发表机构
Adam Mickiewicz University; Poznań University of Economics and Business（亚当·密茨凯维奇大学; 波兹南经济商业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文精确刻画了可匹配半匹配中局部最优与全局最优总完成时间的最坏比率，给出了闭式公式，并证明最坏情况在简单树上即可达到，为有界局部搜索提供了尖锐保证。
AI 中文摘要
将任务分配给服务器的方案可以抵抗每一次小的改进，但仍然使任务等待的时间超过必要。当每个任务需要一单位的服务且资格约束允许所有任务使用不同的服务器时，我们精确地确定了这种方案可能低效的程度。对于每个移动规模 $r$ 和最大当前服务器负载 $K$，我们给出了局部最优与全局最优总完成时间之间最坏比率的闭式公式。这里的局部最优性允许任何改变至多 $r$ 个任务的可行重新分配。每个有限容量界限都在一棵树上达到，其中每个任务至多有两个符合条件的服务器。因此，最坏行为已经出现在简单的资格约束下。在容量上限为 2 时，精确比率为 $1+1/(r+2)$，在具有 $r+2$ 个任务的路径上达到。在没有容量上限的情况下，单任务移动的最坏情况上确界为 $3/2$，双任务移动的约为 $1.294503159$；随着 $r$ 的增长，其超过 1 的量为 $1/(r+2)+O(2^{-r}/r)$。证明使用了比较图上的显式有理势函数和匹配的极值构造。这些结果为有界规模的局部搜索在可匹配半匹配上提供了尖锐的保证，包括在度约束下的精确保证。
英文摘要
An assignment of tasks to servers can resist every small improvement and still make tasks wait longer than necessary. We determine exactly how inefficient such an assignment can be when each task requires one unit of service and the eligibility constraints permit all tasks to use distinct servers. For every move size $r$ and maximum current server load $K$, we give a closed formula for the worst ratio between locally optimal and globally optimal total completion time. Local optimality here allows every feasible reassignment changing at most $r$ tasks. Every finite-cap bound is attained on a tree where each task has at most two eligible servers. Thus the worst behavior already occurs under simple eligibility constraints. At load cap two, the exact ratio is $1+1/(r+2)$, attained on a path with $r+2$ tasks. Without a load cap, the worst-case supremum is $3/2$ for single-task moves and approximately $1.294503159$ for two-task moves; its excess above one is $1/(r+2)+O(2^{-r}/r)$ as $r$ grows. The proof uses an explicit rational potential on a comparison graph and matching extremal constructions. These results give sharp guarantees for bounded-size local search on matchable semi-matchings, including exact guarantees under degree bounds.
Comments11 pages. Verification code and data: https://github.com/mgalazka84/semimatching-locality-gaps
