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 \(t\ge 0\) 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 \({\ge } 1+\frac{1-\epsilon }{2t+3} \approx \frac{4}{3}, \frac{6}{5}, \frac{8}{7}, \ldots \). Thus, in terms of the input size, the result holds for any \(t = 0,1,\ldots ,{\varTheta }(\nu )\) 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).

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Notes

  1. 1.

    Although the term integrality ratio is used in two different senses—one refers to an instance, the other to a relaxation (i.e., all instances)—the context will resolve the ambiguity.

  2. 2.

    By the integral hull we mean the convex hull of the zero-one solutions that are feasible for the original relaxation.

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Acknowledgments

We thank a number of colleagues for useful discussions. We are grateful to Sylvia Boyd and Paul Elliott-Magwood for help with ATSP integrality gaps, and to Levent Tunçel for sharing his knowledge of the area.

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Correspondence to Konstantinos Georgiou.

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An extended abstract of this work appeared in the proceedings of the 40th International Colloquium on Automata, Languages, and Programming (ICALP 2013).

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Cheriyan, J., Gao, Z., Georgiou, K. et al. On integrality ratios for asymmetric TSP in the Sherali–Adams hierarchy. Math. Program. 159, 1–29 (2016). https://doi.org/10.1007/s10107-015-0947-5

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Keywords

  • Asymmetric TSP
  • Sherali–Adams hierarchy
  • Integrality ratios

Mathematics Subject Classification

  • 90C05
  • 90C27