The Complexity of Robot Games on the Integer Line

TL;DR

This paper studies robot games on the integer line, providing an exponential-time algorithm for deciding the winner and establishing a matching lower bound, thus clarifying the computational complexity of the problem.

Contribution

It introduces an exponential-time decision algorithm for robot games on Z and proves a matching lower bound, advancing understanding of their computational complexity.

Findings

Decidable with exponential time complexity

Algorithm matches the proven lower bound

Clarifies the complexity class of robot games on Z

Abstract

In robot games on Z, two players add integers to a counter. Each player has a finite set from which he picks the integer to add, and the objective of the first player is to let the counter reach 0. We present an exponential-time algorithm for deciding the winner of a robot game given the initial counter value, and prove a matching lower bound.