Parameterized Study of the Test Cover Problem

In this paper we carry out a systematic study of a natural covering problem, used for identification across several areas, in the realm of parameterized complexity. In the Test Cover problem we are given a set [n] = {1,. .. , n} of items together with a collection, T , of distinct subsets of these items called tests. We assume that T is a test cover, i.e., for each pair of items there is a test in T containing exactly one of these items. The objective is to find a minimum size subcollection of T , which is still a test cover. The generic parameterized version of Test Cover is denoted by p(k, n, |T |)-Test Cover. Here, we are given ([n], T) and a positive integer parameter k as input and the objective is to decide whether there is a test cover of size at most p(k, n, |T |). We study four parameterizations for Test Cover and obtain the following: (a) k-Test Cover, and (n−k)-Test Cover are fixed-parameter tractable (FPT), i.e., these problems can be solved by algorithms of runtime f (k) · poly(n, |T |), where f (k) is a function of k only. (b) (|T | − k)-Test Cover and (log n + k)-Test Cover are W[1]-hard. Thus, it is unlikely that these problems are FPT.