Current Algorithms for Detecting Subgraphs of Bounded Treewidth Are Probably Optimal
The Subgraph Isomorphism problem is of considerable importance in computer science. We examine the problem when the pattern graph H is of bounded treewidth, as occurs in a variety of applications. This problem has a well-known algorithm via color-coding that runs in time O(n^{tw(H)+1}) [Alon, Yuster, Zwick'95], where n is the number of vertices of the host graph G. While there are pattern graphs known for which Subgraph Isomorphism can be solved in an improved running time of O(n^{tw(H)+1-ε}) or even faster (e.g. for k-cliques), it is not known whether such improvements are possible for all patterns. The only known lower bound rules out time n^{o(tw(H) / log(tw(H)))} for any class of patterns of unbounded treewidth assuming the Exponential Time Hypothesis [Marx'07].
In this paper, we demonstrate the existence of maximally hard pattern graphs H that require time n^{tw(H)+1-o(1)}. Specifically, under the Strong Exponential Time Hypothesis (SETH), a standard assumption from fine-grained complexity theory, we prove the following asymptotic statement for large treewidth t:
For any ε > 0 there exists t ≥ 3 and a pattern graph H of treewidth t such that Subgraph Isomorphism on pattern H has no algorithm running in time O(n^{t+1-ε}).
Under the more recent 3-uniform Hyperclique hypothesis, we even obtain tight lower bounds for each specific treewidth t ≥ 3:
For any t ≥ 3 there exists a pattern graph H of treewidth t such that for any ε > 0 Subgraph Isomorphism on pattern H has no algorithm running in time O(n^{t+1-ε}).
In addition to these main results, we explore (1) colored and uncolored problem variants (and why they are equivalent for most cases), (2) Subgraph Isomorphism for tw < 3, (3) Subgraph Isomorphism parameterized by pathwidth instead of treewidth, and (4) a weighted variant that we call Exact Weight Subgraph Isomorphism, for which we examine pseudo-polynomial time algorithms. For many of these settings we obtain similarly tight upper and lower bounds.
subgraph isomorphism
treewidth
fine-grained complexity
hyperclique
Theory of computation~Design and analysis of algorithms
Theory of computation~Computational complexity and cryptography
40:1-40:16
Track A: Algorithms, Complexity and Games
https://arxiv.org/abs/2105.05062
Karl
Bringmann
Karl Bringmann
Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979).
Jasper
Slusallek
Jasper Slusallek
Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
10.4230/LIPIcs.ICALP.2021.40
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Karl Bringmann and Jasper Slusallek
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