \Phi. Our main result establishes an exhaustive and explicit classification for all hereditary properties, including tight conditional lower bounds under the Exponential Time Hypothesis (ETH): - If a hereditary property \Phi is true for all graphs, or if it is true only for finitely many graphs, then #IndSub(\Phi) is solvable in polynomial time. - Otherwise, #IndSub(\Phi) is #W[1]-complete when parameterised by k, and, assuming ETH, it cannot be solved in time f(k)*|G|^{o(k)} for any function f. This classification features a wide range of properties for which the corresponding detection problem (as classified by Khot and Raman [TCS 02]) is tractable but counting is hard. Moreover, even for properties which are already intractable in their decision version, our results yield significantly stronger lower bounds for the counting problem. As additional result, we also present an exhaustive and explicit parameterised complexity classification for all properties that are invariant under homomorphic equivalence. By covering one of the most natural and general notions of closure, namely, closure under vertex-deletion (hereditary), we generalise some of the earlier results on this problem. For instance, our results fully subsume and strengthen the existing classification of #IndSub(\Phi) for monotone (subgraph-closed) properties due to Roth, Schmitt, and Wellnitz [FOCS 20].

Available Versions of this Item

  • Counting Small Induced Subgraphs with Hereditary Properties. (deposited 26 Jul 2022 12:13) [Currently Displayed]

Actions

Actions (login required)

View Item View Item