s-t path tspresources.mpi-inf.mpg.de/conferences/adfocs-15/material/david-lect… · david p....
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s-t path TSP
David P. WilliamsonCornell University
August 17-21, 2015ADFOCS
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David P. Williamson s-t path TSP
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David P. Williamson s-t path TSP
...come visit us at Cornell!
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David P. Williamson s-t path TSP
The s-t path traveling salesman problem
The s-t Path Traveling Salesman Problem (s-t PathTSP)Input:
• A complete, undirected graph G = (V ,E );• Edge costs c(i , j) ≥ 0 for all e = (i , j) ∈ E ;• Vertices s, t ∈ V .
Goal: Find the min-cost path that starts at s, ends at t, and visitsevery other vertex exactly once.
Costs are symmetric (c(i , j) = c(j , i)) and obey the triangleinequality (c(i , k) ≤ c(i , j) + c(j , k)).
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David P. Williamson s-t path TSP
Hoogeveen’s algorithmLet F be the min-cost spanning tree. Let T be the set of verticeswhose parity needs changing: s iff s has even degree in F , t iff thas even degree in F , and v 6= s, t iff v has odd degree. Then finda minimum-cost T -join J . Find Eulerian path on F ∪ J ; shortcutto an s-t Hamiltonian path.
s
t
s
t
s
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s
t
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David P. Williamson s-t path TSP
Hoogeveen’s algorithmLet F be the min-cost spanning tree. Let T be the set of verticeswhose parity needs changing: s iff s has even degree in F , t iff thas even degree in F , and v 6= s, t iff v has odd degree. Then finda minimum-cost T -join J . Find Eulerian path on F ∪ J ; shortcutto an s-t Hamiltonian path.
s
t
s
t
s
t
s
t
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David P. Williamson s-t path TSP
Hoogeveen’s algorithmLet F be the min-cost spanning tree. Let T be the set of verticeswhose parity needs changing: s iff s has even degree in F , t iff thas even degree in F , and v 6= s, t iff v has odd degree. Then finda minimum-cost T -join J . Find Eulerian path on F ∪ J ; shortcutto an s-t Hamiltonian path.
s
t
s
t
s
t
s
t
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David P. Williamson s-t path TSP
Hoogeveen’s algorithmLet F be the min-cost spanning tree. Let T be the set of verticeswhose parity needs changing: s iff s has even degree in F , t iff thas even degree in F , and v 6= s, t iff v has odd degree. Then finda minimum-cost T -join J . Find Eulerian path on F ∪ J ; shortcutto an s-t Hamiltonian path.
s
t
s
t
s
t
s
t
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David P. Williamson s-t path TSP
Improvements
TheoremHoogeveen’s algorithm is a 5
3 -approximation algorithm.
Recent improvements on Hoogeveen’s algorithm.Hoogeveen (1991) 5
3
An, Kleinberg, Shmoys (2012) 1+√
52 ≈ 1.618
Sebő (2013) 85 = 1.6
Vygen (2015) 1.599
Goal: Understand the An et al. and Sebő algorithm and analysis.
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David P. Williamson s-t path TSP
A Linear Programming Relaxation
Min∑e∈E
cexe
subject to: x(δ(v)) ={
1, v = s, t,2, v 6= s, t,
x(δ(S)) ≥{
1, |S ∩ {s, t}| = 1,2, |S ∩ {s, t}| 6= 1,
0 ≤ xe ≤ 1, ∀e ∈ E ,
where δ(S) is the set of edges with exactly one endpoint in S, andx(E ′) ≡
∑e∈E ′ xe .
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David P. Williamson s-t path TSP
The spanning tree polytope
The spanning tree polytope (convex hull of all spanning trees) isdefined by the following inequalities:
x(E ) = |V | − 1,x(E (S)) ≤ |S| − 1, ∀|S| ⊆ V , |S| ≥ 2,xe ≥ 0, ∀e ∈ E ,
where E (S) is the set of all edges with both endpoints in S.
LemmaAny solution x feasible for the s-t path TSP LP relaxation is in thespanning tree polytope.
