A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
An Explicit Equivalent Positive Semidefinite Program for Nonlinear 0-1 Programs
SIAM Journal on Optimization
A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0--1 Programming
Mathematics of Operations Research
On the Held-Karp relaxation for the asymmetric and symmetric traveling salesman problems
Mathematical Programming: Series A and B
SIAM Journal on Optimization
On The Approximability Of The Traveling Salesman Problem
Combinatorica
New lower bounds for approximation algorithms in the lovasz-schrijver hierarchy
New lower bounds for approximation algorithms in the lovasz-schrijver hierarchy
On the Integrality Ratio for the Asymmetric Traveling Salesman Problem
Mathematics of Operations Research
Linear programming relaxations of maxcut
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Integrality gaps for Sherali-Adams relaxations
Proceedings of the forty-first annual ACM symposium on Theory of computing
An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
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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≥0 and small ε, 0ε≪1, there exists a digraph G on ν=ν(t,ε)=O(t/ε) vertices such that the integrality ratio for level t of the SA system starting with the standard LP on G is ≥ 1+ 1-&epsilon/2t+3 ≈ 4/3, 6/5, 8/7, …. Thus, in terms of the input size, the result holds for any t=0,1,…,Θ(ν) 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 0ε≪1 and the number of vertices is ν(t,ε)=O((t/ε)(t/ε)). 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).