Scaffolding Problems Revisited: Complexity, Approximation and Fixed Parameter Tractable Algorithms, and Some Special Cases

Abstract

This paper is devoted to new results about the scaffolding problem, an integral problem of genome inference in bioinformatics. The problem consists in finding a collection of disjoint cycles and paths covering a particular graph called the "scaffold graph". We examine the difficulty and the approximability of the scaffolding problem in special classes of graphs, either close to trees, or very dense. We propose negative and positive results, exploring the frontier between difficulty and tractability of computing and/or approximating a solution to the problem. Also, we explore a new direction through related problems consisting in finding a family of edges having a strong effect on solution weight. Corollary 2. Let G be a class of graphs such that, for each bipartite graph G there is a supergraph of G in G. Scaffolding is N P-complete on G, even if σ p = 0 and σ c = 1 and ω max = 1. Construction 1 also implies subexponential lower bounds for our problems based on the widely believed complexitytheoretic hypothesis known as the "Exponential-Time Hypothesis" 1 (ETH, see [18, 24]). In fact, the lower bound is established directly from the fact that (planar) Directed Hamiltonian Cycle does not admit a O(2 o(|E(G)|))-time algorithm [19, Theorem 3.5] and that Construction 1 only linearly blows up the instance size. Corollary 3. Let G be a class of graphs such that, for each bipartite graph G there is a supergraph of G in G. Assuming ETH, there is no 2 o(|E(G)|)-time algorithm for Scaffolding on G, even if σ p = 0 and σ c = 1 and ω max = 1. 3.1.3 Weighted cases in sparse graphs The hardness of Scaffolding for dense graphs proved by Theorem 2 motivates the search for tractable cases among classes of sparse graphs. It is known that Scaffolding is polynomial-time solvable on graphs that are close to being a forest (constant treewidth) [22], so we consider a different sparsity measure here. We investigate whether Scaffolding becomes polynomial-time solvable if the result of removing the given perfect matching M * from G forms a forest. We call this class of graphs "quasi forests". Remark that real scaffold graphs are not always quasi-forest, however this is a first step towards their structure. We start off by modifying Construction 1 to make the resulting graph a quasi tree (see Construction 3 and Figure 3). Unfortunately, this requires fixing the length of the sought Hamiltonian cycle. To circumvent this, we present another construction, reducing the N P-complete Weighted 2-SAT to Scaffolding, that does not require fixing the lengths.

Linearizing Genomes: Exact Methods and Local Search

“…In this work, we will not focus on the SCAM problem itself (instead, the reader is referred to Weller et al [14,16,17]). Instead, we assume that we are given a solution S, whose walks then induce a subgraph of the input scaffold graph which we call solution graph (G * , M * , ω, m).…”

“…The scaffolding operation then aims at selecting the best paths in this graph in order to produce longer genomic sequences called scaffolds. Previous work focuses on the production of sequences by solving the so-called Scaffolding problem in this graph [4,14,16]. Scaffolding is a widely studied problem in bioinformatics and can be modeled by numerous, mostly heuristic, methods [8].…”

New Polynomial-Time Algorithm Around the Scaffolding Problem

“…Exact approaches have been explored [9], leading to study sparse cases [8]. The contribution of the present paper is a continuation of [4,10], where special classes of graphs has been studied, from sparse to very dense. Real instances are very sparse, but show some dense regions.…”

New Results About the Linearization of Scaffolds Sharing Repeated Contigs

“…Scaffolding has been studied in the framework of complexity and approximation [4,14,15]. In this case, the produced solution is a collection of disjoint paths alternating between edges from M * (contigs) and edges from E \M * (links between contigs), from which it is easy to infer without any ambiguity a set of nucleotidic sequences by reading contig sequences, and for inter-contig links, either detecting possible overlaps missed by the assembly process, or completing with N's.…”