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Presented by: Instructor: Rahmtin Rotabi Prof. Zarrabi- Z adeh

An O ( log n ) Approximation Ratio for the Asymmetric Traveling Salesman Path Problem Chandra Chekuri Martin P´al. Presented by: Instructor: Rahmtin Rotabi Prof. Zarrabi- Z adeh. Introduction. ATSPP: Asymmetric Traveling Salesman Path Problem Given Info

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Presented by: Instructor: Rahmtin Rotabi Prof. Zarrabi- Z adeh

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  1. An O(logn) Approximation Ratio for theAsymmetric Traveling Salesman PathProblemChandra Chekuri Martin P´al Presented by: Instructor: Rahmtin Rotabi Prof. Zarrabi-Zadeh

  2. Introduction • ATSPP: Asymmetric Traveling Salesman Path Problem • Given Info • Objective • Find optimum - path in • NP Hard

  3. Past works • Metric-TSP • Christofides • ATSP • factor • Best known factor: • Metric-TSPP • best known factor

  4. Past works (cont’d) • ATSPP- our problem • approximation • Proved by Lam and Newman

  5. ATSP (Tour) • -factor for ATSPP -factor for ATSP • Two algorithms for ATSP • Reducing vertices by cycle cover • Factor • Proof is straight forward • Min-Density Cycle Algorithm • Factor • Proof is just like “set cover”

  6. ATSPP- Our work denotes the set of all paths denotes cycle not containing s and t Density?

  7. Density lemma • Assumption: • let be the min-density path of non-trivial path in • Objective: • We can either find the min-density path • Or a cycle in with a lower density • Idea of proof: • Binary search • Bellman-ford

  8. Augmentation lemma • Definitions: • Domination • Extension • Successor • Assumptions: • Let in such that dominates • Objective: • There is a path that dominates , extends

  9. Augmentation lemma proof • Define • Mark some members of with an algorithm • Name them • Obtain P3 from P1 • Replace of by the sub-path

  10. Augmentation lemma proof(cont’d) • The path extends • The path dominates • Straight-forward with following in-equalities • (I1) For we have • (I2) For we have • (I3) For we have • Corollary: Replace with

  11. Algorithm • Start with only one edge • Use proxies • Until we have a spanning path • Use path or cycle augmentation • It will finish after at most iterations • Implemented naively:

  12. Claims and proof • In every iteration, if is the augmenting path or cycle in that iteration, • Use augmentation path lemma • Algorithm factor is . • Step is from k1 to k2 vertices • Path step • Cycle step

  13. Path-constrained ATSPP • Start from • Instead of • Same analysis • Best integrality gap for ATSPP is 2 • Best integrality gap for ATSP is • LP:

  14. Any Question? 

  15. References An O(logn) Approximation Ratio for theAsymmetric Traveling Salesman Path Problem, THEORY OF COMPUTING, Volume 3 (2007), pp. 197–209. Traveling salesman path problem, Mathematical Journal, Volume 113, Issue 1, pp 39-59

  16. Thank you for your time

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