On Extensions of the Unanimous Vote Problem

2026-09-28

AI Summary

This study investigates three stochastic optimization extensions of the unanimous voting problem, encompassing repeatable coin flipping, multi-sided dice rolling, and full-outcome collection scenarios. Methodologically, tight adaptive gaps are established by revealing an intrinsic connection between optimal sequences and mechanical words. Furthermore, efficient algorithms are designed by integrating greedy strategies, mechanical word analysis, and submodular function reduction techniques. The primary contributions include providing periodic optimal solutions for the infinite-flipping model, deriving the first polynomial-time approximation scheme (PTAS) based on combinatorial greedy rules, and achieving both a PTAS and an O(log d) approximation ratio for the d-sided dice problem. Collectively, these advances effectively minimize the expected number of operations required to reach consensus across all considered stochastic settings.

Abstract

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.

Problem

• Unanimous Vote Problem
• stochastic optimization
• biased coins
• d-sided dice
• randomized algorithms

Innovation

• Unanimous Vote Problem
• Greedy Algorithm
• PTAS
• Submodular Ranking
• Adaptivity Gap