ESA 2001: Algorithms — ESA 2001 pp 158-169

# On the Approximability of the Minimum Test Collection Problem

Extended Abstract
• R. Ravi
Conference paper

DOI: 10.1007/3-540-44676-1_13

Part of the Lecture Notes in Computer Science book series (LNCS, volume 2161)
Cite this paper as:
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.