Estimating Population Size via Line Graph Reconstruction

Abstract

Background: We propose a novel graph theoretic method to estimate haplotype population size from genotype data. The method considers only the potential sharing of haplotypes between individuals and is based on transforming the graph of potential haplotype sharing into a line graph using a minimum number of edge and vertex deletions. Results: We show that the resulting line graph deletion problems are NP complete and provide exact integer programming solutions for them. We test our approach using extensive simulations of multiple population evolution and genotypes sampling scenarios. Our results also indicate that the method may be useful in comparing populations and it may be used as a first step in a method for haplotype phasing. Conclusions: Our computational experiments show that when most of the sharings are true sharings the problem can be solved very fast and the estimated size is very close to the true size; when many of the potential sharings do not stem from true haplotype sharing, our method gives reasonable lower bounds on the underlying number of haplotypes. In comparison, a naive approach of phasing the input genotypes provides trivial upper bounds of twice the number of genotypes.

Finding the root graph through minimum edge deletion

“…Since graphs are the media to encode information in diverse domains, problems and notions of Graph Theory usually find application in a whole range of disciplines. Line graphs are not an exception, being relevant in domains other than Graph Theory, such as percolation theory [19], community detection [1,6,11] or computational biology [7], as shown next.…”

“…Further details about equivalence between line-invertible graphs and allelable graphs can be found in [7]. The following example illustrates the relation between the two concepts.…”

“…Halldórsson et al [7] presented an ILP formulation to solve EDPL. The formulation consists of finding a node coloring that satisfies (N1), (N2) and (N3), minimizing the number of pairs of nodes u and v, (u, v) ∈ E, that do not verify (N2), i.e., the deleted edges.…”

“…We present several valid inequalities for our model. Preliminary computational tests showed that dual bounds of our enhanced formulation and the model in [7] were not comparable. In view of these results, we introduce a third approach that combines the previous two formulations and some linking inequalities, which results in a family of tighter formulations.…”

“…The edge-deletion problem for the property of being line-invertible, EDPL from now on, can be seen as a variant of graph coloring. This observation was used by Halldórsson et al [7] to derive an ILP formulation which provides a certain node coloring of the graph resulting from the minimum edge deletion. To our knowledge, this is the only ILP approach for EDPL that has been developed so far.…”