The parameterised complexity of counting even and odd induced subgraphs

TL;DR

This paper investigates the computational complexity of counting even and odd induced subgraphs in graphs, proving both problems are #W[1]-hard but also admitting efficient approximation schemes based on Ramsey theory.

Contribution

It establishes the #W[1]-hardness of counting even/odd induced subgraphs and introduces FPTRAS approximation schemes leveraging structural Ramsey theory insights.

Findings

Both counting problems are #W[1]-hard when parameterized by subgraph size k.

Despite hardness, efficient approximation schemes (FPTRAS) exist for both problems.

Structural results from Ramsey theory underpin the approximation methods.

Abstract

We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that the problems of counting even and odd k-vertex induced subgraphs both admit an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory.