Approximation Schemes for Node-Weighted Geometric Steiner Tree Problems

We provide a randomized approximation scheme for the k-median problem when the input points lie in the d-dimensional Euclidean space.The running time is, which is nearly linear for any fixed ε and d. Moreover our method provides the first polynomial-time approximation scheme for k-median and uncapacitated facility location instances in d-dimensional Euclidean space for any fixed d > 2. Our work extends techniques introduced originally by Arora for the Euclidean TSP. To obtain the improvement we develop a structure theorem to describe hierarchical decomposition of solutions. The theorem is based on an adaptive decomposition scheme, which guesses at every level of the hierarchy the structure of the optimal solution and modifies accordingly the parameters of the decomposition. We believe that our methodology is of independent interest and may find applications to further geometric problems.

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“…Proof: The construction uses some ideas similar to the ones in [26]. We partition the area of R \ S into rectangular moats M 1 , M 2 , .…”

Section: S2 For Each Portalmentioning

confidence: 99%

A Nearly Linear-Time Approximation Scheme for the Euclidean k-Median Problem

Kolliopoulos¹,

Rao

2007

SIAM J. Comput.

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“…Our interest, in this paper, is in exact solutions to the PCEST problem. Such solutions are absent from the literature, however approximation schemes have been discussed in Remy and Steger (2009). The range of problems covered by their methods includes the PCEST problem.…”

Section: Introductionmentioning

confidence: 99%

Solving the prize‐collecting Euclidean Steiner tree problem

Whittle

Brazil

Grossman

et al. 2020

Int Trans Operational Res

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The prize-collecting Euclidean Steiner tree (PCEST) problem is a generalization of the well-known Euclidean Steiner tree (EST) problem. All points given in an EST problem instance are connected by the shortest possible network in a solution. A solution can include additional points called Steiner points. A PCEST problem instance differs from an EST problem instance by the addition of weights for each given point. A PCEST solution connects a subset of the given points in order to maximize the net value of the network (the sum of the selected point weights, less than the length of the network). We present an algorithmic framework for solving the PCEST problem. Included in the framework are efficient methods to determine subsets of points that must be in every solution, and subsets of points that cannot be in any solution. Also included are methods to generate and concatenate full Steiner trees.

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“…Thus we partially settle the approximability for this problem: GAPLESS-MEC is not APX-hard unless NP ⊆ QP (cf. [20]). Thus our result reveals a separation of the hardness of the gapless case and the case where we allow a single gap.…”

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A QPTAS for Gapless MEC

Garg,

Mömke

2018

Preprint

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We consider the problem Minimum Error Correction (MEC). A MEC instance is an n × m matrix M with entries from {0, 1, −}. Feasible solutions are composed of two binary m-bit strings, together with an assignment of each row of M to one of the two strings. The objective is to minimize the number of mismatches (errors) where the row has a value that differs from the assigned solution string. The symbol "−" is a wildcard that matches both 0 and 1. A MEC instance is gapless, if in each row of M all binary entries are consecutive.GAPLESS-MEC is a relevant problem in computational biology, and it is closely related to segmentation problems that were introduced by [Kleinberg-Papadimitriou-Raghavan STOC'98] in the context of data mining.Without restrictions, it is known to be UG-hard to compute an O(1)-approximate solution to MEC. For both MEC and GAPLESS-MEC, the best polynomial time approximation algorithm has a logarithmic performance guarantee. We partially settle the approximation status of GAPLESS-MEC by providing a quasi-polynomial time approximation scheme (QPTAS). Additionally, for the relevant case where the binary part of a row is not contained in the binary part of another row, we provide a polynomial time approximation scheme (PTAS). * Research partially funded by Deutsche Forschungsgemeinschaft grant MO2889/1-1.If M is clear from the context, we sometimes skip the index. A MEC instance is called gapless if in each of the n rows of M , all entries from {0, 1} are consecutive. (As regular expression, a valid row is a word of length m from the language − * {0, 1} * − * ). The MEC problem restricted to gapless instances is GAPLESS-MEC.Our motivation to study GAPLESS-MEC stems from its applications in computational biology. Humans are diploid, and hence there exist two versions of each chromosome. Determining the DNA sequences of these two chromosomal copies -called haplotypes -is important for many applications ranging from population history to clinical questions [21,22]. Many important biological phenomena such as compound heterozygosity, allele-specific events like DNA methylation or gene expression can only be studied when haplotype-resolved genomes are available [14].Existing sequencing technologies cannot read a chromosome from start to end, but instead deliver small pieces of the sequences (called reads). Like in a jigsaw puzzle, the underlying genome sequences are reconstructed from the reads by finding the overlaps between them.The upcoming next-generation sequencing technologies (e.g., Pacific Biosciences) have made the production of relatively long contiguous sequences with sequencing errors feasible, where the sequences come from both copies of chromosome. These sequences are aligned to a reference genome or to a structure called contig. We can formulate the result of this process as a GAPLESS-MEC instance: the sequences are the contiguous strings and the contig determines the columns of the strings.GAPLESS-MEC is a generalization of a problem called BINARY-MEC, the version of MEC with only instances ...

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Approximation Schemes for Node-Weighted Geometric Steiner Tree Problems

Cited by 8 publications

References 23 publications

A Nearly Linear-Time Approximation Scheme for the Euclidean k-Median Problem

A Nearly Linear-Time Approximation Scheme for the Euclidean k-Median Problem

Solving the prize‐collecting Euclidean Steiner tree problem

A QPTAS for Gapless MEC

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