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# The Set Cover Conjecture and Subgraph Isomorphism with a Tree Pattern

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LIPIcs.STACS.2019.45.pdf
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## Cite As

Robert Krauthgamer and Ohad Trabelsi. The Set Cover Conjecture and Subgraph Isomorphism with a Tree Pattern. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.STACS.2019.45

## 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Graph algorithms analysis
• Theory of computation → Discrete optimization
• Theory of computation → Parameterized complexity and exact algorithms
##### Keywords
• Conditional lower bounds
• Hardness in P
• Set Cover Conjecture
• Subgraph Isomorphism

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