lecture 3: asymmetric tsp (cont) - max planck...
TRANSCRIPT
![Page 1: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/1.jpg)
Lecture 3: Asymmetric TSP
(cont)
Ola Svensson
ADFOCS 2015
![Page 2: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/2.jpg)
Asymmetric Traveling Salesman Problem
INPUT: a complete digraph 𝐺 = 𝑉, 𝐸 with pairwise (not necessarily symmetric)
distances that satisfy the triangle inequality
OUTPUT: a tour of minimum weight that visits each vertex once
![Page 3: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/3.jpg)
Held-Karp Relaxation of ATSP
Variables: 𝑥𝑢𝑣 = “indicate whether arc 𝑢, 𝑣 is used in tour”
Minimize: 𝑢𝑣∈𝐸𝑤(𝑢, 𝑣) 𝑥𝑢𝑣
Subject to: 𝒙 𝜹+ 𝒗 = 𝒙 𝜹− 𝒗 = 𝟏 for all 𝑣 ∈ 𝑉
𝒙 𝜹+ 𝑺 ≥ 𝟏 for all S ⊂ 𝑉
𝑥 ≥ 0
Held-Karp Relaxation
![Page 4: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/4.jpg)
TOOLS FOR ROUNDING LP
![Page 5: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/5.jpg)
Circulations
INPUT: a digraph 𝐺 = 𝑉, 𝐸 and for each arc 𝑒 ∈ 𝐸
a lower bound 𝑙 𝑒 ≥ 0 and an upper bound 𝑢 𝑒 ≥ 0
OUTPUT: a circulation 𝑓: 𝐸 → 𝑅+ satisfying
flow conservation: 𝑓 𝛿+ 𝑣 = 𝑓 𝛿− 𝑣 for each 𝑣 ∈ 𝑉
edge bounds: 𝑙 𝑒 ≤ 𝑓 𝑒 ≤ 𝑢 𝑒 for each 𝑒 ∈ 𝐸
Blue edges have lower bound 1 and upper bound 2
Red edges have lower bound 0 and upper bound ∞
![Page 6: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/6.jpg)
Circulations
INPUT: a digraph 𝐺 = 𝑉, 𝐸 and for each arc 𝑒 ∈ 𝐸
a lower bound 𝑙 𝑒 ≥ 0 and an upper bound 𝑢 𝑒 ≥ 0
OUTPUT: a circulation 𝑓: 𝐸 → 𝑅+ satisfying
flow conservation: 𝑓 𝛿+ 𝑣 = 𝑓 𝛿− 𝑣 for each 𝑣 ∈ 𝑉
edge bounds: 𝑙 𝑒 ≤ 𝑓 𝑒 ≤ 𝑢 𝑒 for each 𝑒 ∈ 𝐸
Blue edges have lower bound 1 and upper bound 2
Red edges have lower bound 0 and upper bound ∞
Can be calculated in polytime and also a min cost circulation can be found
![Page 7: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/7.jpg)
When does a circulation exist?
NECESSARY CONDITIONS:
• For every arc 𝑒 ∈ 𝐸 we must have 𝑙 𝑒 ≤ 𝑢(𝑒)
• For every 𝑆 ⊂ 𝑉 we must have 𝑙 𝛿− 𝑆 ≤ 𝑢 𝛿+ 𝑆
HOFFMAN’s CIRCULATION THEOREM:
The above conditions are also sufficient.
Furthermore, if 𝑙 and 𝑢 are integer valued, the circulation 𝑓 can be
chosen to be integral.
Hoffman’60
![Page 8: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/8.jpg)
RANDOMIZED LP
ROUNDING
Basic idea select a subset of edges and make it Eulerian by finding a
circulation
![Page 9: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/9.jpg)
Randomized Round
Find an optimal solution 𝑥∗ to LP relaxation
Scale up 𝑥∗ by taking 𝐾 ≔ 100ln 𝑛 parallel copies of each
edge, each of same LP-value as the original edge
Form 𝐻 by taking each edge with probability equal to its LP-
value
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in 𝐻
![Page 10: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/10.jpg)
Randomized Round
Find an optimal solution 𝑥∗ to LP relaxation
Scale up 𝑥∗ by taking 𝐾 ≔ 100ln 𝑛 parallel copies of each
edge, each of same LP-value as the original edge
Form 𝐻 by taking each edge with probability equal to its LP-
value
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in 𝐻
![Page 11: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/11.jpg)
Randomized Round
Find an optimal solution 𝑥∗ to LP relaxation
Scale up 𝑥∗ by taking 𝐾 ≔ 100ln 𝑛 parallel copies of each
edge, each of same LP-value as the original edge
Form 𝐻 by taking each edge with probability equal to its LP-
value
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in 𝐻
Taken with probability 𝑥𝑒∗
![Page 12: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/12.jpg)
Randomized Round
Find an optimal solution 𝑥∗ to LP relaxation
Scale up 𝑥∗ by taking 𝐾 ≔ 100ln 𝑛 parallel copies of each
edge, each of same LP-value as the original edge
Form 𝐻 by taking each edge with probability equal to its LP-
value
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in 𝐻
![Page 13: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/13.jpg)
Randomized Round
Find an optimal solution 𝑥∗ to LP relaxation
Scale up 𝑥∗ by taking 𝐾 ≔ 100ln 𝑛 parallel copies of each
edge, each of same LP-value as the original edge
Form 𝐻 by taking each edge with probability equal to its LP-
value
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in 𝐻
![Page 14: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/14.jpg)
What’s the problem?
