Extended Islands of Tractability for Parsimony Haplotyping

Abstract

Abstract. Parsimony haplotyping is the problem of finding a smallest size set of haplotypes that can explain a given set of genotypes. The problem is NP-hard, and many heuristic and approximation algorithms as well as polynomial-time solvable special cases have been discovered. We propose improved fixed-parameter tractability results with respect to the parameter "size of the target haplotype set" k by presenting an O * (k 4k )-time algorithm. This also applies to the practically important constrained case, where we can only use haplotypes from a given set. Furthermore, we show that the problem becomes polynomial-time solvable if the given set of genotypes is complete, i.e., contains all possible genotypes that can be explained by the set of haplotypes.

Haplotype Inference Constrained by Plausible Haplotype Data

“…Also, we hope our results will provide insights into the general haplotype inference problem, which has yet to be understood completely. Finally, as a possible future direction of research, it would be useful to extend the results here and in succeeding papers [11,13] to hold for weighted variants of the problems discussed.…”

“…Here we present polynomial-time algorithms with slightly slower running-times for the (α, β) bounded cases of (*,1), (2,*), (5,2), and (3,3). In the second part of the paper we show that like PH [31], CPH is fixed-parameter tractable when parameterized by the number of haplotypes in a minimum size resolution H ⊆ H of G. The running-time of this algorithm was improved recently by Fleischer et al [13].…”

Parameterized Algorithms in Bioinformatics: An Overview

“…A variant where haplotypes can only be picked from a prescribed pool H was considered by Fellows et al [129] who showed a O(k O(k 2 ) n 2 m)-time algorithm. Fleischer et al [130] later presented an O(k 4k+3 m)-time algorithm for the unconstrained version that can also solve the constraint version in O(k 4k+3 m|H | 2 ) time (indeed, these running times can be decreased to O(k 4k+2 m) and O(k 4k+2 m|H |) on average using perfect hashing) as well as a size-O(2 k k 2 ) kernel. Their algorithm can also output all optimal solutions.…”

The Parameterized Complexity of the Rainbow Subgraph Problem

“…This is the only previous fixed-parameter algorithm for MINIMUM RAINBOW SUBGRAPH that we are aware of. There are, however, several results on the parameterized complexity of PARSIMONY HAPLOTYPING [7][8][9]. RAINBOW SUBGRAPH is also a special case of SET COVER WITH PAIRS [10] which, in graph-theoretic terms, corresponds to the case where the input is a multigraph with vertex weights and the aim is to find a minimum-cost rainbow cover.…”