NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs
TL;DR: The approximation schemes for hierarchically specified unit disk graphs presented in this paper are among the first approximation schemes in the literature for natural PSPACE-hard optimization problems.
About: This article is published in Journal of Algorithms.The article was published on 1998-02-01. It has received 345 citations till now. The article focuses on the topics: Indifference graph & Chordal graph.
Citations
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13 Jul 2003
TL;DR: A new geometric routing algorithm is proposed which is outstandingly efficient on practical average-case networks, however is also in theory asymptotically worst-case optimal and the formerly necessary assumption that the distance between network nodes may not fall below a constant value is dropped.
Abstract: All too often a seemingly insurmountable divide between theory and practice can be witnessed. In this paper we try to contribute to narrowing this gap in the field of ad-hoc routing. In particular we consider two aspects: We propose a new geometric routing algorithm which is outstandingly efficient on practical average-case networks, however is also in theory asymptotically worst-case optimal. On the other hand we are able to drop the formerly necessary assumption that the distance between network nodes may not fall below a constant value, an assumption that cannot be maintained for practical networks. Abandoning this assumption we identify from a theoretical point of view two fundamentamentally different classes of cost metrics for routing in ad-hoc networks.
772 citations
Additional excerpts
...Graphs with this restriction have also been called civilized [7] or λ-precision [13] graphs in the literature....
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01 Jan 2001
TL;DR: This paper presents a clustering scheme to create a hierarchical control structure for multi-hop wireless networks and presents an efficient distributed implementation of the clustering algorithm for a set of wireless nodes to create the set of desired clusters.
Abstract: In this paper we present a clustering scheme to create a hierarchical control structure for multi-hop wireless networks. A cluster is defined as a subset of vertices, whose induced graph is connected. In addition, a cluster is required to obey certain constraints that are useful for management and scalability of the hierarchy. All these constraints cannot be met simultaneously for general graphs, but we show how such a clustering can be obtained for wireless network topologies. Finally, we present an efficient distributed implementation of our clustering algorithm for a set of wireless nodes to create the set of desired clusters.
616 citations
Cites background from "NC-Approximation Schemes for NP- an..."
...In this case, the underlying communication graph, is a Unit Disk graph – defined in [20], [21] in terms of “distance” or “proximity” models, which consist of...
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09 Sep 2007
TL;DR: The first NP-completeness proofs in the geometric SINR model, which explicitly uses the fact that nodes are distributed in the Euclidean plane, are presented, which proves two problems to be NP-complete: Scheduling and One-Shot Scheduling.
Abstract: In this paper we study the problem of scheduling wireless links in the geometric SINR 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.
430 citations
Cites methods from "NC-Approximation Schemes for NP- an..."
...To the best of our knowledge, these proofs are either built for the UDG model [15, 19], or for the abstract SINR model (SINRA), and present reductions without a geometric representation....
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29 Sep 2006
TL;DR: It is shown that under a setting with single-hop traffic and no rate control, the maximal scheduling policy can achieve a constant fraction of the capacity region for networks whose connectivity graph can be represented using one of the above classes of graphs.
Abstract: We consider the problem of throughput-optimal scheduling in wireless networks subject to interference constraints. We model the interference using a family of K -hop interference models. We define a K-hop interference model as one for which no two links within K hops can successfully transmit at the same time (Note that IEEE 802.11 DCF corresponds to a 2-hop interference model.) .For a given K, a throughput-optimal scheduler needs to solve a maximum weighted matching problem subject to the K-hop interference constraints. For K=1, the resulting problem is the classical Maximum Weighted Matching problem, that can be solved in polynomial time. However, we show that for K>1,the resulting problems are NP-Hard and cannot be approximated within a factor that grows polynomially with the number of nodes. Interestingly, we show that for specific kinds of graphs, that can be used to model the underlying connectivity graph of a wide range of wireless networks, the resulting problems admit polynomial time approximation schemes. We also show that a simple greedy matching algorithm provides a constant factor approximation to the scheduling problem for all K in this case. We then show that under a setting with single-hop traffic and no rate control, the maximal scheduling policy considered in recent related works can achieve a constant fraction of the capacity region for networks whose connectivity graph can be represented using one of the above classes of graphs. These results are encouraging as they suggest that one can develop distributed algorithms to achieve near optimal throughput in case of a wide range of wireless networks.
398 citations
Cites background or methods from "NC-Approximation Schemes for NP- an..."
...Following the approach in [16], we now show that MWKVMP and, therefore, MKVMP admits a constant factor PTAS when restricted to geometric graphs....
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...For more discussion on such techniques and their analysis, we refer the reader to [16]....
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...Using arguments similar to [16, 15], we then show that the iteration in which the partition yields a K-valid matching of maximum possible weight returns a K-valid matching...
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...NC-approximation schemes for various NP-Hard and PSPACE-Hard problems restricted to geometric graphs are developed in [16]....
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TL;DR: The different ways parameterized complexity can be extended to approximation algorithms, survey results of this type and proposed directions for future research are discussed.
Abstract: Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type and propose directions for future research.
304 citations
Cites background from "NC-Approximation Schemes for NP- an..."
...Maximum Independent Set for unit disks |x|O(1/ ) [71] [83] Maximum Independent Set for unit squares |x|O(1/ ) [71] [83] Minimum Dominating Set for unit disks |x|O(1/ ) [71] [87] Minimum Dominating Set for unit squares |x|O(1/ ) [71] [87]...
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...The first four lines in Table 2 show that if we restrict Maximum Independent Set and Minimum Dominating Set to the intersection graphs of unitradius disks or unit-size squares, then the problem admits a PTAS [71, 94, 93], but does not have an EPTAS [83, 87]....
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...[71] show that Maximum Independent Set for unit disk graphs admits an |x|O(1/ ) time PTAS....
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References
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01 Jan 1979
42,654 citations
Book•
01 Jan 1979
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
Abstract: This is the second edition of a quarterly column the purpose of which is to provide a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’’ W. H. Freeman & Co., San Francisco, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed. Readers having results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this journal.
40,020 citations
Book•
01 Jan 1990TL;DR: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures and presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers.
Abstract: From the Publisher:
The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures. Like the first edition,this text can also be used for self-study by technical professionals since it discusses engineering issues in algorithm design as well as the mathematical aspects.
In its new edition,Introduction to Algorithms continues to provide a comprehensive introduction to the modern study of algorithms. The revision has been updated to reflect changes in the years since the book's original publication. New chapters on the role of algorithms in computing and on probabilistic analysis and randomized algorithms have been included. Sections throughout the book have been rewritten for increased clarity,and material has been added wherever a fuller explanation has seemed useful or new information warrants expanded coverage.
As in the classic first edition,this new edition of Introduction to Algorithms presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers. Further,the algorithms are presented in pseudocode to make the book easily accessible to students from all programming language backgrounds.
Each chapter presents an algorithm,a design technique,an application area,or a related topic. The chapters are not dependent on one another,so the instructor can organize his or her use of the book in the way that best suits the course's needs. Additionally,the new edition offers a 25% increase over the first edition in the number of problems,giving the book 155 problems and over 900 exercises thatreinforcethe concepts the students are learning.
21,651 citations