[Paper Review] On the Complexity of Several Haplotyping Problems

Rudi Cilibrasi, Leo van Iersel+2 more|ArXiv.org|May 13, 2005

Gene expression and cancer classification5 references4 citations

TL;DR

This paper establishes the NP-hardness of Ungapped-Minimum Error Correction (MEC) in haplotyping by reduction from MAX-CUT, resolves an open problem by presenting a polynomial-time algorithm for Pure Parsimony Haplotyping (PPH) when each genotype has at most two ambiguous sites, and introduces a novel bipartite graph formulation to compute PPH via maximum independent set, enabling efficient solution for this constrained case.

ABSTRACT

In this paper we present a collection of results pertaining to haplotyping. The first set of results concerns the combinatorial problem of reconstructing haplotypes from incomplete and/or imperfectly sequenced haplotype data. More specifically, we show that an interesting, restricted case of Minimum Error Correction (MEC) is NP-hard, point out problems in earlier claims about a related problem, and present a polynomial-time algorithm for the ungapped case of Longest Haplotype Reconstruction (LHR). Secondly, we present a polynomial time algorithm for the problem of resolving genotype data using as few haplotypes as possible (the Pure Parsimony Haplotyping Problem, PPH) where each genotype has at most two ambiguous positions, thus solving an open problem posed by Lancia et al in "Haplotyping Populations by Pure Parsimony: Complexity of Exact and Approximation Algorithms."

Motivation & Objective

• To establish the computational complexity of restricted variants of Minimum Error Correction (MEC) in haplotyping, particularly the ungapped case.
• To resolve an open problem posed by Lancia et al. regarding the Pure Parsimony Haplotyping (PPH) problem when each genotype has at most two ambiguous positions.
• To develop a polynomial-time algorithm for PPH under this constraint using a novel bipartite graph formulation.
• To clarify and correct prior claims about related haplotyping problems, particularly those relying on results from [10] that are found to be unsound for this purpose.

Proposed method

• Reduces the optimization version of MAX-CUT to Ungapped-MEC to prove NP-hardness, establishing computational intractability for this restricted MEC variant.
• Constructs a bipartite graph $ B = (V^+ igcup V^-, E) $ where vertices represent haplotypes and resolution indicators, with edges encoding compatibility between genotypes and haplotypes.
• Assigns parity to haplotypes (even/odd) and introduces auxiliary vertex sets $ I_0(g), I_1(g,0), I_1(g,1), I_2(g,+), I_2(g,-) $ to model genotype resolution constraints.
• Uses the maximum independent set (MIS) of $ B $ to minimize the number of distinct haplotypes used in resolving all genotypes, with $ |MaxBIS(B)| = 4n + (|H| - PPH(G)) $.
• Leverages the equivalence between maximum independent set and maximum matching in bipartite graphs to compute MIS in $ O(mn\log n + n^{3/2}) $ time.
• Applies sorting and duplicate-merging techniques to efficiently maintain adjacency information during graph construction, ensuring $ O(1) $ adjacency queries.

Experimental results

Research questions

• RQ1 Is the Ungapped-Minimum Error Correction (MEC) problem NP-hard, and can this be proven without relying on flawed prior results?
• RQ2 Can the Pure Parsimony Haplotyping (PPH) problem be solved in polynomial time when each genotype has at most two ambiguous sites, resolving an open problem?
• RQ3 What is the relationship between the size of the maximum independent set in the constructed bipartite graph and the number of haplotypes required in PPH?
• RQ4 Can a graph-theoretic formulation be used to model and solve the PPH problem efficiently under bounded ambiguity?
• RQ5 Are earlier claims about the complexity of related haplotyping problems, particularly those based on [10], valid for the MEC and PPH settings?

Key findings

• Ungapped-MEC is proven NP-hard via a direct reduction from the optimization version of MAX-CUT, correcting and strengthening prior claims that relied on questionable results from [10].
• The PPH problem is solvable in polynomial time when each genotype has at most two ambiguous sites, resolving an open problem posed by Lancia et al. in [14].
• The proposed bipartite graph formulation allows exact computation of the minimum number of haplotypes needed to explain a set of genotypes, with $ |MaxBIS(B)| = 4n + (|H| - PPH(G)) $.
• The algorithm runs in $ O(mn\log n + n^{3/2}) $ time, combining efficient graph construction with maximum matching and independent set computation.
• The method ensures that all resolution constraints are satisfied by forcing inclusion of $ I $-sets in the MIS, thereby minimizing the number of distinct haplotypes used.
• The construction correctly models genotype resolution by ensuring that only compatible haplotypes can be used, and that unused haplotypes are prioritized in the MIS.