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Sublinear Algorithms for TSP via Path Covers

Authors Soheil Behnezhad , Mohammad Roghani , Aviad Rubinstein , Amin Saberi

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Author Details

Soheil Behnezhad
  • Northeastern University, Boston, MA, USA
Mohammad Roghani
  • Stanford University, CA, USA
Aviad Rubinstein
  • Stanford University, CA, USA
Amin Saberi
  • Stanford University, CA, USA

Funding

Mohammad Roghani and Amin Saberi were supported by NSF award 1812919 and ONR award 141912550. Soheil Behnezhad and Aviad Rubinstein were supported by NSF CCF-1954927, and a David and Lucile Packard Fellowship. Soheil Behnezhad was additionally supported by NSF Awards 1942123, 1812919 and by Moses Charikar’s Simons Investigator Award.

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Soheil Behnezhad, Mohammad Roghani, Aviad Rubinstein, and Amin Saberi. Sublinear Algorithms for TSP via Path Covers. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.19

Abstract

We study sublinear time algorithms for the traveling salesman problem (TSP). First, we focus on the closely related maximum path cover problem, which asks for a collection of vertex disjoint paths that include the maximum number of edges. We show that for any fixed ε > 0, there is an algorithm that (1/2 - ε)-approximates the maximum path cover size of an n-vertex graph in Õ(n) time. This improves upon a (3/8-ε)-approximate Õ(n √n)-time algorithm of Chen, Kannan, and Khanna [ICALP'20].
Equipped with our path cover algorithm, we give an Õ(n) time algorithm that estimates the cost of (1,2)-TSP within a factor of (1.5+ε) which is an improvement over a folklore (1.75 + ε)-approximate Õ(n)-time algorithm, as well as a (1.625+ε)-approximate Õ(n√n)-time algorithm of [CHK ICALP'20]. For graphic TSP, we present an Õ(n) algorithm that estimates the cost of graphic TSP within a factor of 1.83 which is an improvement over a 1.92-approximate Õ(n) time algorithm due to [CHK ICALP'20, Behnezhad FOCS'21]. We show that the approximation can be further improved to 1.66 using n^{2-Ω(1)} time.
All of our Õ(n) time algorithms are information-theoretically time-optimal up to polylog n factors. Additionally, we show that our approximation guarantees for path cover and (1,2)-TSP hit a natural barrier: We show better approximations require better sublinear time algorithms for the well-studied maximum matching problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Sublinear Algorithms
  • Traveling Salesman Problem
  • Approximation Algorithm
  • (1
  • 2)-TSP
  • Graphic TSP

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References

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