发表机构
Toronto Metropolitan University（多伦多都会大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究整数线上两个智能体的对称会合问题，引入无记忆自距离策略，证明已知距离下最优期望会合时间为二次阶，并构造未知距离下近二次的通用策略。
AI 中文摘要
直线上的对称会合是一个搜索问题，其中两个智能体初始相距 $2d$，必须遵循相同的随机策略以尽快相遇。在标准模型中，智能体可以根据整个执行历史来调整其行动，并且已知距离和未知距离两种变体均具有期望会合时间 $\Theta(d)$。我们通过引入无记忆自距离策略来研究记忆的作用，在这种策略中，智能体的决策仅取决于其相对于自身起始位置的位置。对于初始间隔为 $2d$ 的情况，令 $R_d$ 表示已知距离设置中的最优无记忆期望会合时间。我们开发了两种基于吸收马尔可夫链的有限状态框架。截断链通过有限支撑策略给出可计算的上界，而弱窥视链通过揭示信息松弛给出下界。两者结合提供了一种证明最优性的机制。利用该机制，我们精确确定了 $R_1$，并证明它由有限支撑策略达到。对于 $d=2,\ldots,6$，数值优化给出相同的截断结构和目标值，产生低于 $7.83d^2$ 的严格上界。我们不证明计算出的弱窥视最小化器是全局的，但计算的稳定性使我们推测它们是全局的，在这种情况下，相应的截断策略是最优的。我们还证明了 $R_d=\Theta(d^2)$。在未知距离设置中，我们构造了一个与 $d$ 无关的通用策略，对于每个固定的 $\eta>0$，其期望会合时间为 $O(d^{2+\eta})$。因此，在记忆限制下，已知距离的会合时间变为二次，而即使不知道 $d$，近二次性能仍然可能实现。渐近分析使用了生灭马尔可夫链及其电网解释。
英文摘要
Symmetric rendezvous on the line is a search problem in which two agents, initially placed at distance $2d$, must follow the same randomized strategy to meet as quickly as possible. In the standard model, agents may condition their actions on the entire execution history, and both the known- and unknown-distance variants admit expected rendezvous time $Θ(d)$. We study the role of memory by introducing oblivious self-distance strategies, in which an agent's decision depends only on her position relative to her own starting location. For an initial separation of $2d$, let $R_d$ denote the optimal oblivious expected rendezvous time in the known-distance setting. We develop two finite-state frameworks based on absorbing Markov chains. Truncated chains give computable upper bounds through finite-support strategies, while weak-peek chains give lower bounds through a revealed-information relaxation. Together, they provide a mechanism for certifying optimality. Using that mechanism, we determine $R_1$ exactly and prove that it is attained by a finite-support strategy. For $d=2,\ldots,6$, numerical optimization gives the same truncation structure and objective values, yielding rigorous upper bounds below $7.83d^2$. We do not prove that the computed weak-peek minimizers are global, but the stability of the computations leads us to conjecture that they are, in which case the corresponding truncated strategies are optimal. We also prove that $R_d=Θ(d^2)$. In the unknown-distance setting, we construct a universal strategy, independent of $d$, with expected rendezvous time $O(d^{2+η})$ for every fixed $η>0$. Thus, under the memory restriction, the known-distance rendezvous time becomes quadratic, while near-quadratic performance remains possible even without knowing $d$. The asymptotic analysis uses birth-death Markov chains and their electrical-network interpretation.
