Showing papers on "Vertex cover published in 1982"
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].
796 citations
TL;DR: A heuristic is proposed that delivers in O(n^3 ) steps a solution for the set covering problem the value of which does not exceed the maximum number of sets covering an element times the optimal value.
Abstract: We propose a heuristic that delivers in $O(n^3 )$ steps a solution for the set covering problem the value of which does not exceed the maximum number of sets covering an element times the optimal value.
503 citations
TL;DR: It is shown in a umfied manner that there exist hnearume algorithms for many combinatorial problems ff an input graph is restricted to the class of series-parallel graphs.
Abstract: A series-parallel graph can be constructed from a certain graph by recurslvely applying \"series\" and \"parallel\" connections The class of such graphs, which Is a well-known model of series-parallel electrical networks, is a subclass of planar graphs It is shown in a umfied manner that there exist hnearume algorithms for many combinatorial problems ff an input graph is restricted to the class of series-parallel graphs. These include 0) the decision problem with respect to a property characterized by a finite number of forbidden graphs, (u) the mlmmum edge (vertex) deletion problem with respect to the same property as above, and (Ul) the generalized matching problem Consequently, the following problems, among others, prove to be hnear-tlme computable for the class of series-parallel graphs. (I) the minimum vertex cover problem, (2) the maximum outerplanar (reduced) subgraph problem, (3) the minimum feedback vertex set problem, (4) the maximum (induced) hne-subgraph problem, (5) the maximum matching problem, and (6) the maximum disjoint triangle problem.
293 citations
108 citations
05 May 1982
TL;DR: The approximation problem for vertex cover of n-vertex planar graphs is treated and two results are presented: a linear time approximation algorithm and an O(n log n) time approximation scheme.
Abstract: The approximation problem for vertex cover of n-vertex planar graphs is treated. Two results are presented: (1) A linear time approximation algorithm for which the (error) performance bound is 2/3. (2) An O(n log n) time approximation scheme.
51 citations
TL;DR: A branch and bound algorithm for solving the Steiner problem in graphs is presented together with an interesting application to a problem in molecular evolution.
Abstract: The Steiner problem in graphs is concerned with finding a set of edges with minimum total weight which connects a given subset of points in a weighted graph. A branch and bound algorithm for solving this problem is presented together with an interesting application to a problem in molecular evolution. Computational experience gained in using the algorithm compares favorably, for certain classes of graphs, with that of existing methods.
47 citations
TL;DR: This paper presents a polynomial-time approximation algorithm for the maximum independent set problem on planar graphs, and finds an independent set that is necessarily larger in size than half a maximumIndependent set.
Abstract: In this paper we consider the maximum independent set problem in which one would like to find a maximum set of independent (i.e., pairwise nonadjacent) vertices in a given graph. The problem is NP-complete, and still remains so even if we restrict ourselves to the class of planar graphs. It has been conjectured that there exist no polynomial-time exact algorithms for any NP-complete problems. We present a polynomial-time approximation algorithm for the maximum independent set problem on planar graphs. For a given planar graph having any number n of vertices, our algorithm finds, in $O(n\log n)$ time, an independent set that is necessarily larger in size than half a maximum independent set. Thus the absolute worst case ratio of our algorithm is greater than $\tfrac{1}{2}$.
33 citations
TL;DR: The R-domination problem is considered, a generalization of Slater's algorithm, which is applicable for forest graphs and a linear algorithm is presented that solves this problem for block graphs.
15 citations
TL;DR: A fast algorithm for finding minimal feedback vertex sets is presented that uses the principle that any minimal feedback vertices must contain exactly one vertex from each minimal cycle.
Abstract: A fast algorithm for finding minimal feedback vertex sets is presented The algorithm uses the principle that any minimal feedback vertex set must contain exactly one vertex from each minimal cycle Depth-first search is used to implement the algorithm efficiently
1 citations
01 Jan 1982
TL;DR: An overview of the Pattern Recognition problem, with an attention to the class of NP-complete problems, and polynomial problems of interest to P.R.; some hints are given how to escape untractability.
Abstract: The Pattern Recognition (P.R.) problem of the identification of an object from a representation data may be termed as a reduction in the complexity of the representation. Pattern recognizers have to find algorithms which escape the untractability of combinatorial explosion, without being defeated by errors. Thus the recent results obtained in the field of computational complexity are of interest to P.R.. An overview is given, with an attention to the class of NP-complete problems, and polynomial problems of interest to P.R.; some hints are given how to escape untractability.