平面图定向框架，应用于KPlumber和多形填砌
Planar Graph Orientation Frameworks, Applied to KPlumber and Polyomino Tiling
AI总结：
本文提出了一种平面图定向框架，用于解决KPlumber问题，并证明了多形填砌的NP完全性。
摘要：
给定一个图，当可以将边定向以满足顶点处的局部约束时，即每个顶点指定其邻接边的哪些局部定向是允许的？这一类图定向问题是一种特殊的SAT问题，其中每个变量（边定向）恰好出现在两个子句（顶点约束）中——一次为正，一次为负。我们分析了许多自然顶点类型（允许的顶点邻域模式）的复杂性，特别是所有依赖于仅入边数量的对称顶点类型集。在许多场景中，包括具有常数的平面和非平面对称图定向问题中，我们给出了完全二分法，以区分P与NP完全问题类别。我们应用我们的结果，获得新的多项式时间算法，解决了一个20年的KPlumber开放问题；简化现有的NP难证明，用于三连块填砌；并证明了新的NP完全结果，用于四连块填砌。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
