An O(logn) Approximation Ratio for the
Asymmetric Traveling Salesman Path
Problem
Chandra Chekuri
Presented by:
Rahmtin Rotabi
Martin P´al
Instructor:
Prof. Zarrabi-Zadeh
2
Introduction
•
•
ATSPP: Asymmetric Traveling Salesman Path Problem
Given Info
•
•
•
•
𝐺 = 𝑉, 𝐴, 𝑙
𝑙: 𝐴 → 𝑅 +
𝑠, 𝑡 ∈ 𝑉
Objective
•
•
Find optimum 𝑠-𝑡 path in 𝐺
NP Hard
3
Past works
•
•
•
Metric-TSP
•
Christofides
ATSP
•
•
lg 𝑛 factor
Best known factor: 0.842 lg 𝑛
Metric-TSPP
•
5
best known factor
3
4
Past works (cont’d)
•
ATSPP- our problem
•
•
𝑂
𝑛 approximation
Proved by Lam and Newman
5
ATSP (Tour)
•
•
𝛼-factor for ATSPP
?
𝛼-factor for ATSP
Two algorithms for ATSP
•
•
Reducing vertices by cycle cover
•
•
Factor lg 𝑛
Proof is straight forward
Min-Density Cycle Algorithm
•
•
Factor 2𝐻𝑛
Proof is just like “set cover”
6
ATSPP- Our work
•
•
•
𝑃(𝑠, 𝑡) denotes the set of all 𝑠 → 𝑡 paths
𝐶(𝑠, 𝑡) denotes cycle not containing s and t
Density?
7
Density lemma
•
Assumption:
•
•
Objective:
•
•
•
let 𝜆∗ be the min-density path of non-trivial path in 𝑃(𝑠, 𝑡)
We can either find the min-density path
Or a cycle in 𝐶 𝑠, 𝑡 with a lower density
Idea of proof:
•
•
Binary search
Bellman-ford
8
Augmentation lemma
•
•
•
Definitions:
•
•
•
Domination
Extension
Successor
Assumptions:
•
Let P1, P2 in P(s, t) such that P2 dominates P1
Objective:
•
•
There is a path P3 ∈ P(s, t) that dominates P2, extends P1
𝑙 𝑃1 , 𝑃3 ≤ 2𝑙(𝑃2 )
9
Augmentation lemma proof
•
•
Define 𝑋 ∈ 𝑉(𝑃1)
Mark some members of 𝑋 with an algorithm
•
•
•
Name them 𝑔1 , … , 𝑔𝑘
Obtain P3 from P1
Replace 𝑃1 (𝑔𝑖 , 𝑔𝑖+ ) of 𝑃1 by the sub-path 𝑃2 (𝑔𝑖 , 𝑔𝑖+ )
10
Augmentation lemma proof(cont’d)
•
•
•
The path extends 𝑃1
The path dominates 𝑃2
Straight-forward with following in-equalities
•
•
•
•
(I1) For 𝑖 = 1, . . . , 𝑘 − 1, we have 𝑔𝑖 ≤𝑃1 𝑔𝑖+1
(I2) For 𝑖 = 1, . . . , 𝑘 − 1, we have 𝑔𝑖 ≤𝑃2 𝑔𝑖+1 ≤𝑃2 𝑔𝑖+
(I3) For 𝑖 = 1, . . . , 𝑘 − 2, we have 𝑔𝑖+ ≤𝑃2 𝑔𝑖+2
Corollary: Replace 𝑃2 with 𝑂𝑃𝑇
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 𝑉 − 2 iterations
Implemented naively:
•
𝑂(𝑛5 )
12
Claims and proof
•
•
In every iteration, if 𝜋 is the augmenting path or cycle in that iteration,
•
•
𝑙 𝜋 ≤
𝑅 𝜋
𝐶
. 2. 𝑂𝑃𝑇
Use augmentation path lemma
Algorithm factor is max(4𝐻𝑛 − 2,1).
•
•
Step is from k1 to k2 vertices
Path step
•
•
2𝑂𝑃𝑇
𝑘1 −𝑘2
𝑘1
Cycle step
•
2
𝑂𝑃𝑇 𝑘1 −𝑘2 +1
𝐾1
𝑘1 −𝑘2
≤4
𝑂𝑃𝑇
𝑘1
13
Path-constrained ATSPP
• Start from (𝑣1, 𝑣2) , … , (𝑣𝑘−1, 𝑣𝑘 )
•
•
Instead of 𝑠, 𝑡
Same analysis
• Best integrality gap for ATSPP is 2
• Best integrality gap for ATSP is 𝑂(log 𝑛)
• 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
17