Faster algorithms for counting subgraphs in sparse graphs

Marco Bressan

doi:10.1007/s00453-021-00811-0

Abstract

Given a k-node pattern graph H and an n-node host graph G, the subgraph counting problem asks to compute the number of copies of H in G. In this work we address the following question: can we count the copies of H faster if G is sparse? We answer in the affirmative by introducing a novel tree-like decomposition for directed acyclic graphs, inspired by the classic tree decomposition for undirected graphs. This decomposition gives a dynamic program for counting the homomorphisms of H in G by exploiting the degeneracy of G, which allows us to beat the state-of-the-art subgraph counting algorithms when G is sparse enough. For example, we can count the induced copies of any k-node pattern H in time 2^{O(k^2)} O(n^{0.25k + 2} log n) if G has bounded degeneracy, and in time 2^{O(k^2)} O(n^{0.625k + 2} log n) if G has bounded average degree. These bounds are instantiations of a more general result, parameterized by the degeneracy of G and the structure of H, which generalizes classic bounds on counting cliques and complete bipartite graphs. We also give lower bounds based on the Exponential Time Hypothesis, showing that our results are actually a characterization of the complexity of subgraph counting in bounded-degeneracy graphs.

Highlights

We address the following fundamental subgraph counting problem: Input: an n-node graph G and a k-node graph H Output: the number of induced copies of H in GIf no further assumptions are made, the best possible algorithm for this problem is likely to have running time f (k) ⋅ nΘ(k)
In this work we address the following question: can we count the copies of H faster if G is sparse? We answer in the affirmative by introducing a novel tree-like decomposition for directed acyclic graphs, inspired by the classic tree decomposition for undirected graphs
In this work we introduce a novel tree-like graph decomposition, to be applied to the pattern graph H, designed to exploit the degeneracy of G when counting the homomorphisms from H to G

Summary

Introduction

We address the following fundamental subgraph counting problem: Input: an n-node graph G (the host graph) and a k-node graph H (the pattern) Output: the number of induced copies of H in G. If no further assumptions are made, the best possible algorithm for this problem is likely to have running time f (k) ⋅ nΘ(k). The naive brute-force algorithm has running time O(k2nk) , and under the Exponential Time Hypothesis [23] any algorithm for counting k-cliques has running time nΩ(k) [8, 9]. The best algorithm known, which was given over 30 years ago by Nešetřil and Poljak [29] and is based on fast mthaetarilxgomriuthltmiplriucnatsioinn,tiims eonOly(nsl⌊ig3k ⌋h+t2ly) faster where than is. Since ≤ 2.373 [25], this gives a state-of-the-art running time of O(n0.791k+2).

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Conclusion