• With high probability the sampled graph 𝐻 is not even connected
• So we will return an Eulerian graph but it is not connected
![Page 15: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/15.jpg)
Randomized Round
Find an optimal solution 𝑥∗ to LP relaxation
Scale up 𝑥∗ by taking 𝐾 ≔ 1000ln 𝑛 parallel copies of each
edge, each of same LP-value as the original edge
Form 𝐻 by taking each edge with probability equal to its LP-
value
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in 𝐻
Well connected?
Eulerian?
![Page 16: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/16.jpg)
Analyzing 𝐻: out-degree of a vertex
What is the expected out-degree of v?
v
𝐾 = 1000 ln 𝑛
It is 𝐾 ⋅ 𝑥∗ 𝛿+ 𝑣 = 1000 ln 𝑛 =: 𝜇
The number of outgoing edges is the sum of random independent 0/1
variables
Hence, by standard Chernoff bound
Pr 𝛿𝐻+ 𝑣 − 𝜇 ≥
𝜇
3≤ 𝑒−
𝜇30 ≤
1
2𝑛10𝑥∗(𝛿+ 𝑣 )
In words: the number of edges will deviate
from its expectation more than a fraction 1/3 with
probability at most O(1
n10)
![Page 17: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/17.jpg)
Analyzing 𝐻: in-degree of a vertex
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Analyzing 𝐻: in-degree of a vertex
What is the expected in-degree of v?
v
𝐾 = 1000 ln 𝑛
It is 𝐾 ⋅ 𝑥∗ 𝛿− 𝑣 = 1000 ln 𝑛 =: 𝜇
The number of outgoing edges is the sum of random independent 0/1
variables
Hence, by standard Chernoff bound
Pr 𝛿𝐻− 𝑣 − 𝜇 ≥
𝜇
3≤ 𝑒−
𝜇30 ≤
1
2𝑛10𝑥∗(𝛿− 𝑣 )
In words: the number of edges will deviate
from its expectation more than a fraction 1/3 with
probability at most O(1
n10)
![Page 19: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/19.jpg)
Analyzing 𝐻: in/out-degree of a set 𝑆
What is the expected in-degree of 𝑆?
𝐾 = 1000 ln 𝑛 𝑥∗(𝛿− 𝑆 )
It is 𝐾 ⋅ 𝑥∗ 𝛿− 𝑆 = 1000 ln 𝑛 𝑥∗(𝛿−(𝑆)) =: 𝜇
The number of outgoing edges is the sum of random independent 0/1
variables
Hence, by standard Chernoff bound
Pr 𝛿𝐻− 𝑆 − 𝜇 ≥
𝜇
3≤ 𝑒−
𝜇30 ≤
1
2𝑛10𝑥∗(𝛿− 𝑆 )
In words: the number of edges will deviate
from its expectation more than a fraction 1/3 with
probability at most O(1
n10x∗(𝛿− 𝑆 ))
S
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Analyzing 𝐻: bad cuts
We say that a cut 𝑆 is bad in 𝐻 if the incoming and outgoing edges deviate
more than a fraction1/3
𝑆 is bad in 𝐻 if
𝛿𝐻− 𝑆 − 𝜇 ≥
𝜇
3or 𝛿𝐻
+ 𝑆 − 𝜇 ≥𝜇
3
where 𝜇 = 𝑥∗ 𝛿− 𝑆 = 𝑥∗(𝛿+ 𝑆 )
By previous calculations
Pr 𝑆 𝑖𝑠 𝑏𝑎𝑑 ≤Pr 𝛿𝐻− 𝑣 − 𝜇 ≥
𝜇
3+ Pr 𝛿𝐻
+ 𝑣 − 𝜇 ≥𝜇
3
≤1
𝑛10𝑥∗(𝛿+ 𝑆 )
![Page 21: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/21.jpg)
Analyzing 𝐻: bad cuts
We say that a cut 𝑆 is bad in 𝐻 if the incoming and outgoing edges deviate
more than a fraction1/3
𝑆 is bad in 𝐻 if
𝛿𝐻− 𝑆 − 𝜇 ≥
𝜇
3or 𝛿𝐻
+ 𝑆 − 𝜇 ≥𝜇
3
where 𝜇 = 𝑥∗ 𝛿− 𝑆 = 𝑥∗(𝛿+ 𝑆 )
By previous calculations
𝐏𝐫 𝑺 𝒊𝒔 𝒃𝒂𝒅 ≤Pr 𝛿𝐻− 𝑣 − 𝜇 ≥
𝜇
3+ Pr 𝛿𝐻
+ 𝑣 − 𝜇 ≥𝜇
3
≤𝟏
𝒏𝟏𝟎𝒙∗(𝜹+ 𝑺 )
![Page 22: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/22.jpg)
Probability that 𝐻 is good?
𝑯 is good if no cut is bad
We know that a single cut S is bad w.p ≤1
𝑛10𝑥∗ 𝛿+ 𝑆
≤1
𝑛10
But 𝟐𝒏 many cuts so we can’t make union bound…
![Page 23: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/23.jpg)
Remedy: beautiful result by Karger
BOUNDING NUMBER OF SMALL CUTS:
Consider an undirected graph 𝐺 = (𝑉, 𝐸) with edge-weights 𝑤. Let
𝑐 be the value of a min-cut.
Then the number of cuts of value 𝛼𝑐 is at most ≤ 𝑛2𝛼
Karger
But our graph is directed, why can we still use the above theorem?
It is Eulerian (w.r.t. to weights 𝑥∗) so any cut of out-degree/in-degree c
corresponds to a cut of value 2c in the undirected graph
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Probability that 𝐻 is good?
