A complexity and approximation framework for the maximization scaffolding problem

Abstract : 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 NP-complete, even in planar bipartite graphs. Moreover, we show that there is no subexponential-time algorithm for several related problems unless the Exponential-Time Hypothesis fails. Lastly, we also design two polynomial-time approximation algorithms for complete graphs.