A QPTAS for Gapless MEC

Abstract

We consider the problem Minimum Error Correction (MEC). A MEC instance is an n × m matrix M with entries from {0, 1, −}. Feasible solutions are composed of two binary m-bit strings, together with an assignment of each row of M to one of the two strings. The objective is to minimize the number of mismatches (errors) where the row has a value that differs from the assigned solution string. The symbol "−" is a wildcard that matches both 0 and 1. A MEC instance is gapless, if in each row of M all binary entries are consecutive.GAPLESS-MEC is a relevant problem in computational biology, and it is closely related to segmentation problems that were introduced by [Kleinberg-Papadimitriou-Raghavan STOC'98] in the context of data mining.Without restrictions, it is known to be UG-hard to compute an O(1)-approximate solution to MEC. For both MEC and GAPLESS-MEC, the best polynomial time approximation algorithm has a logarithmic performance guarantee. We partially settle the approximation status of GAPLESS-MEC by providing a quasi-polynomial time approximation scheme (QPTAS). Additionally, for the relevant case where the binary part of a row is not contained in the binary part of another row, we provide a polynomial time approximation scheme (PTAS). * Research partially funded by Deutsche Forschungsgemeinschaft grant MO2889/1-1.If M is clear from the context, we sometimes skip the index. A MEC instance is called gapless if in each of the n rows of M , all entries from {0, 1} are consecutive. (As regular expression, a valid row is a word of length m from the language − * {0, 1} * − * ). The MEC problem restricted to gapless instances is GAPLESS-MEC.Our motivation to study GAPLESS-MEC stems from its applications in computational biology. Humans are diploid, and hence there exist two versions of each chromosome. Determining the DNA sequences of these two chromosomal copies -called haplotypes -is important for many applications ranging from population history to clinical questions [21,22]. Many important biological phenomena such as compound heterozygosity, allele-specific events like DNA methylation or gene expression can only be studied when haplotype-resolved genomes are available [14].Existing sequencing technologies cannot read a chromosome from start to end, but instead deliver small pieces of the sequences (called reads). Like in a jigsaw puzzle, the underlying genome sequences are reconstructed from the reads by finding the overlaps between them.The upcoming next-generation sequencing technologies (e.g., Pacific Biosciences) have made the production of relatively long contiguous sequences with sequencing errors feasible, where the sequences come from both copies of chromosome. These sequences are aligned to a reference genome or to a structure called contig. We can formulate the result of this process as a GAPLESS-MEC instance: the sequences are the contiguous strings and the contig determines the columns of the strings.GAPLESS-MEC is a generalization of a problem called BINARY-MEC, the version of MEC with only instances ...