• Cuts of value [1,2] at most 𝑛2⋅2 many
• Cuts of value [2,3] at most 𝑛2⋅3 many
• Cuts of value [3,4] at most 𝑛2⋅4 many
• Cuts of value [4,5] at most 𝑛2⋅5 many
• Cuts of value [n-1, n] at most 𝑛2⋅𝑛 many
Prob. that such a cut is bad 𝑛−10⋅1
Prob. that such a cut is bad 𝑛−10⋅2
Prob. that such a cut is bad 𝑛−10⋅3
Prob. that such a cut is bad 𝑛−10⋅4
Prob. that such a cut is bad 𝑛−10⋅(𝑛−1)
![Page 25: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/25.jpg)
Probability that 𝐻 is good?
• Cuts of value [1,2] at most 𝑛2⋅2 many
• Cuts of value [2,3] at most 𝑛2⋅3 many
• Cuts of value [3,4] at most 𝑛2⋅4 many
• Cuts of value [4,5] at most 𝑛2⋅5 many
• Cuts of value [n-1, n] at most 𝑛2⋅𝑛 many
Prob. that such a cut is bad 𝑛−10⋅1
Prob. that such a cut is bad 𝑛−10⋅2
Prob. that such a cut is bad 𝑛−10⋅3
Prob. that such a cut is bad 𝑛−10⋅4
Prob. that such a cut is bad 𝑛−10⋅(𝑛−1)
By union bound,
Pr 𝐻 𝑖𝑠 𝑔𝑜𝑜𝑑 ≥ 1 −
𝑖=1
𝑛−1
𝑛2⋅ 𝑖+1 −10𝑖 = 1 −
𝑖=1
𝑛−1𝑛2
𝑛8𝑖≥ 1 −
1
𝑛
![Page 26: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/26.jpg)
Probability that 𝐻 is good?
• Cuts of value [1,2] at most 𝑛2⋅2 many
• Cuts of value [2,3] at most 𝑛2⋅3 many
• Cuts of value [3,4] at most 𝑛2⋅4 many
• Cuts of value [4,5] at most 𝑛2⋅5 many
• Cuts of value [n-1, n] at most 𝑛2⋅𝑛 many
Prob. that such a cut is bad 𝑛−10⋅1
Prob. that such a cut is bad 𝑛−10⋅2
Prob. that such a cut is bad 𝑛−10⋅3
Prob. that such a cut is bad 𝑛−10⋅4
Prob. that such a cut is bad 𝑛−10⋅(𝑛−1)
By union bound,
Pr 𝐻 𝑖𝑠 𝑔𝑜𝑜𝑑 ≥ 1 −
𝑖=1
𝑛−1
𝑛2⋅ 𝑖+1 −10𝑖 = 1 −
𝑖=1
𝑛−1𝑛2
𝑛8𝑖≥ 1 −
1
𝑛
![Page 27: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/27.jpg)
Probability that 𝐻 is good?
• Cuts of value [1,2] at most 𝑛2⋅2 many
• Cuts of value [2,3] at most 𝑛2⋅3 many
• Cuts of value [3,4] at most 𝑛2⋅4 many
• Cuts of value [4,5] at most 𝑛2⋅5 many
• Cuts of value [n-1, n] at most 𝑛2⋅𝑛 many
Prob. that such a cut is bad 𝑛−10⋅1
Prob. that such a cut is bad 𝑛−10⋅2
Prob. that such a cut is bad 𝑛−10⋅3
Prob. that such a cut is bad 𝑛−10⋅4
Prob. that such a cut is bad 𝑛−10⋅(𝑛−1)
By union bound,
Pr 𝐻 𝑖𝑠 𝑔𝑜𝑜𝑑 ≥ 1 −
𝑖=1
𝑛−1
𝑛2⋅ 𝑖+1 −10𝑖 = 1 −
𝑖=1
𝑛−1𝑛2
𝑛8𝑖≥ 1 −
1
𝑛
![Page 28: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/28.jpg)
Randomized Round
Find an optimal solution 𝑥∗ to LP relaxation
Scale up 𝑥∗ by taking 𝐾 ≔ 1000ln 𝑛 parallel copies of each
edge, each of same LP-value as the original edge
Form 𝐻 by taking each edge with probability equal to its LP-
value
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in 𝐻
Well connected?
Eulerian?
