Complexity and Polynomial-Time Approximation Algorithms around the Scaffolding Problem

We explore in this paper some complexity issues inspired by the contig scaffolding problem in bioinformatics. We focus on the following problem: given an undirected graph with no loop, and a perfect matching on this graph, find a set of cycles and paths covering every vertex of the graph, with edges alternatively in the matching and outside the matching, and satisfying a given constraint on the numbers of cycles and paths. We show that this problem is \(\mathcal{NP}\)-complete, even in bipartite graphs. We also exhibit non-approximability and polynomial-time approximation results, in the optimization versions of the problem.