The Complexity of Planar Counting Problems

TL;DR

This paper establishes the computational hardness of various counting and satisfiability problems when restricted to planar instances, including #P-hardness, NP-completeness, and D^P-completeness, highlighting the difficulty of approximation in these cases.

Contribution

It provides the first known hardness and completeness results for several counting and satisfiability problems on planar graphs, expanding understanding of their computational complexity.

Findings

#P-hardness of planar counting problems

NP-completeness of ambiguous satisfiability for planar instances

D^P-completeness of unique satisfiability in planar cases

Abstract

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 \begin{romannum} \item[{}] {\sc 3Sat, 1-3Sat, 1-Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Vertex Set, X3C, Partition Into Triangles, and Clique Cover.} \end{romannum} We also prove the {\sf NP}-completeness of the {\sc Ambiguous Satisfiability} problems \cite{Sa80} and the {\sf D$^P$}-completeness (with respect to random polynomial reducibility) of the unique satisfiability problems \cite{VV85} associated with several of the above problems, when restricted to planar instances. Previously, very few {\sf #P}-hardness results, no {\sf NP}-hardness results, and no {\sf D$^P$}-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 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.