The approximability of the String Barcoding problem

The String Barcoding (SBC) problem, introduced by Rash and Gusfield (RECOMB, 2002), consists in finding a minimum set of substrings that can be used to distinguish between all members of a set of given strings. In a computational biology context, the given strings represent a set of known viruses, while the substrings can be used as probes for an hybridization experiment via microarray. Eventually, one aims at the classification of new strings (unknown viruses) through the result of the hybridization experiment. In this paper we show that SBC is as hard to approximate as Set Cover. Furthermore, we show that the constrained version of SBC (with probes of bounded length) is also hard to approximate. These negative results are tight

Approximation algorithms for the test cover problem

In the test cover problem a set of m items is given together with a collection of subsets, called tests. A smallest subcollection of tests is to be selected such that for each pair of items there is a test in the selection that contains exactly one of the two items. It is known that the problem is NP-hard and that the greedy algorithm has a performance ratio O(log m). We observe that, unless P=NP, no polynomial-time algorithm can do essentially better. For the case that each test contains at most k items, we give an O(log k)-approximation algorithm. We pay special attention to the case that each test contains at most two items. A strong relation with a problem of packing paths in a graph is established, which implies that even this special case is NP-hard. We prove APX-hardness of both problems, derive performance guarantees for greedy algorithms, and discuss the performance of a series of local improvement heuristics