Speedup in the Traveling Repairman Problem with Unit Time Windows

The input to the unrooted traveling repairman problem is an undirected metric graph and a subset of nodes, each of which has a time window of unit length. Given that a repairman can start at any location, the goal is to plan a route that visits as many nodes as possible during their respective time windows. A polynomial-time bicriteria approximation algorithm is presented for this problem, gaining an increased fraction of repairman visits for increased speedup of repairman motion. For speedup $s$, we find a $6\gamma/(s + 1)$-approximation for $s$ in the range $1 \leq s \leq 2$ and a $4\gamma/s$-approximation for $s$ in the range $2 \leq s \leq 4$, where $\gamma = 1$ on tree-shaped networks and $\gamma = 2 + \epsilon$ on general metric graphs.Comment: 16 pages, 3 figure

Speedup in the Traveling Repairman Problem with Constrained Time Windows

A bicriteria approximation algorithm is presented for the unrooted traveling repairman problem, realizing increased profit in return for increased speedup of repairman motion. The algorithm generalizes previous results from the case in which all time windows are the same length to the case in which their lengths can range between l and 2. This analysis can extend to any range of time window lengths, following our earlier techniques. This relationship between repairman profit and speedup is applicable over a range of values that is dependent on the cost of putting the input in an especially desirable form, involving what are called "trimmed windows." For time windows with lengths between 1 and 2, the range of values for speedup $s$ for which our analysis holds is $1 \leq s \leq 6$. In this range, we establish an approximation ratio that is constant for any specific value of $s$.Comment: 28 pages, 3 figure