Embedding Planar Graphs into Low-Treewidth Graphs with Applications to Efficient Approximation Schemes for Metric Problems

Fox-Epstein, Eli; Klein, Philip N.; Schild, Aaron

doi:10.1137/1.9781611975482.66

Cited by 19 publications

(43 citation statements)

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“…The above mentioned efficient bicriteria approximation scheme [17] improves on a previous (non-efficient) bicriteria approximation scheme [13], which for any ε > 0 and planar input graph with edge lengths computes a (1 + ε)-approximation with at most (1 + ε)k centers in time n f (ε) for some function f (note that in contrast to above, such an algorithm does not imply a PAS for parameter k). The paper by Demaine et al [10] on the k-Center problem in unweighted planar graphs also considers the so-called class of map graphs, which is a superclass of planar graphs that is not minor-closed.…”

Section: Related Workmentioning

confidence: 91%

The Parameterized Hardness of the k-Center Problem in Transportation Networks

Feldmann

Marx

2020

Algorithmica

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In this paper we study the hardness of the k-Center problem on inputs that model transportation networks. For the problem, a graph G = (V, E) with edge lengths and an integer k are given and a center set C ⊆ V needs to be chosen such that |C| ≤ k. The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and thus it is natural to consider inputs that model transportation networks. Such inputs are often assumed to be planar graphs, low doubling metrics, or bounded highway dimension graphs. For each of these models, parameterized approximation algorithms have been shown to exist. We complement these results by proving that the k-Center problem is W[1]-hard on planar graphs of constant doubling dimension, where the parameter is the combination of the number of centers k, the highway dimension h, and the pathwidth p. Moreover, under the Exponential Time Hypothesis there is no f (k, p, h) · n o(p+ √ k+h) time algorithm for any computable function f . Thus it is unlikely that the optimum solution to k-Center can be found efficiently, even when assuming that the input graph abides to all of the above models for transportation networks at once! Additionally we give a simple parameterized (1+ε)-approximation algorithm for inputs of doubling dimension d with runtime (k k /ε O(kd) ) · n O(1) . This generalizes a previous result, which considered inputs in D-dimensional Lq metrics.

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Section: Related Workmentioning

confidence: 91%

The Parameterized Hardness of the k-Center Problem in Transportation Networks

Feldmann

Marx

2020

Algorithmica

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“…Our new embedding result builds on that of Fox-Epstein et al [13]. The challenge in directly applying their embedding result is that it gives an additive error bound, proportional to the diameter of the graph.…”

Section: Main Contributionsmentioning

confidence: 93%

“…Recently, Fox-Epstein, Klein, and Schild [13] showed how to embed planar graphs into graphs of bounded treewidth, such that distances are preserved up to a small additive error of D, where D is the diameter of the graph. They show how such an embedding can be used to achieve efficient bicriteria approximation schemes for k-Center and d-Independent Set.…”

Section: Metric Embeddingsmentioning

confidence: 99%

A PTAS for Bounded-Capacity Vehicle Routing in Planar Graphs

Becker

Klein

Schild

2019

Lecture Notes in Computer Science

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The Capacitated Vehicle Routing problem is to find a minimum-cost set of tours that collectively cover clients in a graph, such that each tour starts and ends at a specified depot and is subject to a capacity bound on the number of clients it can serve. In this paper, we present a polynomial-time approximation scheme (PTAS) for instances in which the input graph is planar and the capacity is bounded. Previously, only a quasipolynomial-time approximation scheme was known for these instances. To obtain this result, we show how to embed planar graphs into bounded-treewidth graphs while preserving, in expectation, the clientto-client distances up to a small additive error proportional to client distances to the depot.

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“…Other metrics and k-CENTER. For the k-CENTER problem an EPAS exists when parametrizing by both k and the doubling dimension [184], and also for planar graphs there is an EPAS for parameter k, which is implied by the EPTAS of Fox-Epstein et al [185] (cf. [184]).…”

Section: Euclidean K-means With Parameter Dmentioning

confidence: 99%