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A Deterministic Better-than-3/2 Approximation Algorithm for Metric TSP

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Integer Programming and Combinatorial Optimization (IPCO 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13904))

Abstract

We show that the max entropy algorithm can be derandomized (with respect to a particular objective function) to give a deterministic \(3/2-\epsilon \) approximation algorithm for metric TSP for some \(\epsilon > 10^{-36}\).

To obtain our result, we apply the method of conditional expectation to an objective function constructed in prior work which was used to certify that the expected cost of the algorithm is at most \(3/2-\epsilon \) times the cost of an optimal solution to the subtour elimination LP. The proof in this work involves showing that the expected value of this objective function can be computed in polynomial time (at all stages of the algorithm’s execution).

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Notes

  1. 1.

    Note that since we are dealing with irrational numbers, we will not be able to compute this probability exactly. However by doing all calculations with poly(n, N) bits of precision we can ensure our estimate has exponentially small error which will suffice to get the bounds we need later.

References

  1. Applegate, D.L., Bixby, R.E., Chvatal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study (Princeton Series in Applied Mathematics). Princeton University Press, Princeton, NJ, USA (2007)

    Google Scholar 

  2. Asadpour, A., Goemans, M.X., Madry, A., Gharan, S.O., Saberi, A.: An o(log n/ log log n) approximation algorithm for the asymmetric traveling salesman problem. In: SODA, pp. 379–389 (2010)

    Google Scholar 

  3. Nicos Christofides. Worst case analysis of a new heuristic for the traveling salesman problem. Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, PA, 1976

    Google Scholar 

  4. Dantzig, G.B., Fulkerson, D.R., Johnson, S.: On a linear programming combinatorial approach to the traveling salesman problem. OR 7, 58–66 (1959)

    Google Scholar 

  5. Edmonds, J.: Submodular functions, matroids and certain polyhedra. In: Combinatorial Structures and Their Applications, pp. 69–87, New York, NY, USA (1970). Gordon and Breach

    Google Scholar 

  6. Edmonds, J., Johnson, E.L.: Matching, Euler tours and the Chinese postman. Math. Program. 5(1), 88–124 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  7. Goemans, M., Bertsimas, D.: Survivable network, linear programming relaxations and the parsimonious property. Math. Program. 60, 06 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gupta, A., Lee, E., Li, J., Mucha, M., Newman, H., Sarkar, S.: Matroid-based TSP rounding for half-integral solutions. CoRR, abs/2111.09290 (2021)

    Google Scholar 

  9. Held, M., Karp, R.M.: The traveling salesman problem and minimum spanning trees. Oper. Res. 18, 1138–1162 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  10. Haddadan, A., Newman, A.: Towards improving christofides algorithm for half-integer TSP. In: Bender, M.A., Svensson, O., Herman, G., editors, ESA, vol. 144 of LIPIcs, pp. 56:1–56:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

    Google Scholar 

  11. Haddadan, A., Newman, A., Ravi, R.: Shorter tours and longer detours: uniform covers and a bit beyond. Math. Program. 185(1–2), 245–273 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Karlin, A.R., Klein, N., Gharan, S.O.: An improved approximation algorithm for TSP in the half integral case. In: Makarychev, K., Makarychev, Y., Tulsiani, M., Kamath, G., Chuzhoy, J., editors, STOC, pp. 28–39. ACM (2020)

    Google Scholar 

  13. Karlin, A.R., Klein, N., Gharan, S.O.: A (slightly) improved approximation algorithm for metric tsp. In: STOC. ACM (2021)

    Google Scholar 

  14. Karlin, A., Klein, N., Gharan, S.O.: A (slightly) improved bound on the integrality gap of the subtour LP for tsp. In: FOCS, pp. 844–855. IEEE Computer Society (2022)

    Google Scholar 

  15. Karpinski, M., Lampis, M., Schmied, R.: New inapproximability bounds for TSP. J. Comput. Syst. Sci. 81(8), 1665–1677 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Moemke, T., Svensson, O.: Approximating graphic tsp by matchings. In: FOCS, pp. 560–569 (2011)

    Google Scholar 

  17. Mucha, M.: \(\frac{13}{9}\)-approximation for graphic TSP. In: STACS, pp. 30–41 (2012)

    Google Scholar 

  18. Gharan, S.O., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: FOCS, pp. 550–559. IEEE Computer Society (2011)

    Google Scholar 

  19. Serdyukov, A.I.: O nekotorykh ekstremal’nykh obkhodakh v grafakh. Upravlyaemye sistemy 17, 76–79 (1978)

    Google Scholar 

  20. Sebö, A., Vygen, J.: Shorter tours by nicer ears: CoRR abs/1201.1870 (2012)

    Google Scholar 

  21. Traub, V., Vygen, J., Zenklusen, R.: Reducing path TSP to TSP. In: Makarychev, K., Makarychev, Y., Tulsiani, M., Kamath, G., Chuzhoy, J., editors, STOC, pp. 14–27. ACM (2020)

    Google Scholar 

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Authors and Affiliations

  1. University of Washington, Seattle, USA

    Anna R. Karlin, Nathan Klein & Shayan Oveis Gharan

Authors
  1. Anna R. Karlin
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  2. Nathan Klein
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  3. Shayan Oveis Gharan
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Corresponding author

Correspondence to Nathan Klein .

Editor information

Editors and Affiliations

  1. University of Wisconsin-Madison, Madison, WI, USA

    Alberto Del Pia

  2. Otto-von-Guericke-Universität, Magdeburg, Sachsen-Anhalt, Germany

    Volker Kaibel

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Karlin, A.R., Klein, N., Oveis Gharan, S. (2023). A Deterministic Better-than-3/2 Approximation Algorithm for Metric TSP. In: Del Pia, A., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2023. Lecture Notes in Computer Science, vol 13904. Springer, Cham. https://doi.org/10.1007/978-3-031-32726-1_19

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  • DOI: https://doi.org/10.1007/978-3-031-32726-1_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-32725-4

  • Online ISBN: 978-3-031-32726-1

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