Partial sublinear time approximation and inapproximation for maximum coverage

Abstract

We develop a randomized approximation algorithm for the classical maximum coverage problem, which given a list of sets A1,A2,…, Am and integer parameter k, select k sets Ai1,Ai2, …, Aik for maximum union (Formula Presented). In our algorithm, each input set Ai is a black box that can provide its size 0, generate a random element of (Formula Presented), and answer the membership query (Formula Presented) in O(1) time. Our algorithm gives 0 -approximation for maximum coverage problem in (Formula Presented) time, which is independent of the sizes of the input sets. No existing (Formula Presented) time (Formula Presented) -approximation algorithm for the maximum coverage has been found for any function 0 that only depends on the number of sets, where (Formula Presented) (the largest size of input sets). The notion of partial sublinear time algorithm is introduced. For a computational problem with input size controlled by two parameters (Formula Presented) and 0, a partial sublinear time algorithm for it runs in a (Formula Presented) time or (Formula Presented) time. The maximum coverage has a partial sublinear time (Formula Presented) constant factor approximation since (Formula Presented). On the other hand, we show that the maximum coverage problem has no partial sublinear (Formula Presented) time constant factor approximation algorithm.