Showing papers on "Dominating set published in 1985"
TL;DR: A 2-approximation algorithm for the k-center problem with triangle inequality is presented, the key combinatorial object used is called a strong stable set, and the NP-completeness of the corresponding decision problem is proved.
Abstract: In this paper we present a 2-approximation algorithm for the k-center problem with triangle inequality. This result is “best possible” since for any δ < 2 the existence of δ-approximation algorithm would imply that P = NP. It should be noted that no δ-approximation algorithm, for any constant δ, has been reported to date. Linear programming duality theory provides interesting insight to the problem and enables us to derive, in O|E| log |E| time, a solution with value no more than twice the k-center optimal value.
A by-product of the analysis is an O|E| algorithm that identifies a dominating set in G2, the square of a graph G, the size of which is no larger than the size of the minimum dominating set in the graph G. The key combinatorial object used is called a strong stable set, and we prove the NP-completeness of the corresponding decision problem.
992 citations
TL;DR: A polynomial algorithm and a min-max theorem for the cardinality Steiner problem in strongly chordal graphs and a polynometric algorithm for the weighted connected dominating set problem in series-parallel graphs are given.
Abstract: We consider Steiner tree problems and connected dominating set problems for several classes of graphs. We give a polynomial algorithm and a min-max theorem for the cardinality Steiner problem in strongly chordal graphs and a polynomial algorithm for the weighted connected dominating set problem in series-parallel graphs. We establish simple direct transformations between Steiner problems and connected domination problems for several classes of graphs and establish related NP-completeness results.
107 citations
TL;DR: Here the following conjecture of Fink and Jacobson is proved: for any simple graph G and any positive integer k, γ k (G ) ≤ β k ( G ).
61 citations
TL;DR: The domatic number problem is solved in O( n 2 log n ) time when G is a proper circular arc graph, and it is shown NP-complete for general circular arc graphs.
57 citations
Book•
01 Jan 1985
TL;DR: Strongly Equivalent Directed Hypergraphs (G. Ausiello, A. D'Atri and D. Sacca). A Local-Ratio Theorem for Approximating the Weighted Vertex Cover Problem (R. Bar-Yehuda and S. Even). Dynamic Programming Parallel Procedures for SIMD Architectures (P. Bertolazzi). Simulations Among Classes of Random Access Machines and Equivalence Among Numbers Succinctly Represented.
Abstract: Strongly Equivalent Directed Hypergraphs (G. Ausiello, A. D'Atri and D. Sacca). A Local-Ratio Theorem for Approximating the Weighted Vertex Cover Problem (R. Bar-Yehuda and S. Even). Dynamic Programming Parallel Procedures for SIMD Architectures (P. Bertolazzi). Simulations Among Classes of Random Access Machines and Equivalence Among Numbers Succinctly Represented (A. Bertoni, G. Mauri and N. Sabadini). A Realistic Approach to VLSI Relational Data-Base Processing (M.A. Bonucelli et al.). On Counting BECS (R. Casas, J. Diaz and M. Verges). Rigid Extensions of Graph Maps (I.S. Filotti). Algebraic Methods for Trie Statistics (Ph. Flajolet, M. Regnier and D. Sotteau). Easy Solutions for the K-Center Problem or the Dominating Set Problem on Random Graphs (D.S. Hochbaum). Network Design with Multiple Demand: A New Approach (M. Lucertini and G. Paletta). How to Find Long Paths Efficiently (B. Monien). Compact Channel Routing of Multiterminal Nets (M. Sarrafzadeh and F.P. Preparata). Consistency of Quadratic Boolean Equations and the Konig-Egervary Property for Graphs (B. Simeone). On Some Relationships Between Combinatorics and Probabilistic Analysis (M. Talamo, A. Marchetti-Spaccamela and M. Protasi). A Threshold for Multiple Edge Coverings in Random Hypergraphs (C. Vercellis).
38 citations
TL;DR: This paper is concerned with the probabilistic behavior of three NP-complete problems: the K-center, the dominating set, and the set covering problem and the asymptotic values of the optimal solutions and the probability expressions for the interval of values where the optimal solution value lies for every finite problem size.
Abstract: This paper is concerned with the probabilistic behavior of three NP-complete problems: the K-center, the dominating set, and the set covering problem. There are two main contributions: one is the asymptotic values of the optimal solutions and the probability expressions for the interval of values where the optimal solution value lies for every finite problem size. The other contribution is the proof that any random feasible solution is asymptotically optimal with probability that is asymptotically 1.
16 citations
TL;DR: In this article, an O(n 2 )-approximation algorithm for finding a minimum cardinality dominating cycle in a permutation graph is presented, and it is shown that such a cycle always has an even number of vertices unless it is isomorphic to C"3.
14 citations
TL;DR: Several problems are shown to be log space complete, when restricted to bandwidth f(n), for the subclass of NP defined by nondeterministic polynomial time bounded Turing machines with a simultaneous f( n) space restriction, denoted by NTISP(poly, f(N).
Abstract: Several problems are shown to be log space complete, when restricted to bandwidth f(n), for the subclass of NP defined by nondeterministic polynomial time bounded Turing machines with a simultaneous f(n) space restriction, denoted by NTISP(poly, f(n)). These problems are NOT-ALL-EQUAL 3SAT, MONOCHROMATIC TRIANGLE, CUBIC SUBGRAPH, DOMINATING SET, ONE-IN-THREE 3SAT and MONOTONE 3SAT. The problems DOMATIC NUMBER, PARTITION INTO FORESTS and DISJOINT CONNECTING PATHS restricted to bandwidth f(n) are shown to be log space hard for NTISP(poly, f(n)). Their membership in the class NTISP(poly, f(n)) is currently open. As one application of these results, we note that the first six of the problems mentioned are examples of NSPACE(log n) complete problems when restricted to bandwidth log n.
2 citations