On the Algorithmic Complexity of the Mastermind Game with Black-Peg Results

TL;DR

This paper investigates the computational complexity of a simplified version of Mastermind with black pegs, proving NP-completeness for certain decision problems and providing more efficient algorithms for vector discovery.

Contribution

It establishes NP-completeness for the single-color Mastermind problem and introduces an improved algorithm for hidden vector discovery.

Findings

NP-completeness of the decision problem

Development of a more efficient algorithm for vector discovery

Improvement over previous methods by nearly a factor of 2

Abstract

In this paper, we study the algorithmic complexity of the Mastermind game, where results are single-color black pegs. This differs from the usual dual-color version of the game, but better corresponds to applications in genetics. We show that it is NP-complete to determine if a sequence of single-color Mastermind results have a satisfying vector. We also show how to devise efficient algorithms for discovering a hidden vector through single-color queries. Indeed, our algorithm improves a previous method of Chvatal by almost a factor of 2.