# On the Approximability of the Minimum Test Collection Problem

## Abstract

The minimum test collection problem is defined as follows. Given a ground set \(
\mathcal{S}
\) and a collection \(
\mathcal{C}
\) of tests (subsets of \(
\mathcal{S}
\)), find the minimum subcollection \(
\mathcal{C}'
\) of \(
\mathcal{C}
\) such that for every pair of elements (*x, y*) in \(
\mathcal{S}
\) there exists a test in \(
\mathcal{C}'
\) that contains exactly one of *x* and *y*. It is well known that the greedy algorithm gives a 1 + 2lnn approximation for the test collection problem where \(
n = \left| \mathcal{S} \right|
\), 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 \(
\tfrac{7}
{6} + \varepsilon
\) for any fixed ε > 0.

## Keywords

Approximation Algorithm Greedy Algorithm Complete Graph Test Size Performance Guarantee## Preview

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