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David P. Williamson s-t path TSP
A warmup to the improvements
Let OPTLP be the value of an optimal solution x∗ to the LPrelaxation.
Theorem (An, Kleinberg, Shmoys (2012))Hoogeveen’s algorithm returns a solution of cost at most 5
3OPTLP .
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David P. Williamson s-t path TSP
An extremely useful lemma
Let F be a spanning tree, and let T be the vertices whose parityneeds fixing in F .
DefinitionS is an odd set if |S ∩ T | is odd.
LemmaLet S be an odd set. If |S ∩ {s, t}| = 1, then |F ∩ δ(S)| is even. If|S ∩ {s, t}| 6= 1, then |F ∩ δ(S)| is odd.
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David P. Williamson s-t path TSP
T -join LP
The solution to the following linear program is the minimum-costT -join for costs c ≥ 0:
Min∑e∈E
cexe
subject to: x(δ(S)) ≥ 1, ∀S ⊆ V , |S ∩ T | oddxe ≥ 0, ∀e ∈ E .
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David P. Williamson s-t path TSP
Proof of theorem
Theorem (An, Kleinberg, Shmoys (2012))Hoogeveen’s algorithm returns a solution of cost at most 5
3OPTLP .
LemmaLet S be an odd set. If |S ∩ {s, t}| = 1, then |F ∩ δ(S)| is even. If|S ∩ {s, t}| 6= 1, then |F ∩ δ(S)| is odd.
Min∑e∈E
cexe
x(δ(S)) ≥ 1, ∀S ⊆ V , |S ∩ T | oddxe ≥ 0, ∀e ∈ E .
Basic idea: Show that y = 13χF + 1
3x∗ is feasible for T -join LP,
where x∗ is solution to LP relaxation, and χF is characteristicvector for spanning tree F .
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David P. Williamson s-t path TSP
Convex combination
Let x∗ be an optimal LP solution. Let χF be the characteristicvector of a set of edges F , so that
χF (e) ={
1 e ∈ F0 e /∈ F
Since x∗ is in the spanning tree polytope, can write x∗ as a convexcombination of spanning trees F1, . . . ,Fk :
x∗ =k∑
i=1λiχFi ,
such that∑k
i=1 λi = 1, λi ≥ 0.
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David P. Williamson s-t path TSP
Best-of-Many Christofides’ Algorithm
An, Kleinberg, Shmoys (2012) propose the Best-of-ManyChristofides’ algorithm: given optimal LP solution x∗, computeconvex combination of spanning trees
x∗ =k∑
i=1λiχFi .
For each spanning tree Fi , let Ti be the set of vertices whose parityneeds fixing, let Ji be the minimum-cost Ti -join. Find s-tHamiltonian path by shortcutting Fi ∪ Ji . Return the shortest pathfound over all i .
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David P. Williamson s-t path TSP
Best-of-Many Christofides’ Algorithm
x∗ =k∑
i=1λiχFi .
For each spanning tree Fi , let Ti be the set of vertices whose parityneeds fixing, Ji be the minimum-cost Ti -join. Find s-t Hamiltonianpath by shortcutting Fi ∪ Ji . Return the shortest path found overall i .
TheoremThe Best-of-Many Christofides’ algorithm returns a solution of costat most 5
3OPTLP .
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David P. Williamson s-t path TSP
Proof
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David P. Williamson s-t path TSP
Improvement?
To do better, we need to improve the analysis for the costs of theTi -joins; recall that we use that
yi =13χFi +
13x∗
is feasible for the Ti -join LP.
Consideryi = αχFi + βx∗.
Then the cost of the best s-t Hamiltonian path is at most
(1+ α+ β)OPTLP .
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David P. Williamson s-t path TSP
Improvement?
Proof that yi feasible for Ti -join LP had two cases. Assume S odd(|S ∩ Ti | odd).
If |S ∩ {s, t}| 6= 1, then
yi(δ(S)) = α|Fi ∩ δ(S)|+ βx∗(δ(S)) ≥ α+ 2β.
