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