Abstract
A long-standing conjecture in Combinatorial Optimization is that the integrality gap of the Held-Karp LP relaxation for the Asymmetric Traveling Salesman Problem (ATSP) is a constant. In this paper, we give a simpler LP relaxation for the ASTP. The integrality gaps of this relaxation and of the Held-Karp relaxation are within a constant factor of each other. Our LP is simpler in the sense that its extreme solutions have at most 2n − 2 non-zero variables, improving the bound 3n − 2 proved by Vempala and Yannakakis for the extreme solutions of the Held-Karp LP relaxation. Moreover, more than half of these non-zero variables can be rounded to integers while the total cost only increases by a constant factor.
We also show that given a partially rounded solution, in an extreme solution of the corresponding LP relaxation, at least one positive variable is greater or equal to 1/2.
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References
Bansal, N., Khandekar, R., Nagarajan, V.: Additive Guarantees for Degree Bounded Directed Network Design. In: STOC 2008 (2008)
Bläser, M.: A new Approximation Algorithm for the Asymmetric TSP with Triangle Inequality. In: SODA 2002, pp. 638–645 (2002)
Charikar, M., Goemans, M.X., Karloff, H.J.: On the Integrality Ratio for Asymmetric TSP. In: FOCS 2004, pp. 101–107 (2004)
Carr, R., Vempala, S.: On the Held-Karp relaxation for the asymmetric and symmetric TSPs. Mathematical Programming 100(3), 569–587 (2004)
Frank, A.: Personal communication
Frieze, A., Galbiati, G., Maffioli, M.: On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks 12 (1982)
Goemans, M.X.: Minimum Bounded Degree Spanning Trees. In: FOCS 2006, pp. 273–282 (2006)
Gutin, G., Punnen, A.P. (eds.): Traveling Salesman Problem and Its Variations. Springer, Berlin (2002)
Held, M., Karp, R.M.: The traveling salesman problem and minimum spanning trees. Operation Research 18, 1138–1162 (1970)
Hoffman, A.J.: Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In: Proc. Symp. in Applied Mathematics, Amer. Math. Soc., pp. 113–127 (1960)
Jain, K.: A factor 2 approximation for the generalized Steiner network problem. Combinatorica 21, 39–60 (2001)
Kaplan, H., Lewenstein, M., Shafir, N., Sviridenko, M.: Approximation Algorithms for Asymmetric TSP by Decomposing Directed Regular Multidigraphs. In: Proc. of IEEE FOCS, pp. 56–67 (2003)
Schrijver, A.: Combinatorial Optimization - Polyhedra and Efficiency. Springer, Berlin (2003)
Lau, L.C., Naor, J., Salavatipour, M.R., Singh, M.: Survivable network design with degree or order constraints. In: STOC 2007, pp. 651–660 (2007)
Lau, L.C., Singh, M.: Approximating minimum bounded degree spanning trees to within one of optimal. In: STOC 2007, pp. 661–670 (2007)
Lawler, E., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D. (eds.): The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. John Wiley & Sons Ltd., Chichester (1985)
Vempala, S., Yannakakis, M.: A Convex Relaxation for the Asymmetric TSP. In: SODA 1999, pp. 975–976 (1999)
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Nguyen, T. (2008). A Simple LP Relaxation for the Asymmetric Traveling Salesman Problem. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2008 2008. Lecture Notes in Computer Science, vol 5171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85363-3_17
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DOI: https://doi.org/10.1007/978-3-540-85363-3_17
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