Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

Har-Peled, Sariel; Quanrud, Kent

doi:10.1007/978-3-662-48350-3_60

Cited by 27 publications

(24 citation statements)

References 54 publications

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“…Therefore by construction, an r-division of a connected graph G is a collection of connected induced subgraphs of order at most r that cover all edges of G. We will use the terminology from [38]. In particular, the subgraphs in an r-division are termed clusters.…”

Section: Algorithmic Aspects Of R-divisionsmentioning

confidence: 99%

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Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Ducoffe¹,

Habib²,

Viennot³

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Under the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic-time. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many such classes -where the distance VC-dimension of a graph G is defined as the VC-dimension of its ball hypergraph: whose hyperedges are the balls of all possible radius and center in G. In particular for any fixed H, the class of H-minor free graphs has distance VC-dimension at most |V (H)| − 1. number of edges on a shortest path. Beyond its many practical applications, this fundamental problem in Graph Theory has attracted a lot of attention in the fine-grained complexity study of polynomial-time solvable problems [1,4,8,15,18,22,25,32,52]. More precisely, for every n-vertex m-edge unweighted graph the textbook algorithm for computing its diameter runs in time O(nm). In a seminal paper [52] this roughly quadratic running-time was matched by a quadratic lower-bound, assuming the Strong Exponential-Time Hypothesis (SETH). We stress that for graphs with millions of nodes and edges, quadratic time is already too prohibitive.The conditional lower-bound of [52] also holds for sparse graphs i.e., with only m = O(n) edges [1]. However it does not hold for many well-structured graph classes [1,11,13,20,14,21,23,25,31,33,35,49]. Our work proposes some new advances on the characterization of graph families for which we can compute the diameter in truly subquadratic time. Related workBefore we detail our contributions, we wish to mention a few recent (and not so recent) results that are most related to our approach.

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Section: Algorithmic Aspects Of R-divisionsmentioning

confidence: 99%

“…It is easy to prove that an r-division can be computed in polynomial time [38]. Next we use the known connections between strongly sublinear separators and polynomial expansion [27] in order to bound the running-time by some truly subquadratic function.…”

Section: Lemma 13 ( [38]mentioning

confidence: 99%

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Ducoffe¹,

Habib²,

Viennot³

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…More generally, Har-Peled and Quanrud [32] show that local search can be used to obtain PTASs for several problems including independent set, set cover, and dominating set, in graphs with polynomial expansion. These graphs have small separators and therefore r-divisions.…”

Section: R-divisions In Minorsmentioning

confidence: 99%

Local Search Yields Approximation Schemes for k-Means and k-Median in Euclidean and Minor-Free Metrics

Cohen-Addad

Klein

Mathieu

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs; (2) k-median and k-means in edge-weighted planar graphs; (3) k-means in Euclidean space of bounded dimension. Our first and second results extend to minor-closed families of graphs. All our results extend to cost functions that are the p-th power of the shortest-path distance. The algorithm is local search where the local neighborhood of a solution S consists of all solutions obtained from S by removing and adding 1{εOp1q centers.

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“…A very different approach is taken by Cabello and Gajser [3] for proper minor-closed classes and more generally by Har-Peled and Quanrud [20] for classes of graphs with polynomial expansion (which by the result of Dvořák and Norin [13] is equivalent to having strongly sublinear separators). They showed that the trivial local search algorithm (performing bounded-size changes on an initial solution as long as it can be improved by such a change) gives polynomial-time approximation schemes for maximum independent and minimum dominating set, as well as many other related problems.…”

Section: Related Resultsmentioning

confidence: 99%

Baker game and polynomial-time approximation schemes

Dvořák

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Baker [1] devised a technique to obtain approximation schemes for many optimization problems restricted to planar graphs; her technique was later extended to more general graph classes. In particular, using the Baker's technique and the minor structure theorem, Dawar et al. [5] gave Polynomial-Time Approximation Schemes (PTAS) for all monotone optimization problems expressible in the first-order logic when restricted to a proper minor-closed class of graphs. We define a Baker game formalizing the notion of repeated application of Baker's technique interspersed with vertex removal, prove that monotone optimization problems expressible in the first-order logic admit PTAS when restricted to graph classes in which the Baker game can be won in a constant number of rounds, and prove without use of the minor structure theorem that all proper minor-closed classes of graphs have this property.

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Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

Cited by 27 publications

References 54 publications

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Local Search Yields Approximation Schemes for k-Means and k-Median in Euclidean and Minor-Free Metrics

Baker game and polynomial-time approximation schemes

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