必需边环覆盖问题：图问题和谜题的ASP-完备性框架
Required-edge Cycle Cover Problem: an ASP-Completeness Framework for Graph Problems and Puzzles
AI总结：
本文提出必需边环覆盖问题，通过ASP-完备性框架证明了多个谜题的复杂性，解决了CGS的开放问题并加强了Kakuro的复杂性分类。
摘要：
证明笔算谜题的NP完全性通常依赖于从组合问题（如可满足性问题SAT）的归约。尽管这些问题的性质已被广泛研究，但它们纯粹的组合性质往往与谜题的几何约束不匹配。在本文中，我们介绍了必需边环覆盖问题（RCCP）——一种混合图上的环覆盖问题（CCP）变种。混合图上的CCP由Seta（2002）研究，以确立谜题Kakuro（又称Cross Sum）的ASP完全性（即在parsimonious归约下的NP完全性），并且在某些条件下已知为ASP完全。我们证明在某些条件下RCCP的ASP完全性，并通过加强已知的混合图CCP的ASP完全性结果作为推论。利用这些结果，我们解决了约束图可满足性（CGS）在特定情况下的ASP完全性，回答了MIT Hardness Group（2024）提出的一个开放问题。我们还表明，即使可用数字集为{1,2,3}，Kakuro仍保持ASP完全，从而完成了其关于最大可用数字和连续空白单元格最大长度的复杂性分类。它加强了之前已知的结果，即Seta（2002）和Ruepp和Holzer（2010）的结果。此外，我们引入了一个等价于受约束RCCP的流模型；该模型允许在矩形网格上密集地铺设gadget，这使我们能够以parsimonious的方式将RCCP归约为各种笔算谜题。通过应用此框架，我们证明了Chocona、Four Cells、Hinge和Shimaguni等谜题的ASP完全性，并将Choco Banana和Five Cells的现有NP完全性结果加强为ASP完全性。
英文摘要：
Proving the NP-completeness of pencil-and-paper puzzles typically relies on reductions from combinatorial problems such as the satisfiability problem (SAT). Although the properties of these problems are well studied, their purely combinatorial nature often does not align well with the geometric constraints of puzzles. In this paper, we introduce the Required-edge Cycle Cover Problem (RCCP) -- a variant of the Cycle Cover Problem (CCP) on mixed graphs. CCP on mixed graphs was studied by Seta (2002) to establish the ASP-completeness (i.e., NP-completeness under parsimonious reductions) of the puzzle Kakuro (a.k.a. Cross Sum), and is known to be ASP-complete under certain conditions. We prove the ASP-completeness of RCCP under certain conditions, and strengthen known ASP-completeness results of CCP on mixed graphs as a corollary. Using these results, we resolve the ASP-completeness of Constraint Graph Satisfiability (CGS) in a certain case, addressing an open problem posed by the MIT Hardness Group (2024). We also show that Kakuro remains ASP-complete even when the available digit set is $\{1, 2, 3\}$, consequently completing its complexity classification regarding the maximum available digit and the maximum lengths of contiguous blank cells. It strengthens previously known results of Seta (2002) and Ruepp and Holzer (2010). Furthermore, we introduce a flow model equivalent to the constrained RCCP; this model allows gadgets to be tiled densely on a rectangular grid, which enables us to reduce RCCP to various pencil-and-paper puzzles in a parsimonious manner. By applying this framework, we prove the ASP-completeness of several puzzles, including Chocona, Four Cells, Hinge, and Shimaguni, and strengthen existing NP-completeness results for Choco Banana and Five Cells to ASP-completeness.
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