Two robots patrolling on a line: Integer version and approximability

Paper in proceeding, 2020

Suppose that two robots can move at unit speed on a line and must visit certain points called stations infinitely often. Every station allows some maximal waiting time between two visits. The problem is to construct an optimal schedule for the robots. While the one-robot problem is easy to solve in linear time, already for two robots the complexity is open. Chuangpishit, Czyzowicz, Gasieniec, Georgiou, Jurdzinski, and Kranakis (SOFSEM 2018) found a√3-approximation algorithm. Here we provide a PTAS, accomplished by rounding and (perhaps more surpris-ingly) by using the well-quasi ordering of vectors of positive integers. The result is not very practical in the present form, but further investigation of the integer version may make it more usable