Approximation algorithms for NP-complete problems on planar graphs

doi:10.1145/174644.174650

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Approximation algorithms for NP-complete problems on planar graphs

Journal Article•DOI•

Approximation algorithms for NP-complete problems on planar graphs

Brenda S. Baker¹•Institutions (1)

Bell Labs¹

02 Jan 1994-Journal of the ACM (ACM)-Vol. 41, Iss: 1, pp 153-180

TL;DR: A general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs, which includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set.

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Abstract: This paper describes a general technique that can be used to obtain approximation schemes for various NP-complete problems on planar graphs. The strategy depends on decompos- ing a planar graph into subgraphs of a form we call k-outerplanar. For fixed k, the problems of interest are solvable optimally in linear time on k-outerplanar graphs by dynamic programming. For general planar graphs, if the problem is a maximization problem, such as maximum independent set, this technique gives for each k a linear time algorithm that produces a solution whose size is at least k/(k + 1)optimal. If the problem is a minimization problem, such as minimum vertex cover, it gives for each k a linear time algorithm that produces a solution whose size is at most (k + 1)/k optimal. Taking k = (c log log nl or k = (c log nl, where n is the number of nodes and c is some constant, we get polynomial time approximation algorithms whose solution sizes converge toward optimal as n increases. The class of problems for which this approach provides approximation schemes includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominat- ing set, and minimum edge dominating set. For these and certain other problems, the proof of solvability on k-outerplanar graphs also enlarges the class of planar gmphs for which the problems are known to be solvable in polynomial time.

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Book•

Invitation to fixed-parameter algorithms

[...]

Rolf Niedermeier

01 Jan 2006

TL;DR: This paper discusses Fixed-Parameter Algorithms, Parameterized Complexity Theory, and Selected Case Studies, and some of the techniques used in this work.

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Abstract: PART I: FOUNDATIONS 1. Introduction to Fixed-Parameter Algorithms 2. Preliminaries and Agreements 3. Parameterized Complexity Theory - A Primer 4. Vertex Cover - An Illustrative Example 5. The Art of Problem Parameterization 6. Summary and Concluding Remarks PART II: ALGORITHMIC METHODS 7. Data Reduction and Problem Kernels 8. Depth-Bounded Search Trees 9. Dynamic Programming 10. Tree Decompositions of Graphs 11. Further Advanced Techniques 12. Summary and Concluding Remarks PART III: SOME THEORY, SOME CASE STUDIES 13. Parameterized Complexity Theory 14. Connections to Approximation Algorithms 15. Selected Case Studies 16. Zukunftsmusik References Index

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1,730 citations

Book•

Parameterized Algorithms

[...]

Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, Saket Saurabh - Show less +4 more

27 Jul 2015

TL;DR: This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area, providing a toolbox of algorithmic techniques.

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Abstract: This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area. The book covers many of the recent developments of the field, including application of important separators, branching based on linear programming, Cut & Count to obtain faster algorithms on tree decompositions, algorithms based on representative families of matroids, and use of the Strong Exponential Time Hypothesis. A number of older results are revisited and explained in a modern and didactic way. The book provides a toolbox of algorithmic techniques. Part I is an overview of basic techniques, each chapter discussing a certain algorithmic paradigm. The material covered in this part can be used for an introductory course on fixed-parameter tractability. Part II discusses more advanced and specialized algorithmic ideas, bringing the reader to the cutting edge of current research. Part III presents complexity results and lower bounds, giving negative evidence by way of W[1]-hardness, the Exponential Time Hypothesis, and kernelization lower bounds. All the results and concepts are introduced at a level accessible to graduate students and advanced undergraduate students. Every chapter is accompanied by exercises, many with hints, while the bibliographic notes point to original publications and related work.

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1,544 citations

Additional excerpts

...It dates back to the works of Baker [23] and Hochbaum and Maass [265]....
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Journal Article•DOI•

A partial k -arboretum of graphs with bounded treewidth

[...]

Hans L. Bodlaender¹•Institutions (1)

Utrecht University¹

06 Dec 1998-Theoretical Computer Science

TL;DR: This survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class.

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1,197 citations

Cites background from "Approximation algorithms for NP-com..."

...Another, equivalent de nition of the notion `outerplanarity' is the following (see [7])....
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...The maximum distance of a vertex to the outer face is called the outerplanarity [7]....
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Book•DOI•

Complexity and Approximation

[...]