So w.h.p. 𝑯 is well connected and almost Eulerian
We will use these facts to bound the cost of the last step
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Assuming 𝑯 is good, then there exists a circulation on 𝑯 where
each arc has lower bound 𝟏 and upper bound 𝟐
Note that this implies that the cost of a min-cost circulation with lower
bound 1 on each edge in 𝐻 is at most two times the cost of 𝐻
VERIFY CONDITIONS FROM HOFFMAN’s CIRCULATION THM:
• For every arc 𝑒 ∈ 𝐸 we must have 𝑙 𝑒 ≤ 𝑢(𝑒)
• For every 𝑆 ⊂ 𝑉 we must have 𝑙 𝛿− 𝑆 ≤ 𝑢 𝛿+ 𝑆
![Page 30: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/30.jpg)
Assuming 𝑯 is good, then there exists a circulation on 𝑯 where
each arc has lower bound 𝟏 and upper bound 𝟐
Note that this implies that the cost of a min-cost circulation with lower
bound 1 on each edge in 𝐻 is at most two times the cost of 𝐻
VERIFY CONDITIONS FROM HOFFMAN’s CIRCULATION THM:
• For every arc 𝑒 ∈ 𝐸 we must have 𝑙 𝑒 ≤ 𝑢(𝑒)
• For every 𝑆 ⊂ 𝑉 we must have 𝑙 𝛿− 𝑆 ≤ 𝑢 𝛿+ 𝑆
For second condition,
𝑙 𝛿𝐻− 𝑆 ≤
4
3𝐾 𝑥∗ 𝛿− 𝑆 = 2 1 −
1
3𝐾 𝑥∗ 𝛿+ 𝑆 ≤ 𝑢(𝛿𝐻
+ 𝑆 )
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Assuming 𝑯 is good, then there exists a circulation on 𝑯 where
each arc has lower bound 𝟏 and upper bound 𝟐
Note that this implies that the cost of a min-cost circulation with lower
bound 1 on each edge in 𝐻 is at most two times the cost of 𝐻
VERIFY CONDITIONS FROM HOFFMAN’s CIRCULATION THM:
• For every arc 𝑒 ∈ 𝐸 we must have 𝑙 𝑒 ≤ 𝑢(𝑒)
• For every 𝑆 ⊂ 𝑉 we must have 𝑙 𝛿− 𝑆 ≤ 𝑢 𝛿+ 𝑆
For second condition,
𝑙 𝛿𝐻− 𝑆 ≤
4
3𝐾 𝑥∗ 𝛿− 𝑆 = 2 1 −
1
3𝐾 𝑥∗ 𝛿+ 𝑆 ≤ 𝑢(𝛿𝐻
+ 𝑆 )
![Page 32: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/32.jpg)
Randomized Round
Find an optimal solution 𝑥∗ to LP relaxation
Scale up 𝑥∗ by taking 𝐾 ≔ 1000ln 𝑛 parallel copies of each
edge, each of same LP-value as the original edge
Form 𝐻 by taking each edge with probability equal to its LP-
value
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in 𝐻
Expected cost of Tour is at most twice the cost of 𝐻.
• What is the expected cost of 𝑯?
![Page 33: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/33.jpg)
Randomized Round
Find an optimal solution 𝑥∗ to LP relaxation
Scale up 𝑥∗ by taking 𝐾 ≔ 1000ln 𝑛 parallel copies of each
edge, each of same LP-value as the original edge
Form 𝐻 by taking each edge with probability equal to its LP-
value
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in 𝐻
Expected cost of Tour is at most twice the cost of 𝐻.
• What is the expected cost of 𝑯? 𝐾 times the LP cost
![Page 34: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/34.jpg)
Randomized Round
Find an optimal solution 𝑥∗ to LP relaxation
Scale up 𝑥∗ by taking 𝐾 ≔ 1000ln 𝑛 parallel copies of each
edge, each of same LP-value as the original edge
Form 𝐻 by taking each edge with probability equal to its LP-
value
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in 𝐻
Expected cost of Tour is at most twice the cost of 𝐻.
• What is the expected cost of 𝑯? 𝐾 times the LP cost
Okay we are interested in expected cost of 𝐻 conditioned on it being good.
But this is ≤𝐾
1−1/𝑛which between friends is 𝐾 𝑙𝑛 𝑛
![Page 35: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/35.jpg)
Randomized Round
Find an optimal solution 𝑥∗ to LP relaxation
Scale up 𝑥∗ by taking 𝐾 ≔ 1000ln 𝑛 parallel copies of each
edge, each of same LP-value as the original edge
Form 𝐻 by taking each edge with probability equal to its LP-
value
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in 𝐻
Expected cost of Tour is at most twice the cost of 𝐻.
• What is the expected cost of 𝑯? 𝐾 times the LP cost
Okay we are interested in expected cost of 𝐻 conditioned on it being good.
But this is ≤𝐾
1−1/𝑛which between friends is 𝐾
THEOREM:
Randomized round returns an 𝑂(log 𝑛)-approximate tour w.h.p.
Goemans, Harvey, Jain, Singh’10
![Page 36: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/36.jpg)
Main ingredients
• O(log n) guarantee from ensuring
connectivity
• Chernoff bounds ensured
concentration which was useful for
bounding the parity correction cost
• Karger’s result allowed us to apply
the union bound in a smart way
![Page 37: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/37.jpg)
Main ingredients
• O(log n) guarantee from ensuring
connectivity
• Chernoff bounds ensured
concentration which was useful for
bounding the parity correction cost
• Karger’s result allowed us to apply
the union bound in a smart way
SPANNING TREES:
• Always connected
• Negative correlation still allows for
the application of Chernoff’bounds
![Page 38: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/38.jpg)
THIN SPANNING TREES
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Thin Trees
Thin Trees:
Let 𝑇 be a spanning tree of 𝐺 = (𝑉, 𝐸, 𝑤) and 𝑥∗ an optimal LP solution.
𝑻 is 𝜶-thin (w.r.t) 𝒙∗ if for every 𝑺 ⊂ 𝑽
𝜹𝑻 𝑺 ≤ 𝜶𝒙∗(𝜹+ 𝑺 )
Thin Trees to Tours:
Let 𝑇 be an 𝛼-thin spanning tree of 𝐺 = (𝑉, 𝐸, 𝑤) and 𝑥∗ an optimal LP
solution.
Then there is a tour of value at most 𝒘 𝑻 + 𝑶 𝜶 𝑶𝑷𝑻𝑳𝑷
![Page 40: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/40.jpg)
Outline of proof
• The circulation network 𝐺 that for each edge 𝑒 ∈ 𝐸 has
𝑙 𝑒 = 1, 𝑒 ∈ 𝑇0, 𝑒 ∉ 𝑇
and 𝑢 𝑒 = 𝑙 𝑒 + 𝛼𝑥𝑒∗
has a feasible circulation.
