Approximation Algorithms for NP.Complete Problems on Planar Graphs (Preliminary Version

TLDR: In this paper, the authors present a general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs by decomposing a planar graph into subgraphs of a form called k-outerplanar.

Abstract: ABSTRACf This paper describes a general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs. The strategy depends on decom­ posing a planar graph into subgraphs of a form we call k­ outerplanar. For fixed k, the problems of interest are solv­ able 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 in­ dependent set, this technique gives for each k a linear time algorithm that produces a solution whose size is at least (k -1)/k 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+l)/k optimal. Taking k-cloglogn or k-clogn, where n is the number of nodes and c is some constant, we get po­ lynomial time approximation schemes, i.e. algorithms whose solution sizes converge toward optimal as n increases. The class of problems for which this approach provides approxi­ mation schemes includes maximum independent set, max­ imum tile salvage, partition into triangles, maximum H­ matching, minimum vertex cover, minimum dominating 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 graphs for which the problems are known to be solvable.