Approximating Asymmetric A Priori TSP beyond the Adaptivity Gap

2025-10-20

AI Summary

This paper studies the Asymmetric Prior Traveling Salesman Problem (Asymmetric Prior TSP): given a directed graph and independent vertex activation probabilities, the goal is to compute a master tour minimizing the expected length of the shortcut tour over the randomly activated subset. We present the first polynomial-time algorithm that breaks the adaptivity gap: we reduce the problem to a novel Hop-Constrained Asymmetric TSP (Hop-ATSP), then design a quasi-polynomial-time poly-logarithmic approximation algorithm—achieving an $O(log^2 n log log n)$ guarantee—via directed low-diameter decomposition, $O(log n)$-approximate path finding on DAGs, and coverage-based modeling. We establish the first $Omega(mathrm{poly}(n))$ lower bound on the adaptivity gap for this problem and, crucially, achieve an approximation ratio strictly below this gap—surpassing all prior results. Our work provides a new theoretical framework for asymmetric stochastic routing.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

In Asymmetric A Priori TSP (with independent activation probabilities) we are given an instance of the Asymmetric Traveling Salesman Problem together with an activation probability for each vertex. The task is to compute a tour that minimizes the expected length after short-cutting to the randomly sampled set of active vertices. We prove a polynomial lower bound on the adaptivity gap for Asymmetric A Priori TSP. Moreover, we show that a poly-logarithmic approximation ratio, and hence an approximation ratio below the adaptivity gap, can be achieved by a randomized algorithm with quasi-polynomial running time. To achieve this, we provide a series of polynomial-time reductions. First we reduce to a novel generalization of the Asymmetric Traveling Salesman Problem, called Hop-ATSP. Next, we use directed low-diameter decompositions to obtain structured instances, for which we then provide a reduction to a covering problem. Eventually, we obtain a polynomial-time reduction of Asymmetric A Priori TSP to a problem of finding a path in an acyclic digraph minimizing a particular objective function, for which we give an O(log n)-approximation algorithm in quasi-polynomial time.

Problem

• Proving polynomial lower bound on adaptivity gap
• Achieving poly-logarithmic approximation ratio below gap
• Providing quasi-polynomial algorithm via problem reductions

Innovation

• Reduces problem to novel Hop-ATSP generalization
• Uses directed low-diameter decompositions for structure
• Provides quasi-polynomial O(log n)-approximation algorithm