Speedup in the Traveling Repairman Problem with Constrained Time Windows

TL;DR

This paper introduces a bicriteria approximation algorithm for the unrooted traveling repairman problem with variable time window lengths, achieving increased profit through increased speedup, applicable over a range of speedup values.

Contribution

It generalizes previous algorithms to handle variable time window lengths and establishes a constant approximation ratio for a range of speedup values.

Findings

Algorithm achieves increased profit with increased speedup.

Applicable to time windows with lengths between 1 and 2.

Provides a constant approximation ratio within the specified speedup range.

Abstract

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 $s$.