# Seminars & Colloquia Calendar

DIMACS Theory of Computing Seminar

## Set Cover in Sub-linear Time

Location:  CoRE 301
Date & time: Wednesday, 07 March 2018 at 11:00AM - 12:00PM

Abstract:  Given access to a collection of $$m$$ sets over a ground set of $$n$$ elements, the classic set cover problem asks for the minimum number of sets in the collection that cover all the elements. We study this problem from the perspective of sub-linear algorithms, where the input can be accessed by querying either the ith element contained in a set, or the jth set containing an element. In this work, we present sub-linear algorithms for the set cover problem and show that they achieve almost tight query complexities.

More specifically, on the one hand, we show an algorithm that returns an $$alpha$$-approximate cover using $$tilde O(m(n/k)^{1/(alpha-1)} + nk)$$ 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 $$tilde Omega(m(n/k)^{1/(2alpha)})$$. Moreover, we prove that even checking whether a given collection of sets covers all the elements would require $$Omega(nk)$$ 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 $$(1+eps)$$-approximation algorithm with $$tilde O(mn/keps^2)$$ queries. We show that this bound is also essentially tight by establishing a lower bound of $$tilde Omega(mn/k)$$.

This is a joint work with Piotr Indyk, Ronitt Rubinfeld, Ali Vakilian, and Anak Yodpinyanee.

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