• This circulation has cost at most 𝑒∈𝐸 𝑢 𝑒 𝑤(𝑒) ≤ 𝑤 𝑇 + 𝛼𝑂𝑃𝑇𝐿𝑃
• Hence, there is an integral min-cost circulation satisfying the lower bounds
of cost at most 𝑤 𝑇 + 𝛼𝑂𝑃𝑇𝐿𝑃
Remains to prove this!
![Page 41: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/41.jpg)
The circulation network 𝐺 that for each edge 𝑒 ∈ 𝐸 has
𝑙 𝑒 = 1, 𝑒 ∈ 𝑇0, 𝑒 ∉ 𝑇
and 𝑢 𝑒 = 𝑙 𝑒 + 𝛼𝑥𝑒∗
has a feasible circulation.
VERIFY CONDITIONS FROM HOFFMAN’s CIRCULATION THM:
• For every arc 𝑒 ∈ 𝐸 we must have 𝑙 𝑒 ≤ 𝑢(𝑒)
• For every 𝑆 ⊂ 𝑉 we must have 𝑙 𝛿− 𝑆 ≤ 𝑢 𝛿+ 𝑆
For second condition,
𝑙 𝛿− 𝑆 ≤ 𝛼𝑥∗ 𝛿− 𝑆 = 𝛼𝑥∗ 𝛿+ 𝑆 ≤ 𝑢(𝛿+ 𝑆 )
because tree is 𝛼-thin
![Page 42: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/42.jpg)
The circulation network 𝐺 that for each edge 𝑒 ∈ 𝐸 has
𝑙 𝑒 = 1, 𝑒 ∈ 𝑇0, 𝑒 ∉ 𝑇
and 𝑢 𝑒 = 𝑙 𝑒 + 𝛼𝑥𝑒∗
has a feasible circulation.
VERIFY CONDITIONS FROM HOFFMAN’s CIRCULATION THM:
• For every arc 𝑒 ∈ 𝐸 we must have 𝑙 𝑒 ≤ 𝑢(𝑒)
• For every 𝑆 ⊂ 𝑉 we must have 𝑙 𝛿− 𝑆 ≤ 𝑢 𝛿+ 𝑆
For second condition,
𝑙 𝛿− 𝑆 ≤ 𝛼𝑥∗ 𝛿− 𝑆 = 𝛼𝑥∗ 𝛿+ 𝑆 ≤ 𝑢(𝛿+ 𝑆 )
because tree is 𝛼-thin by def. of u
![Page 43: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/43.jpg)
Thin Trees to Tours:
Let 𝑇 be an 𝛼-thin spanning tree of 𝐺 = (𝑉, 𝐸, 𝑤) and 𝑥∗ an optimal LP
solution.
Then there is a tour of value at most 𝒘 𝑻 + 𝑶 𝜶 𝑶𝑷𝑻𝑳𝑷
METHODS FOR FINDING
THIN TREES
![Page 44: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/44.jpg)
Spanning Tree Round
• What is the expected cost of the spanning tree? 1 −1
𝑛𝑂𝑃𝑇𝐿𝑃
• How thin is the tree? We can apply upper Chernoff bound:
Find an optimal solution 𝑥∗ to LP relaxation
Let 𝑧 𝑢𝑣 = (𝑥𝑢𝑣∗ +𝑥𝑣𝑢
∗ ) ⋅ 1 −1
𝑛be a feasible point to the
spanning tree polytope and sample a spanning tree T with
negative correlation satisfying these marginals
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in spanning tree
Pr |𝛿𝑇 𝑆 > 1000log 𝑛
log log 𝑛𝑥∗(𝛿+ 𝑆 )] <
1
𝑛10𝑥∗(𝛿+ 𝑆 )
For any 𝑆 ⊂ 𝑉,
This together with Karger implies that the tree is w.h.p 𝑶𝐥𝐨𝐠 𝒏
𝐥𝐨𝐠 𝐥𝐨𝐠 𝒏-thin
![Page 45: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/45.jpg)
Spanning Tree Round
• What is the expected cost of the spanning tree? 1 −1
𝑛𝑂𝑃𝑇𝐿𝑃
• How thin is the tree? We can apply upper Chernoff bound:
Find an optimal solution 𝑥∗ to LP relaxation
Let 𝑧 𝑢𝑣 = (𝑥𝑢𝑣∗ +𝑥𝑣𝑢
∗ ) ⋅ 1 −1
𝑛be a feasible point to the
spanning tree polytope and sample a spanning tree T with
negative correlation satisfying these marginals
Compute Eulerian graph by finding an (integral) min cost
circulation with lower bound 1 for each arc in spanning tree
Pr |𝛿𝑇 𝑆 > 1000log 𝑛
log log 𝑛𝑥∗(𝛿+ 𝑆 )] <
1
𝑛10𝑥∗(𝛿+ 𝑆 )
For any 𝑆 ⊂ 𝑉,
This together with Karger implies that the tree is w.h.p 𝑶𝐥𝐨𝐠 𝒏
𝐥𝐨𝐠 𝐥𝐨𝐠 𝒏-thin
Theorem:
Spanning tree algorithm is a 𝑶𝒍𝒐𝒈 𝒏
𝒍𝒐𝒈 𝒍𝒐𝒈 𝒏-approximation algorithm for ATSP
Asadpour, Goemans, Madry, Oveis Gharan, Saberi’10
![Page 46: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/46.jpg)
State of the Art of Thin Tree Approach
Theorem:
A randomized polytime algorithm gives a 𝑶𝒍𝒐𝒈 𝒏
𝒍𝒐𝒈 𝒍𝒐𝒈 𝒏-thin tree
Asadpour, Goemans, Madry, Oveis Gharan, Saberi’10
Theorem:
There exists a 𝑶 𝒑𝒐𝒍𝒚𝒍𝒐𝒈𝒍𝒐𝒈 𝒏 -thin tree
Anari, Oveis Gharan’14
These results imply a 𝑂log 𝑛
log log 𝑛-approximation algorithm and a
