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In the  problem we are given a hypergraph  with , and we assume that  is a , i.e. for every pair of vertices , there exists an edge  such that . The objective is to find a minimum subset of  which is a test cover. The problem is used for identification across many areas, and is NP-complete. From a parameterized complexity standpoint, many natural parameterizations of  are either -complete or have no polynomial kernel unless , and thus are unlikely to be solveable efficiently. However, in practice the size of the edges is often bounded. In this paper we study the parameterized complexity of --, the restriction of  in which each edge contains at most  vertices. In contrast to the unbounded case, we show that the following below-bound parameterizations of -- are fixed-parameter tractable with a polynomial kernel: (1) Decide whether there exists a test cover of size , and (2) decide whether there exists a test cover of size , where  is the parameter. In addition, we prove a new lower bound  on the minimum size of a test cover when the size of each edge is bounded by . -- parameterized above this bound is unlikely to be fixed-parameter tractable; in fact, we show that it is para-NP-complete, as it is NP-hard to decide whether an instance of -- has a test cover of size exactly .

Parameterizations of Test Cover with Bounded Test Sizes