We will want α+ 2β ≥ 1, so the Ti -join LP constraint is satisfied.
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David P. Williamson s-t path TSP
Improvement?
If |S ∩ {s, t}| = 1, then
yi(δ(S)) = α|Fi ∩ δ(S)|+ βx∗(δ(S)) ≥ 2α+ βx∗(δ(S)).
Since we assume α+ 2β ≥ 1, we only run into problems if
x∗(δ(S)) < 1− 2αβ
.
Note that α = 0, β = 12 works if x∗(δ(S)) ≥ 2 for all S ⊂ V , and
gives a tour of cost at most 32OPTLP .
So focus on cuts for which x∗(δ(S)) < 2, and add an extra“correction” term to yi to handle these cuts.
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David P. Williamson s-t path TSP
Improvement?
If |S ∩ {s, t}| = 1, then
yi(δ(S)) = α|Fi ∩ δ(S)|+ βx∗(δ(S)) ≥ 2α+ βx∗(δ(S)).
Since we assume α+ 2β ≥ 1, we only run into problems if
x∗(δ(S)) < 1− 2αβ
.
Note that α = 0, β = 12 works if x∗(δ(S)) ≥ 2 for all S ⊂ V , and
gives a tour of cost at most 32OPTLP .
So focus on cuts for which x∗(δ(S)) < 2, and add an extra“correction” term to yi to handle these cuts.
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David P. Williamson s-t path TSP
Improvement?
If |S ∩ {s, t}| = 1, then
yi(δ(S)) = α|Fi ∩ δ(S)|+ βx∗(δ(S)) ≥ 2α+ βx∗(δ(S)).
Since we assume α+ 2β ≥ 1, we only run into problems if
x∗(δ(S)) < 1− 2αβ
.
Note that α = 0, β = 12 works if x∗(δ(S)) ≥ 2 for all S ⊂ V , and
gives a tour of cost at most 32OPTLP .
So focus on cuts for which x∗(δ(S)) < 2, and add an extra“correction” term to yi to handle these cuts.
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David P. Williamson s-t path TSP
τ -Narrow Cuts
DefinitionS is τ -narrow if x∗(δ(S)) < 1+ τ for fixed τ ≤ 1.
Only S such that |S ∩ {s, t}| = 1 are τ -narrow.
DefinitionLet Cτ be all τ -narrow cuts S with s ∈ S.
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David P. Williamson s-t path TSP
τ -Narrow Cuts
The τ -narrow cuts in Cτ have a nice structure.
Theorem (An, Kleinberg, Shmoys (2012))If S1, S2 ∈ Cτ , S1 6= S2, then either S1 ⊂ S2 or S2 ⊂ S1.
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David P. Williamson s-t path TSP
First need to show that
x∗(δ(S1)) + x∗(δ(S2)) ≥ x∗(δ(S1 − S2)) + x∗(δ(S2 − S1)).
S1 S2
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David P. Williamson s-t path TSP
Proof of theorem
Theorem (An, Kleinberg, Shmoys (2012))If S1, S2 ∈ Cτ , S1 6= S2, then either S1 ⊂ S2 or S2 ⊂ S1.
So the τ -narrow cuts look like s ∈ Q1 ⊂ Q2 ⊂ · · · ⊂ Qk ⊂ V .
s . . . t
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David P. Williamson s-t path TSP
Proof of theorem
Theorem (An, Kleinberg, Shmoys (2012))If S1, S2 ∈ Cτ , S1 6= S2, then either S1 ⊂ S2 or S2 ⊂ S1.
So the τ -narrow cuts look like s ∈ Q1 ⊂ Q2 ⊂ · · · ⊂ Qk ⊂ V .
s . . . t
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David P. Williamson s-t path TSP
Correction Factor
Let eQ be the minimum-cost edge in δ(Q). Then consider thefollowing (from Gao (2014)):
yi = αχFi + βx∗ +∑
Q∈Cτ ,|Q∩Ti | odd(1− 2α− βx∗(δ(Q)))χeQ
for α, β, τ ≥ 0 such that
α+ 2β = 1 and τ =1− 2αβ
− 1.
