Revisiting the Set Cover Conjecture
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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. In spite of extensive effort, the fastest algorithm known for the general case runs in time O(mn2^n) [Fomin et al., WG 2004]. In 2012, as progress seemed to halt, Cygan et al. [TALG 2016] have put forth the Set Cover Conjecture (SeCoCo), which asserts that for every fixed ε>0, no algorithm with runtime 2^(1-ε)n poly(m) can solve Set Cover, even if the input sets are of arbitrary large constant size. We propose a weaker conjecture, which we call Log-SeCoCo, that is similar to SeCoCo but allows input sets of size O( n). To support Log-SeCoCo, we show that its failure implies an algorithm that is faster than currently known for the famous Directed Hamiltonicity problem. Even though Directed Hamiltonicity has been studied extensively for over half a century, no algorithm significantly faster than 2^n poly(n) is known for it. In fact, we show a fine-grained reduction to Log-SeCoCo from a generalization of Directed Hamiltonicity, known as the nTree problem, which too can be solved in time 2^n poly(n) [Koutis and Williams, TALG 2016]. We further show an equivalence between solving the parameterized versions of Set Cover and of nTree significantly faster than their current known runtime. Finally, we show that even moderate runtime improvements for Set Cover with bounded-size sets would imply new algorithms for nTree and for Directed Hamiltonicity. Our technical contribution is to reinforce Log-SeCoCo (and arguably SeCoCo) by reductions from other famous problems with known algorithmic barriers, and hope it will lead to more results in this vein, particularly reinforcing the Strong Exponential-Time Hypothesis (SETH) by reductions from other well-known problems.
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