Problem
Research questions and friction points this paper is trying to address.
Symmetric Rendezvous
Oblivious Strategy
Expected Rendezvous Time
Oblivious Self-Distance Symmetric Rendezvous on the Integer Line
🤖 AI Summary
研究通过引入仅依赖于自身位置的无记忆策略解决对称会合问题，利用吸收马尔可夫链框架证明了最优会合时间，并构造了一种未知距离下的通用策略。
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
oblivious self-distance strategy
absorbing Markov chains
truncated chains
weak-peek chains
birth-death Markov chains
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