The complexity of planar counting problems

• Published in: SIAM journal on computing Vol. 27; no. 4; pp. 1142 - 1167
• Main Authors: HUNT, H. B, MARATHE, M. V, RADHAKRISHNAN, V, STEARNS, R. E
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.08.1998
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Approximation; Computer science; Computer science; control theory; systems; Exact sciences and technology; Feedback; Information retrieval. Graph; Theoretical computing; Embedding; NP complete problem; Graph theory; Combinatorial problem; Computational complexity
• ISSN: 0097-5397; 1095-7111
• DOI: 10.1137/S0097539793304601

We prove the #P-hardness of the counting problems associated with various satisfiability, graph, and combinatorial problems, when restricted to planar instances. These problems include 3Sat, 1-3Sat, 1-Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Vertex Set, X3C, Partition Into Triangles, and Clique Cover. We also prove the NP-completeness of the Ambiguous Satisfiability} problems [J. B. Saxe, Two Papers on Graph Embedding Problems, Tech. Report CMU-CS-80-102, Dept. of Computer Science, Carnegie Mellon Univ., Pittsburgh, PA, 1980] and the DP-completeness (with respect to random polynomial reducibility) of the unique satisfiability problems [L. G. Valiant and V. V. Vazirani, NP is as easy as detecting unique solutions, in Proc. 17th ACM Symp. on Theory of Computing, 1985, pp. 458--463] associated with several of the above problems, when restricted to planar instances. Previously, very few #P}-hardness results, no {\sf NP}-hardness results, and no DP-completeness results were known for counting problems, ambiguous satisfiability problems, and unique satisfiability problems, respectively, when restricted to planar instances. Assuming {\sf P \neq $ NP}, one corollary of the above results is that there are no $\epsilon$-approximation algorithms for the problems of maximizing or minimizing a linear objective function subject to a planar system of linear inequality constraints over the integers.