On the complexity of Minimum Path Cover with Subpath Constraints for multi-assembly

Abstract

BackgroundMulti-assembly problems have gathered much attention in the last years, as Next-Generation Sequencing technologies have started being applied to mixed settings, such as reads from the transcriptome (RNA-Seq), or from viral quasi-species. One classical model that has resurfaced in many multi-assembly methods (e.g. in Cufflinks, ShoRAH, BRANCH, CLASS) is the Minimum Path Cover (MPC) Problem, which asks for the minimum number of directed paths that cover all the nodes of a directed acyclic graph. The MPC Problem is highly popular because the acyclicity of the graph ensures its polynomial-time solvability.ResultsIn this paper, we consider two generalizations of it dealing with integrating constraints arising from long reads or paired-end reads; these extensions have also been considered by two recent methods, but not fully solved. More specifically, we study the two problems where also a set of subpaths, or pairs of subpaths, of the graph have to be entirely covered by some path in the MPC. We show that in the case of long reads (subpaths), the generalized problem can be solved in polynomial-time by a reduction to the classical MPC Problem. We also consider the weighted case, and show that it can be solved in polynomial-time by a reduction to a min-cost circulation problem. As a side result, we also improve the time complexity of the classical minimum weight MPC Problem. In the case of paired-end reads (pairs of subpaths), the generalized problem becomes NP-hard, but we show that it is fixed-parameter tractable (FPT) in the total number of constraints. This computational dichotomy between long reads and paired-end reads is also a general insight into multi-assembly problems.

Strain-aware assembly of genomes from mixed samples using flow variation graphs

“…We use the best of both worlds by defining an optimization problem that takes subpath constraints, minimizes node abundance errors, and is polynomially solvable; this establishes a theoretical novelty. Because this novelty gives way to different types of analyses in other settings, and immediately connects to extensively treated theoretical issues [22,21], we feel that it is of value also in its own right.…”

“…Some of the challenges that have to be dealt with in viral quasispecies assembly can also be found in RNA transcipt assembly, where the goal is to reconstruct an unknown number of transcripts and predict the relative transcript abundances. Not surprisingly, many RNA transcipt assemblers define graph optimization problems similar to our flow formulation [21,22,23,24,25]. Although dealing with related problems, these methods cannot be applied in a viral quasispecies setting so easily: they require a collection of reference genomes representing all possible haplotypes as input, which is not available in our setting.…”

“…In [21], node and edge abundance errors are used to define a min-cost flow problem; note that this formulation does not take subpath constraints into account. On the other hand, [22] describes how subpath constraints can be incorporated into a minimum path cover formulation. This results in an optimization problem that is solvable in polynomial time, but does not minimize node abundance errors.…”

Evaluating approaches to find exon chains based on long reads

“…The prediction of full transcripts can be modelled as a combinatorial problem in a splicing graph (Heber et al, 2002), where nodes are exons and arcs are exons consecutive in some read alignment. Rizzi et al (2014) proposed modeling long reads as subpath constraints (exon chains) in a splicing graph. Figure 1 illustrates these concepts.…”

Full-length de novo viral quasispecies assembly through variation graph construction

“…Many RNA transcript assemblers work in a similar way to Virus-VG: first enumerating all possible transcripts, then selecting an 'optimal' set of transcripts using various optimization methods (Feng et al, 2010;Li et al, 2011;Mezlini et al, 2013). This has led to variations of minimum path cover optimization problems that are-regarding a few relevant aspects-similar in spirit to the optimization problem we formulate (Bernard et al, 2014;Pertea et al, 2015;Rizzi et al, 2014;Tomescu et al, 2013;Trapnell et al, 2010). Most importantly, Rizzi et al (2014) introduce node and edge abundance errors and Tomescu et al (2013) show a minimum path cover with subpath constraints to be polynomially solvable.…”