🤖 AI Summary
This study addresses the graph edge orientation problem, which involves assigning directions to edges such that each vertex satisfies a prescribed local out-degree constraint. By modeling the problem as a restricted SAT instance—where each edge variable appears in exactly two vertex constraints—the authors establish a complete computational complexity dichotomy for symmetric vertex types parameterized by required in-degree. They precisely characterize the boundary between polynomial-time solvability and NP-completeness on both planar and non-planar graphs. As a consequence, they resolve the long-standing open problem of KPlumber, presenting a new polynomial-time algorithm for it. Furthermore, their framework simplifies the existing NP-hardness proof for triomino tiling and provides the first proof of NP-completeness 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.