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(mn2ⁿ) [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 2ⁿpoly(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 2ⁿpoly(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 2ⁿpoly(m) (an assumption even weaker than Log-SeCoCo) then kTree cannot be computed significantly faster than 2^kpoly(n), the running time of the Koutis and Williams’ algorithm.