Multiplicative Parameterization Above a Guarantee

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k + g(I) . Here, g(I) is usually a lower bound on the minimum size of a solution. Since its introduction in 1999 for M AX SAT and M AX C UT (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above (or, rather, times) a guarantee: Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k · g(I) . In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, which is the most natural guarantee for this problem, and provide a fixed-parameter algorithm. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ε > 0, multiplicative parameterization above g(I) 1+ε of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of fixed-parameter algorithms as well as kernelization algorithms for additional problems parameterized multiplicatively above girth.

Read-Based Phasing of Related Individuals

Motivation: Read-based phasing deduces the haplotypes of an individual from sequencing reads that cover multiple variants, while genetic phasing takes only genotypes as input and applies the rules of Mendelian inheritance to infer haplotypes within a pedigree of individuals. Combining both into an approach that uses these two independent sources of information -- reads and pedigree -- has the potential to deliver results better than each individually. Results: We provide a theoretical framework combining read-based phasing with genetic haplotyping, and describe a fixed-parameter algorithm and its implementation for finding an optimal solution. We show that leveraging reads of related individuals jointly in this way yields more phased variants and at a higher accuracy than when phased separately, both in simulated and real data. Coverages as low as 2x for each member of a trio yield haplotypes that are as accurate as when analyzed separately at 15x coverage per individual.