Approximation algorithms for the scaffolding problem and its generalizations

Scaffolding is one of the main stages in genome assembly. During this stage, we want to merge contigs assembled from the paired-end reads into bigger chains called scaffolds. For this purpose, the following graph-theoretical problem has been proposed: Given an edge-weighted complete graph G and a perfect matching D of G, we wish to find a Hamiltonian path P in G such that all edges of D appear in P and the total weight of edges in P but not in D is maximized. This problem is NP-hard and the previously best polynomial-time approximation algorithm for it achieves a ratio of 1/2. In this paper, we design a new polynomial-time approximation algorithm achieving a ratio of 5-5ε/9-8ε for any constant 0 < ε < 1. Several generalizations of the problem have also been introduced in the literature and we present polynomial-time approximation algorithms for them that achieve better approximation ratios than the previous bests. In particular, one of the algorithms answers an open question.