Worst-case ration for planar graphs and the method of induction on faces

Abstract

The fact that several inlportant combinatorial optimiz.ation problems are NP-colnplete has motivated research on the worst-case analysis of approximation heuristics for these problems [Jol, GIl, Ch). TIlcse inve.stigations have produced some very interesting results" and considerable insight has been gained by no\v into the power and limitations of existing techniques. The most inlportnnt paradigm in this area is bin packing [JDGGU, J02, Yao, GJ2, GJ3] and its generalizations [GOJY, CGJT). '[his is so because of the elegance and depth of the cOlnbinatorial arguments employed in the proofs of the upper bounds, and the intricate constnlctions of exanlples that achieve them. Despite the presence of the unifying cotlcept of a weighting [unction, the arguments are usually ingenious yet ad hoc, and the construction of worst-case exmnp)es is largely decoupled from the upper bounding process. In this paper we present a fanlily of results concerning certain extrenlal properties of planar graphs. In particular we show the follo\ving: (1) The greedy heuristic (i.e., repeatedly pick the node with smallest degree and delete its neighborhood) applied to a planar graph with n nodes yields an independent set of size at least 4n/21. (2) l'he greedy heuristic yields an independent set at )etlst 23/63 times the optimum. (3) A planar graph with n nodes and minimum degree 3 has ahvays a nzatching with fewer than n/3 free nodes.