On the approximability of the minimum test collection problem (Extended abstract)

Abstract

The minimum test collection problem is defined as follows. Given a ground set S and a collection C of tests (subsets of S), find the minimum subcollection C‘ of C such that for every pair of elements (x; y) in S there exists a test in C‘ that contains exactly one of x and y. It is well known that the greedy algorithm gives a 1 + 2 ln n approximation for the test collection problem where n = |S|, the size of the ground set. In this paper, we show that this algorithm is close to the best possible, namely that there is no o(log n)-approximation algorithm for the test collection problem unless P = NP. We give approximation algorithms for this problem in the case when all the tests have a small cardinality, significantly improving the performance guarantee achievable by the greedy algorithm. In particular, for instances with test sizes at most k we derive an O(log k) approximation. We show APX-hardness of the version with test sizes at most two, and present an approximation algorithm with ratio (formula presented) for any fixed ϵ > 0.