Presented by:Instructor: Rahmtin Rotabi Prof. Zarrabi- Z adeh
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Transcript of Presented by:Instructor: Rahmtin Rotabi Prof. Zarrabi- Z adeh
An O(logn) Approximation Ratio for theAsymmetric Traveling Salesman PathProblem
Chandra Chekuri Martin P´al
Presented by:Instructor:
Rahmtin RotabiProf. Zarrabi-Zadeh
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Introduction
• ATSPP: Asymmetric Traveling Salesman Path Problem• Given Info• •
• Objective• Find optimum - path in • NP Hard
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Past works
• Metric-TSP• Christofides
• ATSP• factor• Best known factor:
• Metric-TSPP• best known factor
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Past works (cont’d)• ATSPP- our problem• approximation• Proved by Lam and Newman
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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”
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ATSPP- Our work
• denotes the set of all paths• denotes cycle not containing s and t• Density?
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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
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Augmentation lemma• Definitions:• Domination• Extension• Successor
• Assumptions:• Let in such that dominates
• Objective:• There is a path that dominates , extends
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Augmentation lemma proof• Define • Mark some members of with an algorithm• Name them • Obtain P3 from P1• Replace of by the sub-path
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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
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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:
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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
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Path-constrained ATSPP
• Start from • Instead of • Same analysis
• Best integrality gap for ATSPP is 2• Best integrality gap for ATSP is • LP:
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Any Question?
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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
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Thank you for your time