A Simple Gap-producing Reduction for the Parameterized Set Cover Problem
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Given an n-vertex bipartite graph I=(S,U,E), the goal of set cover problem is to find a minimum sized subset of S such that every vertex in U is adjacent to some vertex of this subset. It is NP-hard to approximate set cover to within a (1-o(1)) n factor. If we use the size of the optimum solution k as the parameter, then it can be solved in n^k+o(1) time. A natural question is: can we approximate set cover to within an o( n) factor in n^k-ϵ time? In a recent breakthrough result, Karthik, Laekhanukit and Manurangsi showed that assuming the Strong Exponential Time Hypothesis (SETH), for any computable function f, no f(k)· n^k-ϵ-time algorithm can approximate set cover to a factor below ( n)^1/poly(k,e(ϵ)) for some function e. This paper presents a simple gap-producing reduction which, given a set cover instance I=(S,U,E) and two integers k<h< (1-o(1))√( |S|/ |S|), outputs a new set cover instance I'=(S,U',E') with |U'|=|U|^h^k|S|^O(1) in |U|^h^k· |S|^O(1) time such that: (1) if I has a k-sized solution, then so does I'; (2) if I has no k-sized solution, then every solution of I' must contain at least h vertices. Setting h=(1-o(1))√( |S|/ |S|), we show that assuming SETH, for any computable function f, no f(k)· n^k-ϵ-time algorithm can distinguish between a set cover instance with k-sized solution and one whose minimum solution size is at least (1-o(1))·√( n/ n). This improves the result of Karthik, Laekhanukit and Manurangsi.
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