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### Clique Clustering Yields a PTAS for Max-Coloring Interval Graphs

Tim Nonner
2017 Algorithmica
1 2 3 4 4 + 3 + 2 + 1 = 10 Max-Coloring Interval Graphs: Given a weighted interval graph G , find a coloring σ of G that minimizes C ∈σ w C where w C := max I ∈C w I Non-Uniform Resource Requirements  ...  1 2 3 4 4 + 4 + 3 = 11 Max-Coloring Interval Graphs: Given a weighted interval graph G , find a coloring σ of G that minimizes C ∈σ w C where w C := max I ∈C w I Non-Uniform Resource Requirements What  ...  Assume that m = 2 and κ = 2 cost increase = w m m ǫ 2 ω log n = ǫ 2 log n · wm w 1 m · w 1 ω < ǫ · O(1) · OPT clusters in P and still the cost of σ is ≤ (1 + ǫ)OPT.  ...

### Clique Clustering Yields a PTAS for max-Coloring Interval Graphs [chapter]

Tim Nonner
2011 Lecture Notes in Computer Science
This problem, called max-coloring interval graphs, contains the classical problem of coloring interval graphs as a special case for uniform weights, and it arises in many practical scenarios such as memory  ...  , which we call clique clustering.  ...  There is a PTAS for max-coloring interval graphs. Proof. Let P be a clique partition as proposed by Lemma 2.  ...

### Minimum Clique Partition in Unit Disk Graphs

2011 Graphs and Combinatorics
We present two improved approximation algorithms for minimum clique partition in unit disk graphs: (I) A polynomial time approximation scheme (PTAS) running in time n O(1/ε 2 ) .  ...  The minimum clique partition (MCP) problem is that of partitioning the vertex set of a given graph into a minimum number of cliques.  ...  Mutual proximity of nodes in a cluster is therefore a key criterion, and cliques as clusters describe it the best.  ...

### Minimum clique partition in unit disk graphs [article]

2009 arXiv   pre-print
We present two improved approximation algorithms for minimum clique partition in unit disk graphs: (I) A polynomial time approximation scheme (PTAS) running in time n^O(1/^2).  ...  The minimum clique partition (MCP) problem is that of partitioning the vertex set of a given graph into a minimum number of cliques.  ...  Mutual proximity of nodes in a cluster is therefore a key criterion, and cliques as clusters describe it the best.  ...

### Approximate MAP Estimation for Pairwise Potentials via Baker's Technique [article]

Yi-Kai Wang
2018 arXiv   pre-print
We show that computing MAX 2-CSP admits polynomial-time approximation scheme (PTAS) on planar graphs, graphs with bounded local treewidth, H-minor-free graphs, geometric graphs with bounded density and  ...  Our method also gives the first PTAS for computing the ground state of ferromagnetic Edwards-Anderson model without external magnetic field on d-dimensional lattice graphs.  ...  If the graph G is x-colorable, then the value of the minsum S min = min σ i∈V f i must fall into the interval [w x , w x · n].  ...

### On the complexity of the selective graph coloring problem in some special classes of graphs

Marc Demange, Jérôme Monnot, Petrica Pop, Bernard Ries
2014 Theoretical Computer Science
Given an integer k ≥ 1 and a graph G = (V, E) with a partition V 1 , . . . , V p of V , it consists in deciding whether there exists a set V * in G such that V * ∩ V i = 1 for all i ∈ {1, . . . , p}, and  ...  such that the graph induced by V * is k-colorable.  ...  Marc Demange also acknowledges the support by the French Agency for Research under the DEFIS program TODO, ANR-09-EMER-010.  ...

### A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs [chapter]

Imran A. Pirwani, Mohammad R. Salavatipour
2010 Lecture Notes in Computer Science
Our main result is a weakly robust polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a clique partition or (ii) gives a certificate that the graph  ...  We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques.  ...  Acknowledgments We thank Sriram Pemmaraju, Lorna Stewart, and Zoya Svitkina for helpful discussions. Our thanks to an anonymous source for pointing out the result of Capoyleas et al. [4] .  ...

