AI 中文总结
研究在线图探索中随机化能否改善竞争保证，提出随机算法\textsc{RandHeavyTest}，证明其竞争比至多为1.315，与确定性设置分离，给出随机化首个可证优势，还给出相关算法的上下界。
AI 中文摘要
在Kalyanasundaram和Pruhs（1994）引入的在线图探索中，一个智能体必须访问一个初始未知加权图的所有顶点并返回起始位置，图仅在访问的顶点处局部揭示。此前工作仅关注确定性算法，随机策略因探索的固有挑战更难分析。本文给出首个积极结果，表明随机化可改善在线图探索的竞争保证。聚焦循环，给出随机算法\textsc{RandHeavyTest}，证明其竞争比至多为1.315，与确定性设置（最优竞争比约为1.366）严格分离，给出随机化在在线图探索中的首个可证优势。迈向此结果的关键一步是新的简化最优确定性算法\textsc{HeavyTest}，其形式自然引出随机变体。还用下界补充上界，任意随机算法下界为1.115，所谓前向贪心算法（包括\textsc{RandHeavyTest}）自然类的下界为1.207。
英文摘要
In online graph exploration, introduced by Kalyanasundaram and Pruhs (1994), an agent must visit all vertices of an initially unknown weighted graph and return to its starting position, while the graph is revealed only locally at visited vertices. Although the problem has attracted considerable attention, previous work has focused exclusively on deterministic algorithms. Randomized strategies are often substantially harder to analyze because of a fundamental challenge inherent to exploration. In this work, we give the first positive result showing that randomization can improve competitive guarantees in online graph exploration. To this end, we focus on cycles, a simple graph class which nevertheless captures a key difficulty of online exploration. Our main contribution is \(\textsc{RandHeavyTest}\), a randomized algorithm for online exploration of cycles whose competitive ratio we prove to be at most 1.315. This establishes a strict separation from the deterministic setting, where the optimal competitive ratio is $\thickapprox 1.366$, and thus gives the first provable advantage of randomization in online graph exploration. A key step towards this result is a new, simplified optimal deterministic algorithm, \(\textsc{HeavyTest}\), whose formulation naturally suggests the randomized variant. We complement our upper bounds with lower bounds of 1.115 for arbitrary randomized algorithms and 1.207 for the natural class of so-called forward-greedy algorithms, which includes \(\textsc{RandHeavyTest}\).
Comments17 pages, 3 figures
