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DOI: 10.4230/LIPIcs.STACS.2019.45
URL: http://drops.dagstuhl.de/opus/volltexte/2019/10284/
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Krauthgamer, Robert ; Trabelsi, Ohad

The Set Cover Conjecture and Subgraph Isomorphism with a Tree Pattern

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LIPIcs-STACS-2019-45.pdf (0.8 MB)


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}
}

Keywords: Conditional lower bounds, Hardness in P, Set Cover Conjecture, Subgraph Isomorphism
Seminar: 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)
Issue Date: 2019
Date of publication: 12.03.2019


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