Giorgio Ausiello, Alberto Marchetti-Spaccamela, Pierluigi Crescenzi, Giorgio Gambosi, Marco Protasi, Viggo Kann - Show less +2 more

01 Jan 1999

980 citations

Cites background or methods from "Approximation algorithms for NP-com..."

...In [Baker, 1994], by means of a new technique, it is proved a more general result for such problem....
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...Admits a PTAS for planar graphs [Baker, 1994] and for A-precision unit disk graphs [Hunt III et al....
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...Admits a PTAS for planar graphs [Baker, 1994], but does not admit an FPTAS [Berman, Johnson, Leighton, Shor, and Snyder, 1990]....
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...Comment: Admits a PTAS for planar graphs [Baker, 1994] and for A-precision unit disk graphs [Hunt III et al....
[...]
...Admits a PTAS for planar graphs [Baker, 1994] and for unit disk graphs [Hunt III et aI....
[...]

Journal Article•DOI•

Approximation schemes for covering and packing problems in image processing and VLSI

[...]

Dorit S. Hochbaum¹, Wolfgang Maass²•Institutions (2)

University of California, Berkeley¹, University of Illinois at Chicago²

01 Jan 1985-Journal of the ACM

TL;DR: The unified technique that is introduced here, referred to as the shifting strategy, is applicable to numerous geometric covering and packing problems and how it varies with problem parameters is illustrated.

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Abstract: A unified and powerful approach is presented for devising polynomial approximation schemes for many strongly NP-complete problems. Such schemes consist of families of approximation algorithms for each desired performance bound on the relative error e > O, with running time that is polynomial when e is fixed. Though the polynomiality of these algorithms depends on the degree of approximation e being fixed, they cannot be improved, owing to a negative result stating that there are no fully polynomial approximation schemes for strongly NP-complete problems unless NP = P.The unified technique that is introduced here, referred to as the shifting strategy, is applicable to numerous geometric covering and packing problems. The method of using the technique and how it varies with problem parameters are illustrated. A similar technique, independently devised by B. S. Baker, was shown to be applicable for covering and packing problems on planar graphs.

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820 citations

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References

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Johnson: Computers and Intractability-A Guide to the Theory of NP-Completeness

[...]

Michael Randolph Garey

01 Jan 1979

42,654 citations

Book•

Computers and Intractability: A Guide to the Theory of NP-Completeness

[...]

Michael Randolph Garey, David S. Johnson

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.

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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.

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40,020 citations

Book•

Graph theory

[...]

Frank Harary

01 Jan 1969

16,023 citations

Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran

[...]

Michael Randolph Garey, David S. Johnson

01 Jan 1979

3,381 citations

"Approximation algorithms for NP-com..." refers background in this paper

...Given a graph G = (V, E), is G 3-colorable, that is, can the nodes be colored with three colors such that adjacent nodes are always assigned different colors [Garey and Johnson, 1979, problem GT4]?...
[...]
...…mentioned above are solvable in linear time on k-outerplanar graphs for fixed k, as are the following (all problems in this paragraph appear in Garey and Johnson, 1979): minimum maximal matching [problem GTIO], 3-colorability [problem GT4], Hamiltonian circuit [problem GT37], and…...
[...]
...Given a graph G = (V, E) and a positive integer K s IV 1, is there a dominating set of size K or less for G, that is, a subsetV cVwithIV Is Ksuchthatforallu=V V thereisaL= V for which (u, ~ ) E E [Garey and Johnson, 1979, problem GT2]?...
[...]
...Li~?lLlm V37L X COL W, Given a graph G = (V, E) and a positive integer h s I V I , is there a vertex cover of size K or less for G, that is, a subset V c V with IV I K such that for each edge (u, ~) = E at least one of LL and LI belongs to V [Garey and Johnson, 1979, problem GT1]?...
[...]
...Given a graph G = (V, E) and a positive integer K s I V I , can the vertices of G be partitioned into r s K disjoint sets Vl, v,,.. ., ~ such that the subgraph induced by each ~ is a perfect matching (consists entirely of vertices of degree one) [Garey and Johnson, 1979, problem GT16]?...
[...]

Journal Article•DOI•

A Separator Theorem for Planar Graphs

[...]

Richard J. Lipton, Robert E. Tarjan

01 Apr 1979-Siam Journal on Applied Mathematics

TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A, B, C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than ${2n/3}$ vertices, and C contains no more than $2.

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Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than ${2n / 3}$ vertices, and C contains no more than $2\sqrt 2 \sqrt n $ vertices. We exhibit an algorithm which finds such a partition A, B, C in $O( n )$ time.

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1,312 citations