Abstract We consider a graph orientation problem that can be viewed as a generalization of Minimum Graph Coloring. Our problem takes as input an undirected graph $$G = (V, E)$$ G = ( V , E ) in which every edge $$\{u, v\} \in E$$ { u , v } ∈ E has two (potentially different and not necessarily positive) weights representing the lengths of its two possible directions ( u , v ) and ( v , u ), and asks for an orientation, i.e., an assignment of a direction to each edge of G , such that the length of a longest simple directed path in the resulting directed graph is minimized. A longest path in a graph is not always a maximal path when some edges have negative lengths, so the problem has two variants depending on whether all simple directed paths or maximal simple directed paths only are taken into account in the definition. We prove that the problems are NP-hard to approximate even if restricted to subcubic planar graphs, and develop fast polynomial-time algorithms for both problem variants for three classes of graphs: path graphs, cycle graphs, and star graphs.

With the remarkable development in inexpensive sequencing technologies and supporting computational tools, we have the promise of medicine being personalized by knowledge of the individual genome. Current technologies provide high throughput, but short reads. Reconstruction of the donor genome is based either on de novo assembly of the (short) reads, or on mapping donor reads to a standard reference. While such techniques demonstrate high success rates for inferring 'simple' genomic segments, they are confounded by segments with complex duplication patterns, including regions of direct medical relevance, like the HLA and the KIR regions.In this work, we address this problem with a method for assessing the quality of a predicted genome sequence for complex regions of the genome. This method combines two natural types of evidence: sequence similarity of the mapped reads to the predicted donor genome, and distribution of reads across the predicted genome. We define a new scoring function for read-to-genome matchings, which penalizes for sequence dissimilarities and deviations from expected read location distribution, and present an efficient algorithm for finding matchings that minimize the penalty. The algorithm is based on a formal problem, first defined in this paper, called Coverage Sensitive many-to-many min-cost bipartite Matching (CSM). This new problem variant generalizes the standard (one-to-one) weighted bipartite matching problem, and can be solved using network flows. The resulting Java-based tool, called SAGE (Scoring function for Assembled GEnomes), is freely available upon request. We demonstrate over simulated data that SAGE can be used to infer correct haplotypes of the highly repetitive KIR region on the Human chromosome 19.