Planar Formulae and Their Uses
01 May 1982-SIAM Journal on Computing (Society for Industrial and Applied Mathematics)-Vol. 11, Iss: 2, pp 329-343
TL;DR: Using these results, it is able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness forPlanar generalized geography.
Abstract: We define the set of planar boolean formulae, and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is NP-complete. Using these results, we are able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness for planar generalized geography.The NP-completeness of planar node cover and planar Hamiltonian circuit and line were first proved elsewhere [M. R. Garey and D. S. Johnson, The rectilinear Steiner tree is NP-complete, SIAM J. Appl. Math., 32 (1977), pp. 826–834] and [M. R. Garey, D. S. Johnson and R. E. Tarjan, The planar Hamilton circuit problem is NP-complete, SIAM J. Comp., 5 (1976), pp. 704–714].
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TL;DR: In this article, it was shown that probabilistic inference using belief networks is NP-hard and that it seems unlikely that an exact algorithm can be developed to perform inference efficiently over all classes of belief networks and that research should be directed toward the design of efficient special-case, average-case and approximation algorithms.
1,877 citations
Book•
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.
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.
1,544 citations
Cites result from "Planar Formulae and Their Uses"
...The technique of proving NP-hardness of problems on planar graphs via Planar 3SAT was proposed by Lichtenstein [320], and it unified several results that were obtained before....
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TL;DR: It is shown that many standard graph theoretic problems remain NP-complete on unit disks, including coloring, independent set, domination, independent domination, and connected domination; NP-completeness for the domination problem is shown to hold even for grid graphs, a subclass of unit disk graphs.
1,525 citations
TL;DR: This is the fourteenth edition of a quarterly column that provides continuing coverage of new developments in the theory of NP-completeness, and readers who have results they would like mentioned (NP-hardness, PSPACE- hardness, polynomialtime-solvability, etc.), or open problems they wouldlike publicized, should send them to David S. Johnson.
857 citations
Cites background from "Planar Formulae and Their Uses"
...The proof of the NP-completeness of PLANAR 3-SAT has at last made its formal appearance [28], after four years as an ‘‘unpublished manuscript....
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References
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30 Apr 1974
TL;DR: This paper shows that a number of NP-complete problems remain NP- complete even when their domains are substantially restricted, and determines essentially the lowest possible upper bounds on node degree for which the problems remainNP-complete.
Abstract: It is widely believed that showing a problem to be NP-complete is tantamount to proving its computational intractability. In this paper we show that a number of NP-complete problems remain NP-complete even when their domains are substantially restricted. First we show the completeness of SIMPLE MAX CUT (MAX CUT with edge weights restricted to value 1), and, as a corollary, the completeness of the OPTIMAL LINEAR ARRANGEMENT problem. We then show that even if the domains of the NODE COVER and DIRECTED HAMILTONIAN PATH problems are restricted to planar graphs, the two problems remain NP-complete, and that these and other graph problems remain NP-complete even when their domains are restricted to graphs with low node degrees. For GRAPH 3-COLORABILITY, NODE COVER, and UNDIRECTED HAMILTONIAN CIRCUIT, we determine essentially the lowest possible upper bounds on node degree for which the problems remain NP-complete.
648 citations