Isomorphism and similarity for 2-generation pedigrees.
Bottom Line:
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.
ABSTRACT
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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. |
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Mentions: Let . An individual u ∈ I(P) is monogamous if it mates with exactly one partner, i.e., the number of individuals u', u' ≠ u, such that (u, x), (u', x) ∈ E(P) for some x ∈ I(P) is exactly one. A pedigree is monogamous if all the individuals are monogamous. In Figure 2, the sub-pedigree formed by the rightmost component is monogamous while the leftmost component is not. A pedigree P = (I(P), E(P)) is generational if there is a function such that: |
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