We consider the problem of finding patrol schedules for k robots to visit a given set of n sites in a metric space. Each robot has the same maximum speed and the goal is to minimize the weighted maximum latency of any site, where the latency of a site is defined as the maximum time duration between consecutive visits of that site. The problem is NP-hard, as it has the traveling salesman problem as a special case (when k=1 and all sites have the same weight). We present a polynomial-time algorithm with an approximation factor of O(k \log \frac{w_{max}{w_{\min}}} ) to the optimal solution, where w_{ max } and w_{ min } are the maximum and minimum weight of the sites respectively. Further, we consider the special case where the sites are in 1D. When all sites have the same weight, we present a polynomial-time algorithm to solve the problem exactly. When the sites may have different weights, we use dynamic programming to generate an 8-approximate solution, which also runs in polynomial time  .
We consider the problem of finding patrol schedules for k robots to visit a given set of n sites in a metric space. Each robot has the same maximum speed and the goal is to minimize the weighted maximum latency of any site, where the latency of a site is defined as the maximum time duration between consecutive visits of that site. The problem is NP-hard, as it has the traveling salesman problem as a special case (when k=1 and all sites have the same weight). We present a polynomial-time algorithm with an approximation factor of O(k ^2 \log \frac{w_{\max}{w_{\min}}} ) to the optimal solution, where w_{ \max } and w_{ \min } are the maximum and minimum weight of the sites respectively. Further, we consider the special case where the sites are in 1D. When all sites have the same weight, we present a polynomial-time algorithm to solve the problem exactly. If the sites may have different weights, we present a 12-approximate solution, which runs in polynomial time when the number of robots, k, is a constant .
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\log \frac{w_{max
^2 \log \frac{w_{\max
R
max
\max
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min
\min
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When
If
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use dynamic programming to generate an 8-approximate
present a 12-approximate
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also
A
when the number of robots, k, is a constant