Approximating Asymmetric A Priori TSP beyond the Adaptivity Gap

TL;DR

This paper studies the Asymmetric A Priori TSP with probabilistic vertices, proving a lower bound on the adaptivity gap and providing a quasi-polynomial approximation algorithm through a series of reductions.

Contribution

It introduces a novel reduction framework for Asymmetric A Priori TSP and achieves a poly-logarithmic approximation ratio below the adaptivity gap.

Findings

Established a polynomial lower bound on the adaptivity gap.

Developed a quasi-polynomial time algorithm with a poly-logarithmic approximation ratio.

Reduced the problem to a path-finding problem in an acyclic digraph with an O(log n) approximation.

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.