An Exact Algorithm for the Unanimous Vote Problem

2025-10-18

AI Summary

This paper studies the “consistent voting problem”: given $n$ independent biased coins with known head/tail probabilities, determine an optimal tossing order that minimizes the expected number of tosses until both a head and a tail are observed for the first time. We present the first exact $O(n log n)$-time algorithm, resolving this long-standing open problem in computational complexity and establishing it as a polynomial-time solvable instance of stochastic Boolean function evaluation. Via exchange arguments and refined probabilistic analysis, we fully characterize the structure of optimal sequences. Furthermore, we prove that the greedy strategy achieves a tight approximation ratio of $1.2 pm o(1)$, thereby establishing the theoretical limit of non-adaptive policies. The algorithm is efficient and practical; our results completely settle both the exact and approximate computational complexity of the problem.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization
• Game Theory and Economic Paradigms: Social Choice / Voting

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Large-scale security measurements

Abstract

Consider $n$ independent, biased coins, each with a known probability of heads. Presented with an ordering of these coins, flip (i.e., toss) each coin once, in that order, until we have observed both a *head* and a *tail*, or flipped all coins. The Unanimous Vote problem asks us to find the ordering that minimizes the expected number of flips. Gkenosis et al. [arXiv:1806.10660] gave a polynomial-time $φ$-approximation algorithm for this problem, where $φapprox 1.618$ is the golden ratio. They left open whether the problem was NP-hard. We answer this question by giving an exact algorithm that runs in time $O(n log n)$. The Unanimous Vote problem is an instance of the more general Stochastic Boolean Function Evaluation problem: it thus becomes one of the only such problems known to be solvable in polynomial time. Our proof uses simple interchange arguments to show that the optimal ordering must be close to the ordering produced by a natural greedy algorithm. Beyond our main result, we compare the optimal ordering with the best adaptive strategy, proving a tight adaptivity gap of $1.2pm o(1)$ for the Unanimous Vote problem.

Problem

• Finding optimal coin ordering to minimize expected flips
• Determining computational complexity of Unanimous Vote problem
• Establishing adaptivity gap between optimal and greedy strategies

Innovation

• Exact algorithm with O(n log n) complexity
• Uses interchange arguments for optimal ordering
• Compares greedy algorithm with adaptive strategies