Improved guarantees for the a priori TSP

TL;DR

This paper improves approximation guarantees for the a priori TSP problem, providing tighter bounds on solution quality using randomized and deterministic algorithms, which enhances previous results in stochastic routing.

Contribution

The paper introduces stronger approximation bounds for the a priori TSP, including nearly matching lower and upper bounds, and analyzes sampling strategies for master route solutions.

Findings

Randomized sampling achieves less than 3.1 approximation ratio.

Deterministic polynomial-time algorithm achieves less than 5.9 approximation ratio.

Provides bounds that are close to the theoretical limits for the a priori TSP.

Abstract

We revisit the a priori TSP (with independent activation) and prove stronger approximation guarantees than were previously known. In the a priori TSP, we are given a metric space $(V,c)$ and an activation probability $p(v)$ for each customer $v\in V$. We ask for a TSP tour $T$ for $V$ that minimizes the expected length after cutting $T$ short by skipping the inactive customers. All known approximation algorithms select a nonempty subset $S$ of the customers and construct a master route solution, consisting of a TSP tour for $S$ and two edges connecting every customer $v\in V\setminus S$ to a nearest customer in $S$. We address the following questions. If we randomly sample the subset $S$, what should be the sampling probabilities? How much worse than the optimum can the best master route solution be? The answers to these questions (we provide almost matching lower and upper bounds) lead to improved approximation guarantees: less than 3.1 with randomized sampling, and less than 5.9 with a deterministic polynomial-time algorithm.