关于一致投票问题扩展的研究
On Extensions of the Unanimous Vote Problem
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中文总结 AI 辅助
本文研究一致投票问题的三个扩展，证明无限翻转变体存在贪心最优序列并刻画周期性，给出骰子变体的PTAS和对数近似算法。
中文摘要 AI 辅助
一致投票问题旨在确定一个固定的顺序来翻转n枚有偏硬币，每枚硬币只能翻转一次，使得在观察到正面和反面各一次（或翻转完所有硬币）之前，期望的翻转次数最小化。Duman Keles等人（arXiv:2510.16678 [cs.DS]）给出了该问题的一个O(n log n)时间算法。一致投票问题的扩展是随机优化问题的丰富来源。我们关注三个扩展：（1）一个变体，其中每枚硬币可以被任意多次翻转（因此解是一个无限的硬币选择序列）；（2）一个推广，使用d面骰子，每个骰子只能掷一次，必须掷骰子直到观察到两个不同的结果（或所有骰子都已掷完）；（3）另一个推广，使用d面骰子，必须掷骰子直到观察到所有d个结果。对于（1），我们证明存在一个遵循简单贪心规则的最优序列；同样的规则仅对原始问题（arXiv:2510.16678 [cs.DS]）给出1-加性近似。该规则还产生了特定最优序列与相关机械词之间的对应关系，我们利用这一点来刻画该最优序列为周期性的条件。我们建立了该变体的紧乘性和加性自适应间隙。对于（2），我们证明来自（arXiv:2510.16678 [cs.DS]）的贪心规则的两种不同推广可以结合以获得PTAS。对于（3），我们通过将问题归约为子模排序（arXiv:1007.2503 [cs.DS]）给出O(log d)近似算法；相同的归约技术可用于为其他随机探测问题生成近似算法。最后，我们提出了一些相关的开放问题。
英文摘要
The Unanimous Vote problem is to determine a fixed order in which to flip each of $n$ biased coins, where each coin can be flipped only once, such that the expected number of flips until seeing both a head and a tail (or flipping all coins) is minimized. Duman Keles et al. (arXiv:2510.16678 [cs.DS]) gave an $\mathcal{O}(n \log n)$-time algorithm for this problem. Extensions of the Unanimous Vote problem are a rich source of stochastic optimization problems. We focus on three: (1) a variant in which each coin can be flipped arbitrarily many times (a solution is thus an infinite sequence of coin choices), (2) a generalization with $d$-sided dice, that can each be rolled once, where dice must be rolled until two different outcomes are observed (or all dice have been rolled), and (3) a different generalization with $d$-sided dice, where dice must be rolled until all $d$ outcomes have been observed. For (1), we show that there is an optimal sequence which follows a simple greedy rule; the same rule only gives a 1-additive approximation for the original problem (arXiv:2510.16678 [cs.DS]). The rule also yields a correspondence between a particular optimal sequence and a related mechanical word, which we exploit to characterize the conditions under which this optimal sequence is periodic. We establish tight multiplicative and additive adaptivity gaps for this variant. For (2), we show that two different generalizations of the greedy rule from (arXiv:2510.16678 [cs.DS]) can be combined to obtain a PTAS. For (3), we give an $\mathcal{O}(\log d)$-approximation algorithm by reducing the problem to Submodular Ranking (arXiv:1007.2503 [cs.DS]); the same reduction technique can be used to yield approximation algorithms for other stochastic probing problems. Finally, we pose a number of related open questions.
发表机构
- New York University（纽约大学）
- Rutgers Business School（罗格斯商学院）
机构由 AI 辅助整理，请以论文原文为准。
