Till Nierhoff

Approximation Algorithms for the Steiner Tree Problem in Graphs

Given a graph G = (V, E), a set R \(R \subseteq V\) V, and a length function on the edges, a Steiner tree is a connected subgraph of G that spans all vertices in R. (It might use vertices in V \ R as well.) The Steiner tree problem in graphs is to find a shortest Steiner tree, i.e., a Steiner tree whose total edge length is minimum. This problem is well known to be NP-hard [19] and therefore we cannot expect to find polynomial time algorithms for solving it exactly. This motivates the search for good approximation algorithms for the Steiner tree problem in graphs, i. e., algorithms that have polynomial running time and return solutions that are not far from an optimum solution.

Haplotyping with missing data via perfect path phylogenies

Computational methods for inferring haplotype information from genotype data are used in studying the association between genomic variation and medical condition. Recently, Gusfield proposed a haplotype inference method that is based on perfect phylogeny principles. A fundamental problem arises when one tries to apply this approach in the presence of missing genotype data, which is common in practice. We show that the resulting theoretical problem is NP-hard even in very restricted cases. To cope with missing data, we introduce a variant of haplotyping via perfect phylogeny in which a path phylogeny is sought. Searching for perfect path phylogenies is strongly motivated by the characteristics of human genotype data: 70% of real instances that admit a perfect phylogeny also admit a perfect path phylogeny. Our main result is a fixed-parameter algorithm for haplotyping with missing data via perfect path phylogenies. We also present a simple linear-time algorithm for the problem on complete data.

Accelerating screening of 3D protein data with a graph theoretical approach

MOTIVATION The Dictionary of Interfaces in Proteins (DIP) is a database collecting the 3D structure of interacting parts of proteins that are called patches. It serves as a repository, in which patches similar to given query patches can be found. The computation of the similarity of two patches is time consuming and traversing the entire DIP requires some hours. In this work we address the question of how the patches similar to a given query can be identified by scanning only a small part of DIP. The answer to this question requires the investigation of the distribution of the similarity of patches. RESULTS The score values describing the similarity of two patches can roughly be divided into three ranges that correspond to different levels of spatial similarity. Interestingly, the two iso-score lines separating the three classes can be determined by two different approaches. Applying a concept of the theory of random graphs reveals significant structural properties of the data in DIP. These can be used to accelerate scanning the DIP for patches similar to a given query. Searches for very similar patches could be accelerated by a factor of more than 25. Patches with a medium similarity could be found 10 times faster than by brute-force search.

Steiner trees in uniformly quasi-bipartite graphs

The area of approximation algorithms for the Steiner tree problem in graphs has seen continuous progress over the last years. Currently the best approximation algorithm has a performance ratio of 1.550. This is still far away from 1.0074, the largest known lower bound on the achievable performance ratio. As all instances resulting from known lower bound reductions are uniformly quasi-bipartite, it is interesting whether this special case can be approximated better than the general case. We present an approximation algorithm with performance ratio 73/60 > 1.217 for the uniformly quasi-bipartite case. This improves on the previously known ratio of 1.279 of Robins and Zelikovsky. We use a new method of analysis that combines ideas from the greedy algorithm for set cover with a matroid-style exchange argument to model the connectivity constraint. As a consequence, we are able to provide a tight instance.

A Hard Dial-a-Ride Problem that is Easy on Average

In the dial-a-ride-problem (Darp) objects have to be moved between given sources and destinations in a transportation network by means of a server. The goal is to find the shortest transportation for the server. We study the Darp when the underlying transportation network forms a caterpillar. This special case is strongly NP-hard in the worst case. We prove that in a probabilistic setting there exists a polynomial time algorithm that finds an optimal solution with high probability. Moreover, with high probability the optimality of the solution found can be certified efficiently. In addition, we examine the complexity of the Darp in a semirandom setting and in the unweighted case.

Perfect Path Phylogeny Haplotyping with Missing Data Is Fixed-Parameter Tractable

Haplotyping via perfect phylogeny is a method for retrieving haplotypes from genotypes. Fast algorithms are known for computing perfect phylogenies from complete and error-free input instances—these instances can be organized as a genotype matrix whose rows are the genotypes and whose columns are the single nucleotide polymorphisms under consideration. Unfortunately, in the more realistic setting of missing entries in the genotype matrix, even restricted forms of the perfect phylogeny haplotyping problem become NP-hard. We show that haplotyping via perfect phylogeny with missing data becomes computationally tractable when imposing additional biologically motivated constraints. Firstly, we focus on asking for perfect phylogenies that are paths, which is motivated by the discovery that yin-yang haplotypes span large parts of the human genome. A yin-yang haplotype implies that every corresponding perfect phylogeny has to be a path. Secondly, we assume that the number of missing entries in every column of the input genotype matrix is bounded. We show that the perfect path phylogeny haplotyping problem is fixed-parameter tractable when we consider the maximum number of missing entries per column of the genotype matrix as parameter. The restrictions we impose are met by a majority of the problem instances encountered in publicly available human genome data.

On the complexity of SNP block partitioning under the perfect phylogeny model

Recent technologies for typing single nucleotide polymorphisms (SNPs) across a population are producing genome-wide genotype data for tens of thousands of SNP sites. The emergence of such large data sets underscores the importance of algorithms for large-scale haplotyping. Common haplotyping approaches first partition the SNPs into blocks of high linkage-disequilibrium, and then infer haplotypes for each block separately. We investigate an integrated haplotyping approach where a partition of the SNPs into a minimum number of non-contiguous subsets is sought, such that each subset can be haplotyped under the perfect phylogeny model. We show that finding an optimum partition is NP-hard even if we are guaranteed that two subsets suffice. On the positive side, we show that a variant of the problem, in which each subset is required to admit a perfect path phylogeny haplotyping, is solvable in polynomial time.

A heuristic for the Stacker Crane Problem on trees which is almost surely exact

Given an edge-weighted transportation network G and a list of transportation requests L, the stacker crane problem is to find a minimum-cost tour for a server along the edges of G that serves all requests. The server has capacity one, and starts and stops at the same vertex. In this paper, we consider the case that the transportation network G is a tree, and that the requests are chosen randomly according to a certain class of probability distributions. We show that a polynomial time algorithm by Frederickson and Guan [11], which guarantees a 4/3-approximation in the worst case, on almost all inputs finds a minimum-cost tour, along with a certificate of the optimality of its output.

Greedily Approximating the r-independent Set and k-center Problems on Random Instances

In this paper we analyse the performance of the greedy algorithm for r-independent set on random graphs. We show that for almost all instances The greedy algorithm has a performance ratio of 2+o(1). The greedy algorithm yields a 1+o(1) approximation of the r-dominating set problem. The k-center problem can be solved optimally.

Optimal flow distribution among multiple channels with unknown capacities

Consider a simple network flow problem in which a flow of value D must be split among n channels directed from a source to a sink. The initially unknown channel capacities can be probed by attempting to send a flow of at most D units through the network. If the flow is not feasible, we are told on which channels the capacity was exceeded (binary feedback) and possibly also how many units of flow were successfully sent on these channels (throughput feedback). For throughput feedback we present optimal protocols for minimizing the number of rounds needed to find a feasible flow and for minimizing the total amount of wasted flow. For binary feedback we present an asymptotically optimal protocol.