𝑂(𝑝𝑜𝑙𝑦 log log 𝑛) bound on the integrality gap
![Page 47: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/47.jpg)
Two Approaches
Easy to Find Eulerian graph
Repeatedly find cycle-covers to get
log2 𝑛-approximation
[Frieze, Galbiati, Maffiolo’82]
0.99 log2 𝑛-approximation[Bläser’03]
0.84 log2 𝑛-approximation [Kaplan, Lewenstein, Shafrir, Sviridenko’05]
0.67 log2 𝑛-approximation
[Feige, Singh’07]
Easy to Find Connected Graph
𝑂(log 𝑛 / log log 𝑛)-approximation[Asadpour, Goemans, Madry, Oveis Gharan, Saberi’10]
𝑂(1)-approximation for planar and
bounded genus graphs
[Oveis Gharan, Saberi’11]
O(𝑝𝑜𝑙𝑦 𝑙𝑜𝑔 log 𝑛) bound on integrality gap (generalization of Kadison-Singer)
[Anari, Oveis Gharan’14]
![Page 48: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/48.jpg)
To summarize the two approaches
Best approximation algorithm: 𝑂(log 𝑛
log log 𝑛)
Best upper bound on integrality gap: 𝑂(𝑝𝑜𝑙𝑦 log log 𝑛)
Best lower bound on integrality gap: 2This is believed to be
close to the truth
No better guarantees for shortest path metrics on unweighted
graphs for which there was recent improvements for the symmetric TSP
Oveis Gharan, Saberi & Singh
Mömke & S
Mucha
1,33
1,38
1,43
1,48
Dec'10 April'11 Sep'11 Jan'12
Sebö & Vygen
![Page 49: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/49.jpg)
To summarize the two approaches
Best approximation algorithm: 𝑂(log 𝑛
log log 𝑛)
Best upper bound on integrality gap: 𝑂(𝑝𝑜𝑙𝑦 log log 𝑛)
Best lower bound on integrality gap: 2This is believed to be
close to the truth
No better guarantees for shortest path metrics on unweighted
graphs for which there was recent improvements for the symmetric TSP
• Main difficulty in cycle cover approach is to bound #iterations
• In thin-tree approach we reduce ATSP to an unweighted problem
![Page 50: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/50.jpg)
A NEW APPROACH
Find Eulerian graph with some connectivity requirements
![Page 51: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/51.jpg)
Relaxing Connectivity
𝒙 𝜹+ 𝒗 = 𝒙(𝜹− 𝒗 ) for all 𝑣 ∈ 𝑉
𝒙 𝜹+ 𝑺 ≥ 𝟏 for all 𝐒 ⊂ 𝑽
𝑥 ≥ 0
Instead of
𝒙 𝜹+ 𝒗 = 𝒙(𝜹− 𝒗 ) for all 𝑣 ∈ 𝑉
𝒙 𝜹+ 𝑺 ≥ 𝟏 for those 𝐒 ∈ 𝑪
𝑥 ≥ 0
Do for smart C
![Page 52: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/52.jpg)
Application of approach
THEOREM:
For ATSP on node-weighted graphs, the integrality gap is at most 15.
Moreover, there is a 27-approximation algorithm.
![Page 53: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/53.jpg)
Application of approach
THEOREM:
For ATSP on node-weighted graphs, the integrality gap is at most 15.
Moreover, there is a 27-approximation algorithm.
2
5
1
Exist 𝑓: 𝑉 → 𝑅+ s.t
𝑤(𝑢, 𝑣) = 𝑓(𝑢) for all 𝑢, 𝑣 ∈ 𝐸
2
5
1
5
5 1
7
1
![Page 54: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/54.jpg)
Application of approach
THEOREM:
For ATSP on node-weighted graphs, the integrality gap is at most 15.
Moreover, there is a 27-approximation algorithm.
2
5
1
Exist 𝑓: 𝑉 → 𝑅+ s.t
𝑤(𝑢, 𝑣) = 𝑓(𝑢) for all 𝑢, 𝑣 ∈ 𝐸
2
5
1
5
5 11
![Page 55: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/55.jpg)
Application of approach
THEOREM:
For ATSP on node-weighted graphs, the integrality gap is at most 15.
Moreover, there is a 27-approximation algorithm.
