Parameterized Study of the Test Cover Problem

TL;DR

This paper systematically analyzes the parameterized complexity of the Test Cover problem, identifying which parameterizations are fixed-parameter tractable and which are W[1]-hard, thus clarifying the computational boundaries.

Contribution

It provides a comprehensive parameterized complexity classification for four different parameterizations of the Test Cover problem.

Findings

$k$-Test Cover and $(n-k)$-Test Cover are fixed-parameter tractable.

$(|\mathcal{T}|-k)$-Test Cover and $(\log n + k)$-Test Cover are W[1]-hard.

Abstract

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 {\sc Test Cover} problem we are given a set $[n]=\{1,...,n\}$ of items together with a collection, $\cal T$, of distinct subsets of these items called tests. We assume that $\cal T$ is a test cover, i.e., for each pair of items there is a test in $\cal T$ containing exactly one of these items. The objective is to find a minimum size subcollection of $\cal T$, which is still a test cover. The generic parameterized version of {\sc Test Cover} is denoted by $p(k,n,|{\cal T}|)$-{\sc Test Cover}. Here, we are given $([n],\cal{T})$ and a positive integer parameter $k$ as input and the objective is to decide whether there is a test cover of size at most $p(k,n,|{\cal T}|)$. We study four parameterizations for {\sc Test Cover} and obtain the following: (a) $k$-{\sc Test Cover}, and $(n-k)$-{\sc Test Cover} are fixed-parameter tractable (FPT). (b) $(|{\cal T}|-k)$-{\sc Test Cover} and $(\log n+k)$-{\sc Test Cover} are W[1]-hard. Thus, it is unlikely that these problems are FPT.