Star Partitions of Perfect Graphs

Bevern, René van; Bredereck, Robert; Bulteau, Laurent; Froese, Vincent; Niedermeier, Rolf; Woeginger, Gerhard J.

doi:10.1007/978-3-662-43948-7_15

Cited by 9 publications

(4 citation statements)

References 28 publications

(30 reference statements)

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“…Question: Is there a partition of the vertices of G in such a way that each part induces a path on three vertices? Bevern et al [26] proved that this problem is N P -complete.…”

Section: Complexity When the Radius Is Fixedmentioning

confidence: 99%

Identification of points using disks

Gledel¹,

Parreau²

2019

Discrete Mathematics

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We consider the problem of identifying n points in the plane using disks, i.e., minimizing the number of disks so that each point is contained in a disk and no two points are in exactly the same set of disks. This problem can be seen as an instance of the test covering problem with geometric constraints on the tests. We give tight lower and upper bounds on the number of disks needed to identify any set of n points of the plane. In particular, we prove that if there are no three colinear points nor four cocyclic points, then 2⌈n/6⌉ + 1 disks are enough, improving the known bound of ⌈(n + 1)/2⌉ when we only require that no three points are colinear. We also consider complexity issues when the radius of the disks is fixed, proving that this problem is NP-complete. In contrast, we give a linear-time algorithm computing the exact number of disks if the points are colinear.

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“…Question: Is there a partition of the vertices of G in such a way that each part induces a path on three vertices? Bevern et al [26] proved that this problem is N P -complete.…”

Section: Complexity When the Radius Is Fixedmentioning

confidence: 99%

Identification of points using disks

Gledel¹,

Parreau²

2019

Discrete Mathematics

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“…Computing star partitions is known to be NP-hard even on subcubic grid graphs and split graphs [6]. By Proposition 1 in Section 3.2 it follows that Dissolution is also NP-hard on planar graph because grid graphs are planar.…”

Section: Planar Graphsmentioning

confidence: 99%

Network-Based Vertex Dissolution

Bevern¹,

Bredereck²,

Froese³

et al. 2015

SIAM J. Discrete Math.

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We introduce a graph-theoretic vertex dissolution model that applies to a number of redistribution scenarios such as gerrymandering in political districting or work balancing in an online situation. The central aspect of our model is the deletion of certain vertices and the redistribution of their load to neighboring vertices in a completely balanced way. We investigate how the underlying graph structure, the knowledge of which vertices should be deleted, and the relation between old and new vertex loads influence the computational complexity of the underlying graph problems. Our results establish a clear borderline between tractable and intractable cases.Comment: Version accepted at SIAM Journal on Discrete Mathematic

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“…In Gledel and Parreau (2019), it was shown that Continuous-G-Min-Disc-Code for bounded-radius disks is NP-complete. The same proof technique, a reduction from the NP-complete P 3 -Partition-Grid problem van Bevern et al (2014), can be adapted to show the following. Given an instance G of P 3 -Partition-Grid, we construct an instance P G of Continuous-G-Min-Disc-Code as follows.…”

Section: A Ptas For the 1d Unit Interval Casementioning

confidence: 99%

Discriminating Codes in Geometric Setups

Dey¹,

Foucaud²,

Nandy³

et al. 2020

Preprint

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We study two geometric variations of the discriminating code problem. In the discrete version, a finite set of points P and a finite set of objects S are given in R d . The objective is to choose a subset S * ⊆ S of minimum cardinality such that the subsets S * i ⊆ S * covering p i , satisfy S * i = ∅ for each i = 1, 2, . . . , n, and S * i = S * j for each pair (i, j), i = j. In the continuous version, the solution set S * can be chosen freely among a (potentially infinite) class of allowed geometric objects.In the 1-dimensional case (d = 1), the points are placed on some fixed-line L, and the objects in S and S * are finite sub-segments of L (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. This is also in contrast with most geometric covering problems, which are usually polynomial-time solvable in 1D. We then design a polynomial-time 2-approximation algorithm for the 1-dimensional discrete case. We also design

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Star Partitions of Perfect Graphs

Cited by 9 publications

References 28 publications

Identification of points using disks

Identification of points using disks

Network-Based Vertex Dissolution

Discriminating Codes in Geometric Setups

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