A 2-Approximation Algorithm for the Complementary Maximal Strip Recovery Problem

The Maximal Strip Recovery problem (MSR) and its complementary (CMSR) are well-studied NP-hard problems in computational genomics. The input of these dual problems are two signed permutations. The goal is to delete some gene markers from both permutations, such that, in the remaining permutations, each gene marker has at least one common neighbor. Equivalently, the resulting permutations could be partitioned into common strips of length at least two. Then MSR is to maximize the number of remaining genes, while the objective of CMSR is to delete the minimum number of gene markers. In this paper, we present a new approximation algorithm for the Complementary Maximal Strip Recovery (CMSR) problem. Our approximation factor is 2, improving the currently best 7/3-approximation algorithm. Although the improvement on the factor is not huge, the analysis is greatly simplified by a compensating method, commonly referred to as the non-oblivious local search technique. In such a method a substitution may not always increase the value of the current solution (it sometimes may even decrease the solution value), though it always improves the value of another function seemingly unrelated to the objective function

Complexity and Approximation Results for the Min-Sum and Min-Max Disjoint Paths Problems

Given a graph G=(V, E) and k source-sink pairs (s1, t1), …, (sk, tk) with each si, ti  V, the Min-Sum Disjoint Paths problem asks to find k disjoint paths connecting all the source-sink pairs with minimized total length, while the Min-Max Disjoint Paths problem asks for k disjoint paths connecting all the source-sink pairs with minimized length of the longest path. We show that the weighted Min-Sum Disjoint Paths problem is FPNP-complete in general graphs, and the unweighted Min-Sum Disjoint Paths problem and the unweighted Min-Max Disjoint Paths problem cannot be approximated within m(m1-1) for any constant   > 0 even in planar graphs, assuming P P NP, where m is the number of edges in G. We give for the first time a simple bicriteria approximation algorithm for the unweighted Min-Max Edge-Disjoint Paths problem and the weighted Min-Sum Edge-Disjoint Paths problem, w

Isomorphism and Similarity for 2-Generation Pedigrees

We consider the emerging problem of comparing the similarity between (unlabeled) pedigrees. More specifically, we focus on the simplest pedigrees, namely, the 2-generation pedigrees. We show that the isomorphism testing for two 2-generation pedigrees is GI-hard. If the 2-generation pedigrees are monogamous (i.e., each individual at level-1 can mate with exactly one partner) then the isomorphism testing problem can be solved in polynomial time. We then consider the problem by relaxing it into an NP-complete decomposition problem which can be formulated as the Minimum Common Integer Pair Partition (MCIPP) problem, which we show to be FPT by exploiting a property of the optimal solution. While there is still some difficulty to overcome, this lays down a solid foundation for this research

Genomic Scaffold Filling Revisited

The genomic scaffold filling problem has attracted a lot of attention recently. The problem is on filling an incomplete sequence (scaffold) I into I\u27, with respect to a complete reference genome G, such that the number of adjacencies between G and I\u27 is maximized. The problem is NP-complete and APX-hard, and admits a 1.2-approximation. However, the sequence input I is not quite practical and does not fit most of the real datasets (where a scaffold is more often given as a list of contigs). In this paper, we revisit the genomic scaffold filling problem by considering this important case when, (1) a scaffold S is given, the missing genes X = c(G) - c(S) can only be inserted in between the contigs, and the objective is to maximize the number of adjacencies between G and the filled S\u27 and (2) a scaffold S is given, a subset of the missing genes X\u27 subset X = c(G) - c(S) can only be inserted in between the contigs, and the objective is still to maximize the number of adjacencies between G and the filled S\u27\u27. For problem (1), we present a simple NP-completeness proof, we then present a factor-2 greedy approximation algorithm, and finally we show that the problem is FPT when each gene appears at most d times in G. For problem (2), we prove that the problem is W[1]-hard and then we present a factor-2 FPT-approximation for the case when each gene appears at most d times in G

IsoTree: A New Framework for De novo Transcriptome Assembly from RNA-seq Reads

High-throughput sequencing of mRNA has made the deep and efficient probing of transcriptome more affordable. However, the vast amounts of short RNA-seq reads make de novo transcriptome assembly an algorithmic challenge. In this work, we present IsoTree, a novel framework for transcripts reconstruction in the absence of reference genomes. Unlike most of de novo assembly methods that build de Bruijn graph or splicing graph by connecting $k-mers$ which are sets of overlapping substrings generated from reads, IsoTree constructs splicing graph by connecting reads directly. For each splicing graph, IsoTree applies an iterative scheme of mixed integer linear program to build a prefix tree, called isoform tree. Each path from the root node of the isoform tree to a leaf node represents a plausible transcript candidate which will be pruned based on the information of paired-end reads. Experiments showed that in most cases IsoTree performs better than other leading transcriptome assembly programs. IsoTree is available at https://github.com/Jane110111107/IsoTree