On Integrality Ratios for Asymmetric TSP in the Sherali-Adams Hierarchy

TL;DR

This paper investigates the limitations of the Sherali-Adams hierarchy when applied to the asymmetric TSP, demonstrating that certain LP relaxations remain far from integral solutions even after multiple rounds, thus highlighting inherent integrality gaps.

Contribution

The authors identify structural properties of digraphs that produce large integrality ratios for the Sherali-Adams hierarchy on ATSP, extending results to the Path-ATSP and simplifying analysis for the balanced LP.

Findings

Existence of digraphs with large integrality ratios at any level of Sherali-Adams hierarchy

Construction of hard instances based on structural properties of digraphs

Extension of results to Path-ATSP and the balanced LP relaxation

Abstract

We study the ATSP (Asymmetric Traveling Salesman Problem), and our focus is on negative results in the framework of the Sherali-Adams (SA) Lift and Project method. Our main result pertains to the standard LP (linear programming) relaxation of ATSP, due to Dantzig, Fulkerson, and Johnson. For any fixed integer $t\geq 0$ and small $\epsilon$, $0<\epsilon\ll{1}$, there exists a digraph $G$ on $\nu=\nu(t,\epsilon)=O(t/\epsilon)$ vertices such that the integrality ratio for level~$t$ of the SA system starting with the standard LP on $G$ is $\ge 1+\frac{1-\epsilon}{2t+3} \approx \frac43, \frac65, \frac87, \dots$. Thus, in terms of the input size, the result holds for any $t = 0,1,\dots,\Theta(\nu)$ levels. Our key contribution is to identify a structural property of digraphs that allows us to construct fractional feasible solutions for any level~$t$ of the SA system starting from the standard~LP. Our hard instances are simple and satisfy the structural property. There is a further relaxation of the standard LP called the balanced LP, and our methods simplify considerably when the starting LP for the SA system is the balanced~LP; in particular, the relevant structural property (of digraphs) simplifies such that it is satisfied by the digraphs given by the well-known construction of Charikar, Goemans and Karloff (CGK). Consequently, the CGK digraphs serve as hard instances, and we obtain an integrality ratio of $1 +\frac{1-\epsilon}{t+1}$ for any level~$t$ of the SA system, where $0<\epsilon\ll{1}$ and the number of vertices is $\nu(t,\epsilon)=O((t/\epsilon)^{(t/\epsilon)})$. Also, our results for the standard~LP extend to the Path-ATSP (find a min cost Hamiltonian dipath from a given source vertex to a given sink vertex).