License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2019.45
URN: urn:nbn:de:0030-drops-102840
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10284/
Krauthgamer, Robert ;
Trabelsi, Ohad
The Set Cover Conjecture and Subgraph Isomorphism with a Tree Pattern
Abstract
In the Set Cover problem, the input is a ground set of n elements and a collection of m sets, and the goal is to find the smallest sub-collection of sets whose union is the entire ground set. The fastest algorithm known runs in time O(mn2^n) [Fomin et al., WG 2004], and the Set Cover Conjecture (SeCoCo) [Cygan et al., TALG 2016] asserts that for every fixed epsilon>0, no algorithm can solve Set Cover in time 2^{(1-epsilon)n} poly(m), even if set sizes are bounded by Delta=Delta(epsilon). We show strong connections between this problem and kTree, a special case of Subgraph Isomorphism where the input is an n-node graph G and a k-node tree T, and the goal is to determine whether G has a subgraph isomorphic to T.
First, we propose a weaker conjecture Log-SeCoCo, that allows input sets of size Delta=O(1/epsilon * log n), and show that an algorithm breaking Log-SeCoCo would imply a faster algorithm than the currently known 2^n poly(n)-time algorithm [Koutis and Williams, TALG 2016] for Directed nTree, which is kTree with k=n and arbitrary directions to the edges of G and T. This would also improve the running time for Directed Hamiltonicity, for which no algorithm significantly faster than 2^n poly(n) is known despite extensive research.
Second, we prove that if p-Partial Cover, a parameterized version of Set Cover that requires covering at least p elements, cannot be solved significantly faster than 2^n poly(m) (an assumption even weaker than Log-SeCoCo) then kTree cannot be computed significantly faster than 2^k poly(n), the running time of the Koutis and Williams' algorithm.
BibTeX - Entry
@InProceedings{krauthgamer_et_al:LIPIcs:2019:10284,
author = {Robert Krauthgamer and Ohad Trabelsi},
title = {{The Set Cover Conjecture and Subgraph Isomorphism with a Tree Pattern}},
booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
pages = {45:1--45:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-100-9},
ISSN = {1868-8969},
year = {2019},
volume = {126},
editor = {Rolf Niedermeier and Christophe Paul},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10284},
doi = {10.4230/LIPIcs.STACS.2019.45},
annote = {Keywords: Conditional lower bounds, Hardness in P, Set Cover Conjecture, Subgraph Isomorphism}
}
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Keywords: |
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Conditional lower bounds, Hardness in P, Set Cover Conjecture, Subgraph Isomorphism |
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Collection: |
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36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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12.03.2019 |
