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 subcollection 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^{(1epsilon)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 nnode graph G and a knode tree T, and the goal is to determine whether G has a subgraph isomorphic to T.
First, we propose a weaker conjecture LogSeCoCo, that allows input sets of size Delta=O(1/epsilon * log n), and show that an algorithm breaking LogSeCoCo 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 pPartial 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 LogSeCoCo) 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 = {45:145:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771009},
ISSN = {18688969},
year = {2019},
volume = {126},
editor = {Rolf Niedermeier and Christophe Paul},
publisher = {Schloss DagstuhlLeibnizZentrum 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 
Collection: 

36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019) 
Issue Date: 

2019 
Date of publication: 

12.03.2019 