Set Cover in Sub-linear Time
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We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of m sets over n elements in the query model, we show that sub-linear algorithms derived from existing techniques have almost tight query complexities. On one hand, first we show an adaptation of the streaming algorithm presented in Har-Peled et al. [2016] to the sub-linear query model, that returns an α-approximate cover using Õ(m(n/k)^1/(α-1) + nk) queries to the input, where k denotes the value of a minimum set cover. We then complement this upper bound by proving that for lower values of k, the required number of queries is Ω̃(m(n/k)^1/(2α)), even for estimating the optimal cover size. Moreover, we prove that even checking whether a given collection of sets covers all the elements would require Ω(nk) queries. These two lower bounds provide strong evidence that the upper bound is almost tight for certain values of the parameter k. On the other hand, we show that this bound is not optimal for larger values of the parameter k, as there exists a (1+ε)-approximation algorithm with Õ(mn/kε^2) queries. We show that this bound is essentially tight for sufficiently small constant ε, by establishing a lower bound of Ω̃(mn/k) query complexity.
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