发表机构
CNRS; Université Lyon 1; INSA Lyon; University of Waterloo; American University of Beirut（法国国家科学研究中心; 里昂第一大学; 里昂国立应用科学学院; 滑铁卢大学; 贝鲁特美国大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究单步移动标记令牌路由问题的复杂性，证明在网格和平面图上NP完全，在树上以边重数为参数W[1]难，但以最大度和候选重数为参数可固定参数求解。
AI 中文摘要
在中性原子量子计算机中，原子沿着空位路径移动到目标位置，且一个目标位置可能为一种原子保留。受此任务启发，我们引入单步移动标记令牌路由问题：图的每个源顶点和目标顶点都被赋予一组标签，令牌占据源顶点。一个解包含一个匹配，将每个源顶点分配给一个兼容的目标顶点（其标签集与源顶点的标签集相交），为每个匹配对指定一条路径，以及一个移动顺序，使得在移动令牌时，其路径上不包含其他令牌。已知当每个源顶点与每个目标顶点都兼容时，该问题可在多项式时间内求解；而当每个源顶点恰好与一个目标顶点兼容时，该问题在网格图上是NP完全的。我们证明，即使在某个解具有两两边不相交的路径的情况下，后者在网格图和最大度为四的平面图上仍然是NP完全的。在树上，已知该问题即使对于最大度为三的情况也是NP完全的。我们通过解边重数（一个解中共享一条边的路径的最大数量）和候选边重数（共享一条边的兼容对的最大数量）来研究树。我们证明，在最大度为三的树上，即使给定移动顺序，该问题以解边重数为参数是W[1]难的；在无界度的树上，即使候选边重数至多为八，该问题也是NP完全的。我们证明在树上，该问题以最大度和候选边重数为参数是固定参数可解的，也以候选顶点重数（顶点处的相同计数）为参数是固定参数可解的。除非P=NP，否则最大度和候选边重数都不能省略。
英文摘要
In neutral-atom quantum computers, atoms are moved to target positions along paths of empty positions, and a target position may be reserved for one species of atom. Motivated by this task, we introduce Single-Move Labeled Token Routing: every source and every target vertex of a graph is assigned a set of labels, and tokens occupy the sources. A solution consists of a matching that assigns each source to a compatible target (one whose label set intersects its own), a route for each matched pair, and a movement order in which, when a token is moved, its route contains no other token. The problem is known to be polynomial-time solvable when every source is compatible with every target, and $\mathsf{NP}$-complete on grid graphs when each source is compatible with exactly one target. We prove that the latter case remains $\mathsf{NP}$-complete on grids and on planar graphs of maximum degree four even when some solution has pairwise edge-disjoint routes. On trees, the problem is known to be $\mathsf{NP}$-complete even for maximum degree three. We study trees through the solution edge multiplicity, the largest number of routes of a solution sharing an edge, and the candidate edge multiplicity, the largest number of compatible pairs whose paths share an edge. We prove that on trees of maximum degree three, the problem is $\mathsf{W}[1]$-hard parameterized by a bound on the solution edge multiplicity, even when a movement order is given, and that on trees of unbounded degree, it is $\mathsf{NP}$-complete even when the candidate edge multiplicity is at most eight. We show that on trees the problem is fixed-parameter tractable parameterized by the maximum degree together with the candidate edge multiplicity, and also by the candidate vertex multiplicity, the same count at vertices. Unless $\mathsf{P}=\mathsf{NP}$, neither the maximum degree nor the candidate edge multiplicity can be omitted.
Comments53 pages, 8 figures
