Price of Locality in Permutation Mastermind: Are TikTok influencers Chaotic Enough?

2026-01-27

AI Summary

This study investigates the impact of locality constraints on query efficiency in Permutation Mastermind, focusing on two models: ℓk-locality (consecutive guesses differ in at most k positions) and wk-locality (changes confined to a sliding window of length k). By integrating combinatorial optimization, computational complexity theory, and randomized algorithm analysis, this work provides the first systematic quantification of the query complexity overhead imposed by local strategies. It establishes that any local strategy requires Θ(n²) queries in the worst case—significantly worse than the O(n log n) achievable by unrestricted strategies. Furthermore, it proves that the ℓ3-local satisfiability problem is NP-hard, while an efficient randomized polynomial-time algorithm exists for ℓ2-locality, thereby forging a theoretical link between locality constraints and computational hardness and elucidating the inherent limitations of human-preferred solving strategies.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

In the permutation Mastermind game, the goal is to uncover a secret permutation $\sigma^\star \colon [n] \to [n]$ by making a series of guesses $\pi_1, \ldots, \pi_T$ which must also be permutations of $[n]$, and receiving as feedback after guess $\pi_t$ the number of positions $i$ for which $\sigma^\star(i) = \pi_t(i)$. While the existing literature on permutation Mastermind suggests strategies in which $\pi_t$ and $\pi_{t+1}$ might be widely different permutations, a resurgence in popularity of this game as a TikTok trend shows that humans (or at least TikTok influencers) use strategies in which consecutive guesses are very similar. For example, it is common to see players attempt one transposition at a time and slowly see their score increase. Motivated by these observations, we study the theoretical impact of two forms of"locality"in permutation Mastermind strategies: $\ell_k$-local strategies, in which any two consecutive guesses differ in at most $k$ positions, and the even more restrictive class of $w_k$-local strategies, in which consecutive guesses differ in a window of length at most $k$. We show that, in broad terms, the optimal number of guesses for local strategies is quadratic, and thus much worse than the $O(n \lg n)$ guesses that suffice for non-local strategies. We also show NP-hardness of the satisfiability version for $\ell_3$-local strategies, whereas in the $\ell_2$-local variant the problem admits a randomized polynomial algorithm.

Problem

• Permutation Mastermind
• locality
• query complexity
• NP-hardness
• combinatorial search

Innovation

• locality
• permutation Mastermind
• query complexity
• NP-hardness
• combinatorial search