Parallel approximation schemes for a class of planar and near planar combinatorial optimization problems

  • Authors:

  • Harry B. Hunt, III;Madhav V. Marathe;Venkatesh Radhakrishnan;S. S. Ravi;Daniel J. Rosenkrantz;Richard E. Stearns

  • Affiliations:

  • Univ. at Albany---SUNY, New York;Los Alamos National Laboratory, Los Alamos, New Mexico;Hewlett-Packard Company, Cupertino, California;Univ. at Albany---SUNY, New York;Univ. at Albany---SUNY, New York;Univ. at Albany---SUNY, New York

  • Venue:
  • Information and Computation
  • Year:
  • 2002

Quantified Score

Hi-index 0.00

Visualization

Abstract

Define a (δ, g)-almost planar graph to be a graph G(V,E)consisting of vertex set V and a genus g layout with at mostδ. \V\ crossover nodes. We study a class of combinatorialoptimization problems formulated as follows. Let X =[x1, x2,xn} be a set of variableseach of which has a finite domain D = {o, 1, ,, poly(n)}. Also, letS be a fixed finite set of finte arity relations {R1,lllRq}. The optimization problem Max-Relation(S) is thefollowing:Given a set of terms {t1,t2,,tm}, where each term ti is ofthe form f(xi) for some f &egr; S, assignvalues to each Xi, 1 n, so as to maximize thenumber of satisfied tems. We show that for each fixed finite set Sand fixed δ, g0, there is an NC-approximation scheme(NCAS) for the problem MAX-RELATION(S) when restricted to instanceswhose bipartite graphs (that represent the variable-termrelationship) are (δ, g)-almost planar. This result inconjundtion with approximation---prserving reductions toMAX-RELATION(S) enables us to obtain NCASs for a number of graphtheoretic and satisfiability problems when restricted to (δ,g)---almost planar instances. Our results provide acharacterization of a class of problems having an NCAS almostplanar instances. Our results provide a characterization of a classof problems having an NCAS (and hence a PTAS).