The Set Cover Conjecture and Subgraph Isomorphism with a Tree Pattern

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) [Fomin et al., WG 2004], and the Set Cover Conjecture (SeCoCo) [Cygan et al., TALG 2016] asserts that for every fixed ε > 0, no algorithm can solve Set Cover in time 2(1−ε)npoly(m), even if set sizes are bounded by ∆ = ∆(ε). 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 ∆ = O(1/ε · log n), and show that an algorithm breaking Log-SeCoCo would imply a faster algorithm than the currently known 2poly(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 2poly(n) is known despite extensive research. Second, we prove that if Set Cover cannot be solved significantly faster than 2poly(m) (an assumption even weaker than Log-SeCoCo), then kTree cannot be computed significantly faster than 2poly(n), the running time of the Koutis and Williams’ algorithm. Applying the same techniques to the p-Partial Cover problem, a parameterized version of Set Cover that requires covering at least p elements, we obtain a new algorithm with running time (2+ε)(m+n) for arbitrary ε > 0, which improves previous work and is nearly optimal assuming say LogSeCoCo.