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(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.

Supported by a Merck Computational Biology and Chemistry Program Graduate Fellowship from the Merck Company Foundation.

Supported in part by subcontract No. 16082-RFP-00-2C in the area of “Combinatorial Optimization in Biology (XAXE),” Los Alamos National Laboratories.