发表机构
Université de Sherbrooke（舍布鲁克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过从3-SAT的归约证明梳状不等式分离问题是NP难的，即使输入受限时亦然，并讨论了其对优化与近似的影响。
AI 中文摘要
梳状不等式是对称旅行商问题的重要割平面，然而其精确分离的复杂性一直是多面体组合学中一个长期悬而未决的问题。我们提出了一个从3-SAT的归约，证明判定梳状不等式是否被违反是NP完全的，且相应的分离问题是NP难的。即使当输入向量属于子回路消除多面体、每条边的值为零、二分之一或一，且支撑图是非平面且最大度为四时，该结果依然成立。该归约构造了一个图，其中六顶点梯子小工具编码布尔关系，三次图强制每个逻辑变量出现之间的一致性。对于具有$v$个变量和$m$个子句的命题逻辑公式，构造的图有$40v+70m+30$个顶点和线性数量的正边。该归约还证明了当每个允许的齿有两个或四个顶点且齿的一半在手柄中时，硬度依然成立。我们讨论了其对近似最大梳状违反、梳状松弛优化的影响，并解释了为什么分离的硬度不会自动转移到更大的不等式族。
英文摘要
Comb inequalities are important cutting planes for the symmetric travelling salesman problem, yet the complexity of their exact separation has remained a longstanding question in polyhedral combinatorics. We present a reduction from 3-SAT proving that deciding whether a comb inequality is violated is NP-complete and that the corresponding separation problem is NP-hard. This result holds even when the input vector belongs to the subtour elimination polytope, every edge value is zero, one half or one and the support graph is nonplanar with maximum degree four. The reduction constructs a graph in which six-vertex ladder gadgets encode Boolean relations and cubic graphs enforce consistency among occurrences of each logical variable. For a propositional logic formula with $v$ variables and $m$ clauses, the constructed graph has $40v+70m+30$ vertices and a linear number of positive edges. The reduction also proves hardness when every permitted tooth has two or four vertices with half of the tooth in the handle. We discuss consequences for approximating the maximum comb violation, optimization over the comb relaxation and explain why hardness of separation does not automatically transfer to larger inequality families.
