在线度量旅行商问题：突破√n 壁垒
Online Metric TSP: Beyond the $\sqrt{n}$ Barrier
浏览论文内容
中文总结 AI 辅助
该研究针对在线度量TSP，提出用m=(1+ε)n空间的确定性算法，竞争比达O(log³n/ε)，突破m=n时Θ(√n)的壁垒，同时证明m=n^(1+ε)时的下界，明确空间与竞争力的权衡关系。
中文摘要 AI 辅助
我们研究旅行商问题（TSP）的在线变体，其中n个点按顺序到达，必须插入不断演化的路径中。在允许任意插入的经典场景中，自20世纪70年代以来就已知存在O(log n)竞争比的算法（Rosenkrantz、Stearns和Lewis 1977；Imase和Waxman 1991）。最近，Abrahamsen、Bercea、Beretta、Klausen和Kozma[ESA 2024]提出了在线度量TSP，这是一种更严格的模型：每个到达的点必须分配到大小为m≥n的数组的不同单元格中，最终路径顺序由非空单元格决定；参数m表征算法的空间使用情况。当m=2ⁿ时，该模型可恢复任意插入场景，因此存在O(log n)竞争比的算法。相比之下，当m=n（即每个点的位置在到达时固定）时，Bertram[ESA 2025]最近证明竞争比为Θ(√n)。我们研究这两个极端之间空间使用与竞争力的权衡。我们注意到，作者此前已针对在线排序问题探索过这种权衡，而在线排序问题是线度量下在线度量TSP的特殊情况[SODA 2026]。我们的主要结果是，对于任意ε≤1，使用m=(1+ε)n空间的确定性在线度量TSP算法可达到O(log³n/ε)的竞争比。特别地，将空间从n增加到2n可使竞争比从Θ(√n)提升至O(log³n)。我们补充了一个下界：对于m=n^(1+ε)，当ε≥Ω(log log n / log n)时，任何确定性算法的竞争比为Ω(1/ε)。因此，即使m=O(n·polylog(n))，确定性算法也无法达到常数竞争比。
英文摘要
We study an online variant of the Traveling Salesperson Problem (TSP) in which $n$ points arrive sequentially and must be inserted into an evolving tour. In the classical setting where arbitrary insertions are allowed, an $O(\log n)$-competitive algorithm has been known since the 1970s (Rosenkrantz, Stearns and Lewis 1977, Imase and Waxman 1991). Recently, Abrahamsen, Bercea, Beretta, Klausen, and Kozma [ESA 2024] introduced online metric TSP, a stricter model in which each arriving point must be assigned to a distinct cell of an array of size $m \ge n$, with the final tour order induced by the non-empty cells; the parameter $m$ captures the space usage of the algorithm. When $m = 2^{n}$, this model recovers arbitrary insertions and therefore admits an $O(\log n)$-competitive algorithm. In contrast, when $m = n$, i.e., when each point's position is fixed on arrival, Bertram [ESA 2025] recently showed that the competitive ratio is $Θ(\sqrt{n})$. We investigate the tradeoff between space usage and competitiveness between these extremes. We note that this tradeoff was previously explored by the authors [SODA 2026] for the online sorting problem, which is the special case of online metric TSP on a line metric. Our main result is a deterministic online metric TSP algorithm using $m = (1+ε) n$ space that achieves a competitive ratio of $O(\log^{3} n / ε)$, for any $ε\le 1$. In particular, increasing the space from $n$ to $2n$ improves the competitive ratio from $Θ(\sqrt{n})$ to $O(\log^{3} n)$. We complement this with a lower bound showing that for $m = n^{1+ε}$, any deterministic algorithm has a competitive ratio $Ω(1/ε)$, for all $ε\ge Ω(\log \log n / \log n)$. Consequently, even with $m = O(n \cdot \mathrm{polylog}(n))$, deterministic algorithms cannot achieve a constant competitive ratio.