Exist 𝑓: 𝑉 → 𝑅+ s.t
𝑤(𝑢, 𝑣) = 𝑓(𝑢) for all 𝑢, 𝑣 ∈ 𝐸
2
5
1
5
5 11
2
5
1Value = 2+5+1+5+1=14
Generalizes shortest path metrics considered for symmetric TSP
![Page 56: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/56.jpg)
OUR APPROACH
Cousin to repeated cycle cover approach
Distant relative to thin tree approach
![Page 57: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/57.jpg)
Relaxing Connectivity
𝒙 𝜹+ 𝒗 = 𝒙(𝜹− 𝒗 ) for all 𝑣 ∈ 𝑉
𝒙 𝜹+ 𝑺 ≥ 𝟏 for all 𝐒 ⊂ 𝑽
𝑥 ≥ 0
Instead of
𝒙 𝜹+ 𝒗 = 𝒙(𝜹− 𝒗 ) for all 𝑣 ∈ 𝑉
𝒙 𝜹+ 𝑺 ≥ 𝟏 for those 𝐒 ∈ 𝑪
𝑥 ≥ 0
Do for smart C
![Page 58: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/58.jpg)
Local Connectivity ATSP
INPUT: an edge-weighted digraph 𝐺 = 𝑉, 𝐸,𝑤 , a partition 𝑉1 ∪⋯∪ 𝑉𝑘 of 𝑉
![Page 59: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/59.jpg)
Local Connectivity ATSP
INPUT: an edge-weighted digraph 𝐺 = 𝑉, 𝐸,𝑤 , a partition 𝑉1 ∪⋯∪ 𝑉𝑘 of 𝑉
OUTPUT: an Eulerian multisubset of edges 𝐹 such that
each cut (𝑉𝑖 , 𝑉𝑖) is “covered” by 𝐹
![Page 60: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/60.jpg)
Local Connectivity ATSP
INPUT: an edge-weighted digraph 𝐺 = 𝑉, 𝐸,𝑤 , a partition 𝑉1 ∪⋯∪ 𝑉𝑘 of 𝑉
OUTPUT: an Eulerian multisubset of edges 𝐹 such that
each cut (𝑉𝑖 , 𝑉𝑖) is “covered” by 𝐹
![Page 61: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/61.jpg)
Local Connectivity ATSP
An algorithm is 𝜶-light if it always outputs a solution 𝐹 s.t.
for each component in (𝑽, 𝑭), #𝒆𝒅𝒈𝒆𝒔
#𝒗𝒆𝒓𝒕𝒊𝒄𝒆𝒔≤ 𝜶,
6
5
1
Note: designing an 𝜶-light algorithm is “easier” than an 𝜶-approximation for ATSP
![Page 62: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/62.jpg)
Our main technical result
THEOREM:
If there is an 𝛼-light algorithm A for Local-Connectivity ATSP then the
integrality gap for ATSP is at most 5𝛼.
![Page 63: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/63.jpg)
Our main technical result
THEOREM:
If there is an 𝛼-light algorithm A for Local-Connectivity ATSP then the
integrality gap for ATSP is at most 5𝛼.
Moreover, a 9𝛼-approximate tour can be found in polynomial time if A runs
in polynomial time.
The problems are equivalent up to a small constant factor
There is an easy 3-light algorithm for node-weighted metric
(only part where special metric is used)
![Page 64: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/64.jpg)
PROOF IDEA OF MAIN THM
![Page 65: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/65.jpg)
Repeatedly solve Local-Connectivity ATSP
with the current connected subgraphs as partitions
![Page 66: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/66.jpg)
Repeatedly solve Local-Connectivity ATSP
with the current connected subgraphs as partitions
![Page 67: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/67.jpg)
Repeatedly solve Local-Connectivity ATSP
with the current connected subgraphs as partitions
Cost of red edges ≤ 𝜶𝑶𝑷𝑻
![Page 68: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/68.jpg)
Cost of red edges ≤ 𝜶𝑶𝑷𝑻
Repeatedly solve Local-Connectivity ATSP
with the current connected subgraphs as partitions
![Page 69: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/69.jpg)
Repeatedly solve Local-Connectivity ATSP
with the current connected subgraphs as partitions
Cost of red edges ≤ 𝜶𝑶𝑷𝑻
![Page 70: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/70.jpg)
Repeatedly solve Local-Connectivity ATSP
with the current connected subgraphs as partitions
Cost of red edges ≤ 𝜶𝑶𝑷𝑻
Cost of blue edges≤ 𝜶𝑶𝑷𝑻
![Page 71: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/71.jpg)
Repeatedly solve Local-Connectivity ATSP
with the current connected subgraphs as partitions
Cost of red edges ≤ 𝜶𝑶𝑷𝑻
Cost of blue edges≤ 𝜶𝑶𝑷𝑻
Total cost ≤ #𝒊𝒕𝒆𝒓𝒂𝒕𝒊𝒐𝒏𝒔 ⋅ 𝜶𝑶𝑷𝑻 ≤ 𝒍𝒐𝒈 𝒏 ⋅ 𝜶𝑶𝑷𝑻
![Page 72: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/72.jpg)
Lexicographic Initialization
![Page 73: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/73.jpg)
𝑯𝟏∗ = largest connected component that has 𝟐𝜶-light tour
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Lexicographic Initialization
![Page 74: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/74.jpg)
𝑯𝟏∗ = largest connected component that has 𝟐𝜶-light tour
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Lexicographic Initialization
If red tour is 𝛼-light then |F| ≤ 𝛼|𝑉1∗|
Why? F
![Page 75: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/75.jpg)
𝑯𝟏∗ = largest connected component that has 2𝜶-light tour
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Lexicographic Initialization
If red tour is 𝛼-light then |F| ≤ 𝛼|𝑉1∗|
Why? Otherwise 𝐹 spans more than
𝑉1∗ vertices and this should be
our largest component!
