We give simple, deterministic, distributed algorithms for computing maximal matchings, maximal independent sets and colourings. We show that edge colourings with at most 2 ; 1 colours, and maximal matchings can be computed within O(log n + ) deterministic rounds, where is the maximum degree of the network. We also show h o w to nd maximal independent sets and ( + 1)-vertex colourings within O(log n + 2 ) deterministic rounds. All hidden constants are very small and the algorithms are very simple.
Cycles in graphs play an important role in many applications, e.g., analysis of electrical networks, analysis of chemical and biological pathways, periodic scheduling, and graph drawing. From a mathematical point of view, cycles in graphs have a rich structure. Cycle bases are a compact description of the set of all cycles of a graph. In this paper, we survey results on cycle bases and prove new ones. We introduce different kinds of cycle bases, characterize them in terms of their cycle matrix, and prove structural results about them, in particular, a-priori length bounds. We give polynomial algorithms for the minimum cycle basis problem for some of the classes and prove APX -hardness for others. We also discuss three applications and show that they require different kinds of cycle bases.
I n this paper we address the pure parsimony haplotyping problem: Find a minimum number of haplotypes that explains a given set of genotypes. We prove that the problem is APX-hard and present a 2 k−1 -approximation algorithm for the case in which each genotype has at most k ambiguous positions. We further give a new integer-programming formulation that has (for the first time) a polynomial number variables and constraints. Finally, we give approximation algorithms, not based on linear programming, whose running times are almost linear in the input size.
BackgroundThrough transcription and alternative splicing, a gene can be transcribed into different RNA sequences (isoforms), depending on the individual, on the tissue the cell is in, or in response to some stimuli. Recent RNA-Seq technology allows for new high-throughput ways for isoform identification and quantification based on short reads, and various methods have been put forward for this non-trivial problem.ResultsIn this paper we propose a novel radically different method based on minimum-cost network flows. This has a two-fold advantage: on the one hand, it translates the problem as an established one in the field of network flows, which can be solved in polynomial time, with different existing solvers; on the other hand, it is general enough to encompass many of the previous proposals under the least sum of squares model. Our method works as follows: in order to find the transcripts which best explain, under a given fitness model, a splicing graph resulting from an RNA-Seq experiment, we find a min-cost flow in an offset flow network, under an equivalent cost model. Under very weak assumptions on the fitness model, the optimal flow can be computed in polynomial time. Parsimoniously splitting the flow back into few path transcripts can be done with any of the heuristics and approximations available from the theory of network flows. In the present implementation, we choose the simple strategy of repeatedly removing the heaviest path.ConclusionsWe proposed a new very general method based on network flows for a multiassembly problem arising from isoform identification and quantification with RNA-Seq. Experimental results on prediction accuracy show that our method is very competitive with popular tools such as Cufflinks and IsoLasso. Our tool, called Traph (Transcrips in gRAPHs), is available at: http://www.cs.helsinki.fi/gsa/traph/.
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