Approximation Algorithms for the Traveling Repairman and Speeding Deliveryman Problems

Constant-factor, polynomial-time approximation algorithms are presented for two variations of the traveling salesman problem with time windows. In the first variation, the traveling repairman problem, the goal is to find a tour that visits the maximum possible number of locations during their time windows. In the second variation, the speeding deliveryman problem, the goal is to find a tour that uses the minimum possible speedup to visit all locations during their time windows. For both variations, the time windows are of unit length, and the distance metric is based on a weighted, undirected graph. Algorithms with improved approximation ratios are given for the case when the input is defined on a tree rather than a general graph. The algorithms are also extended to handle time windows whose lengths fall in any bounded range.

💡 Research Summary

The paper studies two time‑window variants of the classic traveling salesman problem (TSP): the Traveling Repairman Problem (TRP) and the Speeding Deliveryman Problem (SDP). In the TRP the objective is to maximize the number of service requests (or total profit) that can be visited within their prescribed unit‑length time windows; in the SDP the objective is to minimize the speed‑up factor required for a single tour that visits every request within its window. Both problems are defined on a weighted undirected graph, and the authors assume that all time windows have the same length (unit‑time) and that each request yields the same profit. The paper’s main contributions are constant‑factor polynomial‑time approximation algorithms for both problems, with improved ratios when the underlying network is a tree rather than a general graph. The authors also extend their techniques to handle time windows whose lengths lie within any bounded range.

Key technical ideas.
The central technique is trimming: the time line is partitioned into intervals of length ½ (0, 0.5, 1, 1.5, …). Because each request’s window is exactly one unit long, at least half of the window lies entirely inside a single interval. The algorithm discards the part of the window that falls outside this “target interval” and keeps only the portion that is fully contained. This operation may lose some profit for the repairman or increase the required speed for the deliveryman, but the loss is bounded by a constant factor.

Two theorems formalize these bounds. The Limited‑Loss Theorem shows that after trimming, there exists a feasible repairman tour whose profit is at least one‑third of the optimal untrimmed profit. The proof uses a simple “best‑of‑three” argument: an optimal tour must have at least one‑third of its service events in target intervals, early intervals, or late intervals; shifting the schedule by ±0.5 time units yields a feasible trimmed tour with at least that many events. The Small‑Speedup Theorem proves that for the deliveryman, a trimmed schedule can be realized with a speed‑up factor of at most 4. The construction runs the optimal tour forward and backward at four times the optimal speed in a patterned fashion (forward for one interval, backward for 0.75, forward for 0.25), guaranteeing that every trimmed request is still served within its shortened window.

Tree case.
When the underlying graph is a tree, trimming reduces each request to a single interval, after which the authors can solve the repairman problem exactly using dynamic programming. They contract the unique path between the start and end nodes into a single “super‑node” and then sweep the tree from the leaves upward. For each node u they compute a table L_u