Approximation Schemes Using L-Reductions

We present parallel approximation schemes for a large class of problems in logic and graph theory, when these problems are restricted to planar graphs and delta-near-planar graphs. Our results are based on the positive use of L-reductions, and the decomposition of a given graph into subgraphs for which the given problem can be solved optimally in polynomial time. The problems considered include MAX 3SAT, MAX SAT(S), maximum independent set, and minimum dominating set. Our NC-approximation schemes exhibit the same time versus performance tradeoff as those of Baker [Ba83]. For delta-near-planar graphs, this is the first time that approximation schemes of any kind have been obtained for these problems.