Arkin et al. in 2002 introduced a scheduling-like problem called Freeze-Tag Problem (FTP) motivated by robot swarm activation. The input consists of the locations of n mobile punctual robots in some metric space or graph. Only one begins "active", while the others are initially "frozen". All active robots can move at unit speed and, upon reaching a frozen one's location, activates it. The goal is to activate all the robots in the minimum amount of time, the so-called makespan. Until 2017 the hardness of this problem in metric spaces was still open, but then Yu et al. proved it to be NP-Hard in the Euclidian plane, and in the same year, Demaine and Roudoy demonstrated that the FTP is also hard in 3D with any $L_p$ distance (with p > 1). However, we still don't know whether Demaine's and Roudoy's result could be translated to the plane. This paper fills the p=1 gap by showing that the FTP is NP-Hard in 3D with $L_1$ distance.
翻译:Arkin等人在2002年引入了一个类似日程安排的问题,称为“冻结塔格问题 ” ( FTP), 其动机是机器人的振动。 输入内容包括在某些公制空间或图形中移动的守时机器人的位置。 只有一个开始“ 活动 ”, 而其他的最初是“ 冻结 ” 。 所有活动机器人都可以以单位速度移动, 到达冷冻地点后, 激活它。 目标是在最小时间范围内激活所有机器人, 即所谓的黑板 。 直到2017年, 公制空间中这一问题的坚硬性仍然开放, 但后来Yu 等人证明它在Euclidian 飞机中是 NP- Hard 。 同年, Demaine 和 Roudoy 显示, FTP 在3D 中也很困难, 距离为$L_ p( p > ) 。 然而, 我们仍不知道 Demaine 和 Rodoy 的结果是否能被翻译成飞机。 。 该文件填补了 p=1 差距1, 显示 FTP 在 $L_ 距离 3D 。