We provide the rst nontrivial approximation algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH, the NP-Hard problem of nding a heaviest planar subgraph in an edge-weighted graph G. This problem has applications in circuit layout, facility layout, and graph drawing. No previous algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH had performance ratio exceeding 1=3, which is obtained by any algorithm that produces a maximum weight spanning tree in G. Based on the Berman-Ramaiyer Steiner tree algorithm, the new algorithm has performance ratio at least 1=3 + 1=72. We also show that if G is complete and its edge weights satisfy the triangle inequality, then the performance ratio is at least 3=8. Furthermore, we derive the rst nontrivial performance ratio (7=12 instead of 1=2) for the NP-Hard MAXIMUM WEIGHT OUTERPLANAR SUBGRAPH problem. 1
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