Approximation algorithms for maximum independent set of pseudo-disks

Chan, Timothy M.; Har-Peled, Sariel

doi:10.1145/1542362.1542420

Cited by 102 publications

(179 citation statements)

References 51 publications

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“…In the rest of this section, we assume we have a feasible solution y to (6)- (9), and ignore the fact that we only satisfy (8) up to a factor of e e−1 , since it affects the approximation ratio by a constant.…”

Section: The Lp Relaxationmentioning

confidence: 99%

“…We use the ellipsoid algorithm to find a feasible solution to the above LP. Given y ∈ R n the separation oracle needs to check if constraint (8) is satisfied (the other constraints are easy to check). For each k, 1 ≤ k ≤ m, we consider an instance I k of the weighted maximum-coverage problem with m sets {L i } m i=1 and weights w j = β k (X ′ j ) · y j on each task j ∈ [n].…”

Section: B Missing Proofs From Sectionmentioning

confidence: 99%

“…We now check the α-packable condition. Chan and Har-Peled [8] gave an O(1) approximation algorithm for rounding the LP relaxation (2) for such set systems when they have linear "union complexity". Many fat objects, such as disks and rectangles with constant aspect-ratio, are known to satisfy the linear union complexity condition.…”

mentioning

confidence: 99%

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Stochastic Makespan Minimization in Structured Set Systems (Extended Abstract)

Gupta

Kumar

Nagarajan

et al. 2020

Lecture Notes in Computer Science

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We study stochastic combinatorial optimization problems where the objective is to minimize the expected maximum load (a.k.a. the makespan). In this framework, we have a set of n tasks and m resources, where each task j uses some subset of the resources. Tasks have random sizes X j , and our goal is to non-adaptively select t tasks to minimize the expected maximum load over all resources, where the load on any resource i is the total size of all selected tasks that use i. For example, given a set of intervals in time, with each interval j having random load X j , how do we choose t intervals to minimize the expected maximum load at any time? Our technique is also applicable to other problems with some geometric structure in the relation between tasks and resources; e.g., packing paths, rectangles, and "fat" objects. Specifically, we give an O(log log m)-approximation algorithm for all these problems.Our approach uses a strong LP relaxation using the cumulant generating functions of the random variables. We also show that this LP has an Ω(log * m) integrality gap even for the problem of selecting intervals on a line. Moreover, we show logarithmic gaps for problems without geometric structure, showing that some structure is needed to get good results using these techniques.

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Section: The Lp Relaxationmentioning

confidence: 99%

Section: B Missing Proofs From Sectionmentioning

confidence: 99%

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Stochastic Makespan Minimization in Structured Set Systems (Extended Abstract)

Gupta

Kumar

Nagarajan

et al. 2020

Lecture Notes in Computer Science

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“…In fact, and perhaps surprisingly, such a situation is not unique. For the problem of computing a maximum independent set of pseudo-disks, local search yields a PTAS in the unweighted case and it remains an important open problem as whether a PTAS exists for the weighted case [2]. Thus, obtaining a PTAS for the "weighted" version of some problems seems a much harder task than for the unweighted case.…”

Section: Introductionmentioning

confidence: 99%

A Polynomial-Time Approximation Scheme for Facility Location on Planar Graphs

Cohen-Addad

Pilipczuk

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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We consider the classic F L problem on planar graphs (non-uniform, uncapacitated). Given an edge-weighted planar graph G, a set of clients C ⊆ V (G), a set of facilities F ⊆ V (G), and opening costs open : F → R 0 , the goal is to nd a subset D of F that minimizesThe F L problem remains one of the most classic and fundamental optimization problem for which it is not known whether it admits a polynomial-time approximation scheme (PTAS) on planar graphs despite signi cant e ort for obtaining one. We solve this open problem by giving an algorithm that for any ε > 0, computes a solution of cost at most (1 + ε) times the optimum in time. * This work is a part of projects CUTACOMBS (Ma. Pilipczuk) and TOTAL (Mi. Pilipczuk) that have received funding

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“…For the MIS problem, Fox and Pach [15] gave an algorithm with an approximation factor of n when the input consists of a set of curves, any two intersecting at most a constant number of times. The MIS problem has been studied on the intersection graph of other geometric objects such as line segments [1], disks and squares [13], rectangles [8] and pseudo-disks [10].…”

Section: Introductionmentioning

confidence: 99%

Computing Maximum Independent Set on Outerstring Graphs and Their Relatives

Bose

Carmi

Keil

et al. 2019

Lecture Notes in Computer Science

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A graph G with n vertices is called an outerstring graph if it has an intersection representation of a set of n curves inside a disk such that one endpoint of every curve is attached to the boundary of the disk. Given an outerstring graph representation, the Maximum Independent Set (MIS) problem of the underlying graph can be solved in O(s 3 ) time, where s is the number of segments in the representation (Keil et al., Comput. Geom., 60:19-25, 2017). If the strings are of constant size (e.g., line segments, L-shapes, etc.), then the algorithm takes O(n 3 ) time.In this paper, we examine the fine-grained complexity of the MIS problem on some well-known outerstring representations. We show that solving the MIS problem on grounded segment and grounded square-L representations is at least as hard as solving MIS on circle graph representations. Note that no O(n 2−δ )-time algorithm, δ > 0, is known for the MIS problem on circle graphs. For the grounded string representations where the strings are y-monotone simple polygonal paths of constant length with segments at integral coordinates, we solve MIS in O(n 2 ) time and show this to be the best possible under the strong exponential time hypothesis (SETH). For the intersection graph of n L-shapes in the plane, we give a (4 · log OPT)-approximation algorithm for MIS (where OPT denotes the size of an optimal solution), improving the previously best-known (4 · log n)-approximation algorithm of Biedl and Derka (WADS 2017).

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Approximation algorithms for maximum independent set of pseudo-disks

Cited by 102 publications

References 51 publications

Stochastic Makespan Minimization in Structured Set Systems (Extended Abstract)

Stochastic Makespan Minimization in Structured Set Systems (Extended Abstract)

A Polynomial-Time Approximation Scheme for Facility Location on Planar Graphs

Computing Maximum Independent Set on Outerstring Graphs and Their Relatives

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