log n Approximation Ratio for the Asymmetric Traveling Salesman Path Problem Chandra Chekuri Martin Pal Presented by Instructor Rahmtin Rotabi Prof ID: 526756
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An
O(log
n) Approximation Ratio for theAsymmetric Traveling Salesman PathProblemChandra Chekuri Martin
P´al
Presented by: Instructor:
Rahmtin Rotabi Prof. Zarrabi-Zadeh
2Slide3
Introduction
ATSPP:
Asymmetric Traveling Salesman Path ProblemGiven Info
Objective
Find optimum
-
path in
NP Hard
3Slide4
Past works
Metric-TSP
ChristofidesATSP
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” 6Slide7
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 densityIdea of proof:Binary searchBellman-ford 8Slide9
Augmentation lemma
Definitions
:DominationExtensionSuccessorAssumptions:
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
P
3 from
P1Replace
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 pathUse path or cycle augmentationIt 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
39-5916Slide17
Thank you for your time
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