Halldórsson B.V., Halldórsson M.M., Ravi R. (2001) On the Approximability of the Minimum Test Collection Problem. In: auf der Heide F.M. (eds) Algorithms — ESA 2001. ESA 2001. Lecture Notes in Computer Science, vol 2161. Springer, Berlin, Heidelberg

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(logn)-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(logk) 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.