Planar Graph Orientation Frameworks, Applied to KPlumber and Polyomino Tiling

TL;DR

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, is analyzed.

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.

Figures (27)

• Figure 1: Left: The two satisfying assignments of a synchronizer vertex. Right: The two satisfying assignments of a $6$-alternator vertex.
• Figure 2: Left: $f$-equalizers simulate a synchronizer. Right: Degree-$3$ or -$4$ duplicators of net flow $\pm1$ or $\pm2$ also simulate a synchronizer.
• Figure 3: Left: Linking copies of a duplicator with synchronizers. Right: Simulating any duplicator of net flow $\pm f$. A synchronizer is placed at every red dot.
• Figure 4: Constructions used in Lemma \ref{['lem:crossover']}. Vertices labeled $S$ denote synchronizers, $a/b$ denote $a$-in-$b$, $=$ denote equalizers.
• Figure 5: The structure in reduction rule \ref{['item:rule']}. Green vertices/faces denote degree constraints.

Theorems & Definitions (75)

• Definition 2.1
• Theorem 2.1: Schaefer's dichotomy schaefer1978
• Theorem 2.2: Planar Schaefer's dichotomy delta-matroid-planar-schaefer
• Corollary 2.3: Schaefer's dichotomy with constants
• Definition 2.2
• Theorem 2.4: Symmetric $\Gamma$-SAT-E$2$ dichotomy S-in-k-SAT-2
• Lemma 3.1
• proof
• Lemma 3.2
• proof : Proof sketch