NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs

Hunt, Harry B.; Marathe, Madhav V.; Radhakrishnan, Venkatesh Babu; Ravi, S.; Rosenkrantz, Daniel J.; Stearns, Richard E.

doi:10.1006/jagm.1997.0903

Cited by 254 publications

(149 citation statements)

References 45 publications

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“…As has already been argued in the introduction, there have been various NP-completeness proofs for wireless networks. To the best of our knowledge, these proofs are either built for the UDG model [15,19], or for the abstract SIN R model (SIN RA), and present reductions without a geometric representation. A typical such proof establishes an arbitrary gain matrix between the participating nodes, which results in a standard graph.…”

Section: Related Workmentioning

confidence: 99%

Complexity in geometric SINR

Goussevskaia

Oswald

Wattenhofer

2007

Proceedings of the 8th ACM International Symposium on Mobile Ad Hoc Networking and Computing

347

363

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In this paper we study the problem of scheduling wireless links in the geometric SIN R model, which explicitly uses the fact that nodes are distributed in the Euclidean plane. We present the first NP-completeness proofs in such a model. In particular, we prove two problems to be NP-complete: Scheduling and One-Shot Scheduling. The first problem consists in finding a minimum-length schedule for a given set of links. The second problem receives a weighted set of links as input and consists in finding a maximum-weight subset of links to be scheduled simultaneously in one shot. In addition to the complexity proofs, we devise an approximation algorithm for each problem.

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Section: Related Workmentioning

confidence: 99%

Complexity in geometric SINR

Goussevskaia

Oswald

Wattenhofer

2007

Proceedings of the 8th ACM International Symposium on Mobile Ad Hoc Networking and Computing

347

363

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“…6 It is also of interest to determine the complexity of Id Code and Loc-Dom Set for bipartite permutation graphs and unit interval graphs. Finally, we remark that Min Dominating Set admits PTAS algorithms for planar graphs [5] and for unit disk graphs [38,50]. Does the same hold for Min Id Code and Min Loc-Dom Set?…”

Section: Open Problemsmentioning

confidence: 88%

“…[25] for an online database, and [20,35,36] for surveys and summaries). The log-APX-completeness of Min Dominating Set is known to hold even for bipartite graphs and split graphs [19], however it does not hold for planar graphs or unit disk graphs (for which Min Dominating Set admits PTAS algorithms [5,38,50]) or in (bipartite) graphs of bounded maximum degree (at least 3), where it is APX-complete [19].…”

Section: Related Workmentioning

confidence: 99%

Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Foucaud

2015

Journal of Discrete Algorithms

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An identifying code is a subset of vertices of a graph with the property that each vertex is uniquely determined (identified) by its nonempty neighbourhood within the identifying code. When only vertices out of the code are asked to be identified, we get the related concept of a locating-dominating set. These notions are closely related to a number of similar and well-studied concepts such as the one of a test cover. In this paper, we study the decision problems Identifying Code and Locating-Dominating Set (which consist in deciding whether a given graph admits an identifying code or a locating-dominating set, respectively, with a given size) and their minimization variants Minimum Identifying Code and Minimum LocatingDominating Set. These problems are known to be NP-hard, even when the input graph belongs to a number of specific graph classes such as planar bipartite graphs. Moreover, it is known that they are approximable within a logarithmic factor, but hard to approximate within any sub-logarithmic factor. We extend the latter result to the case where the input graph is bipartite, split or co-bipartite: both problems remain hard in these cases. Among other results, we also show that for bipartite graphs of bounded maximum degree (at least 3), the two problems are hard to approximate within some constant factor, a question which was open. We summarize all known results in the area, and we compare them to the ones for the related problem Dominating Set. In particular, our work exhibits important graph classes for which Dominating Set is efficiently solvable, but Identifying Code and Locating-Dominating Set are hard (whereas in all previous works, their complexity was the same). We also introduce graph classes for which the converse holds, and for which the complexities of Identifying Code and Locating-Dominating Set differ.

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“…Several polynomial time approximation schemes exist as well [14,21,26,35,37,38]. Most of these schemes have in common that they use the so called shifting technique.…”

Section: Previous Workmentioning

confidence: 99%

Approximation Algorithms for Unit Disk Graphs

Leeuwen

2005

Graph-Theoretic Concepts in Computer Science

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Mobile ad hoc networks are frequently modeled by unit disk graphs. We consider several classical graph theoretic problems on unit disk graphs (Maximum Independent Set, Minimum Vertex Cover, and Minimum (Connected) Dominating Set), which are relevant to such networks.We propose two new notions for unit disk graphs: thickness and density. The thickness of a graph is the number of disk centers in any width 1 slab. If the thickness of a graph is bounded, then the considered problems can be solved in polynomial time. We prove this both indirectly by presenting a relation between unit disk graphs of bounded thickness and the pathwidth of such graphs, and directly by giving dynamic programming algorithms. This result implies that the problems are fixed-parameter tractable in the thickness.We then consider unit disk graphs of bounded density. The density of a graph is the number of disk centers in any 1-by-1 box. We present a new approximation scheme for the considered problems, which uses the bounded thickness results mentioned above and the so called shifting technique. We show that the scheme is an asymptotic FPTAS and that this result is optimal, in the sense that no FPTAS can exist (unless P=NP). The scheme for Minimum Connected Dominating Set is the first FPTAS ∞ for this problem. The analysis that is applied can also be used to improve existing results, which among others implies the existence of an FPTAS ∞ for MCDS on planar graphs.

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NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs

Cited by 254 publications

References 45 publications

Complexity in geometric SINR

Complexity in geometric SINR

Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Approximation Algorithms for Unit Disk Graphs

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