F
![Page 76: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/76.jpg)
𝑯𝟏∗ = largest connected component that has 𝟐𝜶-light tour
𝑯𝟐∗ = largest disjoint connected component that has 2𝜶-light tour
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Lexicographic Initialization
F
Intersects 𝐻2∗ but not 𝐻1
∗:
𝐹 ≤ 𝛼|𝑉2∗|
𝐻2∗
![Page 77: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/77.jpg)
𝑯𝟏∗ = largest connected component that has 𝟐𝜶-light tour
𝑯𝟐∗ = largest disjoint connected component that has 𝟐𝜶-light tour
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Lexicographic Initialization
𝐻2∗
![Page 78: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/78.jpg)
𝑯𝟏∗ = largest connected component that has 𝟐𝜶-light tour
𝑯𝟐∗ = largest disjoint connected component that has 𝟐𝜶-light tour
𝑯𝒌∗ = largest disjoint connected component that has 𝟐𝜶-light tour
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Lexicographic Initialization
𝐻2∗
…
𝐻3∗
𝐻4∗
𝐻5∗ 𝐻6
∗ 𝐻7∗
![Page 79: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/79.jpg)
Cost of initialization at most 𝟐𝜶𝑶𝑷𝑻 since each component is 𝟐𝜶-light
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Wish List for Merging Step
𝐻2∗ 𝐻3
∗
𝐻4∗
𝐻5∗ 𝐻6
∗ 𝐻7∗
![Page 80: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/80.jpg)
Cost of initialization at most 𝟐𝜶𝑶𝑷𝑻 since each component is 𝟐𝜶-light
Suppose we can connect graph by adding 𝜶-light components so that
we charge at most one to each component of initialization
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Wish List for Merging Step
𝐻2∗ 𝐻3
∗
𝐻4∗
𝐻5∗ 𝐻6
∗ 𝐻7∗
![Page 81: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/81.jpg)
Cost of initialization at most 𝟐𝜶𝑶𝑷𝑻 since each component is 𝟐𝜶-light
Suppose we can connect graph by adding 𝜶-light components so that
we charge at most one to each component of initialization
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Wish List for Merging Step
𝐻2∗ 𝐻3
∗
𝐻4∗
𝐻5∗ 𝐻6
∗ 𝐻7∗
𝑐𝑜𝑠𝑡 ≤ 𝛼|𝑉1∗|
![Page 82: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/82.jpg)
Cost of initialization at most 𝟐𝜶𝑶𝑷𝑻 since each component is 𝟐𝜶-light
Suppose we can connect graph by adding 𝜶-light components so that
we charge at most one to each component of initialization
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Wish List for Merging Step
𝐻2∗ 𝐻3
∗
𝐻4∗
𝐻5∗ 𝐻6
∗ 𝐻7∗
𝑐𝑜𝑠𝑡 ≤ 𝛼|𝑉1∗|
≤ 𝛼|𝑉2∗|
![Page 83: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/83.jpg)
Cost of initialization at most 𝟐𝜶𝑶𝑷𝑻 since each component is 𝟐𝜶-light
Suppose we can connect graph by adding 𝜶-light components so that
we charge at most one to each component of initialization
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Wish List for Merging Step
𝐻2∗ 𝐻3
∗
𝐻4∗
𝐻5∗ 𝐻6
∗ 𝐻7∗
𝑐𝑜𝑠𝑡 ≤ 𝛼|𝑉1∗|
≤ 𝛼|𝑉2∗|
≤ 𝛼|𝑉3∗|
![Page 84: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/84.jpg)
Cost of initialization at most 𝟐𝜶𝑶𝑷𝑻 since each component is 𝟐𝜶-light
Suppose we can connect graph by adding 𝜶-light components so that
we charge at most one to each component of initialization
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Wish List for Merging Step
𝐻2∗ 𝐻3
∗
𝐻4∗
𝐻5∗ 𝐻6
∗ 𝐻7∗
𝑐𝑜𝑠𝑡 ≤ 𝛼|𝑉1∗|
≤ 𝛼|𝑉2∗|
≤ 𝛼|𝑉3∗|≤ 𝛼|𝑉4
∗|
![Page 85: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/85.jpg)
Cost of initialization at most 𝟐𝜶𝑶𝑷𝑻 since each component is 𝟐𝜶-light
Suppose we can connect graph by adding 𝜶-light components so that
we charge at most one to each component of initialization
Then total cost ≤ 𝟑 𝜶𝑶𝑷𝑻
𝐻1∗ = (𝑉1
∗, 𝐸1∗)
Wish List for Merging Step
𝐻2∗ 𝐻3
∗
𝐻4∗
𝐻5∗ 𝐻6
∗ 𝐻7∗
𝑐𝑜𝑠𝑡 ≤ 𝛼|𝑉1∗|
≤ 𝛼|𝑉2∗|
≤ 𝛼|𝑉3∗|≤ 𝛼|𝑉4
∗|
![Page 86: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/86.jpg)
Some open problems…
![Page 87: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/87.jpg)
Node-Weighted Symmetric TSP
• distance of 𝑢, 𝑣 ∈ 𝐸 is w 𝑢 + 𝑤 𝑣
1
3
13
23
5
5
4
4
4
Can you do better than Christofides (1.5)?
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Constant for ATSP on General Metrics?
THIN TREE:
Does any k-edge connected graph have a 𝑂(1)-thin tree, i.e., a tree T
such that for each 𝑆 ⊂ 𝑉,
𝛿𝑇 𝑆 ≤𝛼
𝑘|𝛿𝐺 𝑆 | ?
LOCAL-CONNECTIVITY ATSP:
Can you design a 𝑂 1 -light algorithm for general metrics?
![Page 89: Lecture 3: Asymmetric TSP (cont) - Max Planck Societyresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect3.pdf · Lecture 3: Asymmetric TSP (cont) Ola Svensson ADFOCS](https://reader033.vdocuments.us/reader033/viewer/2022051813/6030f29fb702d30a7a671133/html5/thumbnails/89.jpg)
Summary
• Overview of Existing Approaches
• New approach that relaxes connectivity
• Integrality gap for node-weighted metrics is ≤ 13
• Nice Open Questions both for Symmetric and Asymmetric TSP
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