Fence patrolling by mobile agents with distinct speeds

TL;DR

This paper investigates the problem of patrolling a line segment with multiple mobile agents of different speeds, disproves a previous conjecture about maximum patrol length, and provides new bounds for specific cases.

Contribution

The paper disproves Czyzowicz et al.'s conjecture for k=6 agents and confirms its validity for k=2 and 3, advancing understanding of patrol strategies.

Findings

Counterexample for k=6 agents disproves the conjecture.

Conjecture holds true for k=2 and 3 agents.

Provides bounds and strategies for different numbers of agents.

Abstract

Suppose we want to patrol a fence (line segment) using k mobile agents with given speeds v_1, ..., v_k so that every point on the fence is visited by an agent at least once in every unit time period. Czyzowicz et al. conjectured that the maximum length of the fence that can be patrolled is (v_1 + ... + v_k)/2, which is achieved by the simple strategy where each agent i moves back and forth in a segment of length v_i/2. We disprove this conjecture by a counterexample involving k = 6 agents. We also show that the conjecture is true for k = 2, 3.