### A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

Imran A. Pirwani, Mohammad R. Salavatipour
2011 Algorithmica
Our main result is a weakly robust polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a clique partition or (ii) gives a certificate that the graph  ...  We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques.  ...  Acknowledgments We thank Sriram Pemmaraju, Lorna Stewart, and Zoya Svitkina for helpful discussions. Our thanks to an anonymous source for pointing out the result of Capoyleas et al. [4] .  ...

### Diversity maximization in doubling metrics [article]

Alfonso Cevallos, Friedrich Eisenbrand, Sarah Morell
2018 arXiv   pre-print
Finally, we provide a proof of NP-hardness for remote-clique with squared distances in doubling metric spaces.  ...  The existence of such a PTAS however was left open.  ...  [3] that for any constants 0 < ε ≤ 2 3 and δ > 0, it is hard to distinguish between a) an n-vertex graph containing a clique of size k = n 1−ε , and b) an n-vertex graph where each k-vertex subset induces  ...

### Independent Sets of Dynamic Rectangles: Algorithms and Experiments [article]

Sujoy Bhore, Guangping Li, Martin Nöllenburg
2020 arXiv   pre-print
For uniform rectangles, we first give an algorithm that maintains a 4-approximate MAX-IS with O(1) update time.  ...  We present the first deterministic algorithm for maintaining a MIS (and thus a 4-approximate MAX-IS) of a dynamic set of uniform rectangles with amortized sub-logarithmic update time.  ...  Moreover, the Max-IS problem admits a polynomial-time approximation scheme (PTAS) for unit square intersection graphs [20] and pseudo disks [14] .  ...

### Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs [article]

Sariel Har-Peled, Kent Quanrud
2016 arXiv   pre-print
We prove that such graphs have small separators. Next, we present efficient (1+ε)-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others.  ...  We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density.  ...  We are particularly grateful to the anonymous referee of a previous version of this paper who pointed out the connection of our work to graphs with bounded expansion.  ...

### Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

Sariel Har-Peled, Kent Quanrud
2017 SIAM journal on computing (Print)
For example, the new algorithms yield the only PTAS known for covering points by fat triangles (that are shallow).  ...  This family is quite general, includes planar graphs, and in particular is a subset of the family of graphs that have polynomial expansion.  ...  The authors thank Mark de Berg for useful discussions related to the problems studied in this paper. In particular, he pointed out an improved bound for Lemma 2.2.7 p9 , detailed in the Appendix A.2.  ...

### Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs [chapter]

Sariel Har-Peled, Kent Quanrud
2015 Lecture Notes in Computer Science
For example, the new algorithms yield the only PTAS known for covering points by fat triangles (that are shallow).  ...  This family is quite general, includes planar graphs, and in particular is a subset of the family of graphs that have polynomial expansion.  ...  The authors thank Mark de Berg for useful discussions related to the problems studied in this paper. In particular, he pointed out an improved bound for Lemma 2.2.7 p9 , detailed in the Appendix A.2.  ...

### Flexible Resource Allocation for Clouds and All-Optical Networks [article]

Dmitriy Katz, Baruch Schieber, Hadas Shachnai
2016 arXiv   pre-print
We further show that the contiguous variant admits a (5/4 + ε)-approximation algorithm, for any fixed ε > 0, on instances whose job intervals form a proper interval graph.  ...  to competing job intervals in a flexible manner.  ...  Acknowledgments: We thank Magnús Halldórsson and Viswanath Nagarajan for valuable discussions.  ...

### Real-time scheduling to minimize machine busy times

Rohit Khandekar, Baruch Schieber, Hadas Shachnai, Tami Tamir
2014 Journal of Scheduling
Better bounds and exact algorithms are derived for several special cases, including proper interval graphs, intervals forming a clique and laminar families of intervals.  ...  We are given a set of jobs, each having a release time, a due date, a processing time and demand for machine capacity.  ...  Acknowledgment: We thank Dor Arad for helpful discussions on an earlier version of the paper. We are also grateful to the two anonymous referees for valuable comments and suggestions.  ...
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