2026-07-22T12:53:30Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:14109 2024-03-06T10:53:29Z ddc:004 open_access

Current Algorithms for Detecting Subgraphs of Bounded Treewidth Are Probably Optimal Bringmann, Karl Slusallek, Jasper subgraph isomorphism treewidth fine-grained complexity hyperclique 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Karl Bringmann and Jasper Slusallek 2021 Is Part Of LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2021.40 urn:nbn:de:0030-drops-141095 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.40 eng https://creativecommons.org/licenses/by/4.0/legalcode