Planar Graph Orientation Frameworks, Applied to KPlumber and Polyomino Tiling

TL;DR

This paper studies graph orientation problems with local vertex constraints, providing complexity classifications, polynomial algorithms for KPlumber, and new NP-completeness results for polyomino tiling.

Contribution

It offers a comprehensive complexity dichotomy for symmetric graph orientation problems and applies these results to solve open problems in tiling and KPlumber.

Findings

Full P vs. NP-complete dichotomy for symmetric vertex types

Polynomial algorithms for KPlumber problem

New NP-completeness results for tetromino tiling

Abstract

Given a graph, when can we orient the edges to satisfy local constraints at the vertices, where each vertex specifies which local orientations of its incident edges are allowed? This family of graph orientation problems is a special kind of SAT problem, where each variable (edge orientation) appears in exactly two clauses (vertex constraints) -- once positively and once negatively. We analyze the complexity of many natural vertex types (patterns of allowed vertex neighborhoods), most notably all sets of symmetric vertex types which depend on only the number of incoming edges. In many scenarios, including Planar and Non-Planar Symmetric Graph Orientation with constants, we give a full dichotomy characterizing P vs. NP-complete problem classes. We apply our results to obtain new polynomial-time algorithms, resolving a 20-year-old open problem about KPlumber; to simplify existing NP-hardness proofs for tiling with trominoes; and to prove new NP-completeness results for tiling with tetrominoes.