Strongly Sublinear Separators and Polynomial Expansion

Dvořák, Zdeněk; Norin, Sergey

doi:10.1137/15m1017569

Cited by 60 publications

(70 citation statements)

References 11 publications

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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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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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“…It was recently shown that any graph with strongly sublinear hereditary separators has polynomial expansion [DN15]. In conjunction with the preceding separator (for low-density objects), this yields a second proof that the intersection graphs of low-density objects have polynomial expansion, however with weaker bounds.…”

Section: Separatorsmentioning

confidence: 90%

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

Har-Peled

Quanrud

2015

Algorithms - ESA 2015

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We investigate the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, includes planar graphs, and in particular is a subset of the family of graphs that have polynomial expansion.We present efficient (1 + ε)-approximation algorithms for polynomial expansion graphs, for Independent Set, Set Cover, and Dominating Set problems, among others, and these results seem to be new. Naturally, PTAS's for these problems are known for subclasses of this graph family. These results have immediate applications in the geometric domain. For example, the new algorithms yield the only PTAS known for covering points by fat triangles (that are shallow).We also prove corresponding hardness of approximation for some of these optimization problems, characterizing their intractability with respect to density. For example, we show that there is no PTAS for covering points by fat triangles if they are not shallow, thus matching our PTAS for this problem with respect to depth.

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“…Using the existence of good 3-regular expanders, Dvořák and Norin [22,Corollary 7] find a family of 3-regular graphs on k edges and n = Instead of relying on Proposition 1.6 to count homomorphisms in Theorem 1.1, we can use more sophisticated methods where available. For instance, we can use fast matrix multiplication to count homomorphisms from patterns H of treewidth at most two, thus proving Theorem 1.3.…”

Section: To Obtain #Sub(h → G)mentioning

confidence: 99%

Homomorphisms are a good basis for counting small subgraphs

Curticapean

Dell

Marx

2017

Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

182

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We introduce graph motif parameters, a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs. Classical works by Lovász show that many interesting quantities have this form, including, for fixed graphs H, the number of H-copies (induced or not) in an input graph G, and the number of homomorphisms from H to G.Using the framework of graph motif parameters, we obtain faster algorithms for counting subgraph copies of fixed graphs H in host graphs G: For graphs H on k edges, we show how to count subgraph copies ofby a surprisingly simple algorithm. This improves upon previously known running times, such as O(n 0.91k+c ) time for k-edge matchings or O(n 0.46k+c ) time for k-cycles. Furthermore, we prove a general complexity dichotomy for evaluating graph motif parameters: Given a class C of such parameters, we consider the problem of evaluating f ∈ C on input graphs G, parameterized by the number of induced subgraphs that f depends upon. For every recursively enumerable class C, we prove the above problem to be either FPT or #W[1]-hard, with an explicit dichotomy criterion. This allows us to recover known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms in a uniform and simplified way, together with improved lower bounds.Finally, we extend graph motif parameters to colored subgraphs and prove a complexity trichotomy:

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Strongly Sublinear Separators and Polynomial Expansion

Cited by 60 publications

References 11 publications

Baker game and polynomial-time approximation schemes

Baker game and polynomial-time approximation schemes

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

Homomorphisms are a good basis for counting small subgraphs

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