On integrality ratios for asymmetric TSP in the Sherali–Adams hierarchy

Abstract

We study the Asymmetric Traveling Salesman Problem (ATSP), 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 linear programming (LP) relaxation of ATSP, due to Dantzig, Fulkerson, and Johnson. For any fixed integer $$t\ge 0$$tź0 and small $$\epsilon $$∈, $$0<\epsilon \ll {1}$$0<∈ź1, there exists a digraph G on $$\nu =\nu (t,\epsilon )=O(t/\epsilon )$$ź=ź(t,∈)=O(t/∈) vertices such that the integrality ratio for level t of the SA system starting with the standard LP on G is $${\ge } 1+\frac{1-\epsilon }{2t+3} \approx \frac{4}{3}, \frac{6}{5}, \frac{8}{7}, \ldots $$ź1+1-∈2t+3ź43,65,87,ź. Thus, in terms of the input size, the result holds for any $$t = 0,1,\ldots ,{\varTheta }(\nu )$$t=0,1,ź,ź(ź) 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 $$1 +\frac{1-\epsilon }{t+1}$$1+1-∈t+1 for any level t of the SA system, where $$0<\epsilon \ll {1}$$0<∈ź1 and the number of vertices is $$\nu (t,\epsilon )=O((t/\epsilon )^{(t/\epsilon )})$$ź(t,∈)=O((t/∈)(t/∈)). 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).