Theoremyi is feasible for the Ti -join LP.
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David P. Williamson s-t path TSP
Proof
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David P. Williamson s-t path TSP
Two Lemmas
Recall x∗ =∑k
i=1 λiχFi , with∑k
i=1 λi = 1 and λi ≥ 0. So λi is aprobability distribution on the trees Fi ; probability of Fi is λi .
LemmaLet F be a randomly sampled tree Fi , and T the correspondingvertices Ti . Let Q ∈ Cτ be a τ -narrow cut. Then
Pr[|δ(Q) ∩ F| = 1] ≥ 2− x∗(δ(Q))
Pr[|Q ∩ T | odd] ≤ x∗(δ(Q))− 1.
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David P. Williamson s-t path TSP
Two LemmasRecall eQ is the cheapest edge crossing a τ -narrow cut Q ∈ Cτ .
Lemma ∑Q∈Cτ
ceQ ≤∑e∈E
cex∗e .
s
Q1 Q2
v . . . t
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David P. Williamson s-t path TSP
An-Kleinberg-Shmoys
Theorem (An, Kleinberg, and Shmoys (2012))
Best-of-Many Christofides’ is a 1+√
52 -approximation algorithm for
s-t path TSP.
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David P. Williamson s-t path TSP
Proof of AKSFor the proof, recall that eQ is min-cost edge in δ(Q), Cτ are thecuts Q with x∗(δ(Q)) < 1+ τ ,
yi = αχFi + βx∗ +∑
Q∈Cτ ,|Q∩Ti | odd(1− 2α− βx∗(δ(Q)))χeQ
is feasible for the Ti -join LP, and
LemmaLet F be a randomly sampled tree Fi , and T the correspondingvertices Ti . Let Q ∈ Cτ be a τ -narrow cut. Then
Pr[|δ(Q) ∩ F| = 1] ≥ 2− x∗(δ(Q))
Pr[|Q ∩ T | odd] ≤ x∗(δ(Q))− 1.
Lemma ∑Q∈Cτ
ceQ ≤∑e∈E
cex∗e .
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David P. Williamson s-t path TSP
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David P. Williamson s-t path TSP
Sebő’s Improvement
Sebő (2013) gives a tighter analysis of the Best-of-ManyChristofides’ algorithm. For spanning tree Fi , let F st
i be the set ofedges in the s-t path in Fi . Recall from the proof of Hoogeven’salgorithm that Fi − F st
i is also a Ti -join, so c(Ji) ≤ c(Fi − F sti ).
s
t
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David P. Williamson s-t path TSP
Sebő’s Improvement
Sebő (2013) gives a tighter analysis of the Best-of-ManyChristofides’ algorithm. For spanning tree Fi , let F st
i be the set ofedges in the s-t path in Fi . Recall from the proof of Hoogeven’salgorithm that Fi − F st
i is also a Ti -join, so c(Ji) ≤ c(Fi − F sti ).
s
t
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David P. Williamson s-t path TSP
One More Lemma
Let F be a random spanning tree (tree Fi with probability λi), andF st its associated s-t path. Let c(F st) be the cost of this path.Recall that
Pr[|F ∩ δ(Q)| = 1] ≥ 2− x∗(δ(Q))
for a τ -narrow cut Q.
Lemma (Sebő (2013))∑Q∈Cτ
(2− x∗(δ(Q)))ceQ ≤ E [c(F st)].
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David P. Williamson s-t path TSP
Sebő (2013)
Theorem (Sebő (2013))Best-of-Many Christofides’ is an 8
5 -approximation algorithm.
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David P. Williamson s-t path TSP
Proof of SebőFor the proof, recall that eQ is min-cost edge in δ(Q), Cτ are thecuts Q with x∗(δ(Q)) < 1+ τ ,
yi = αχFi + βx∗ +∑
Q∈Cτ ,|Q∩Ti | odd(1− 2α− βx∗(δ(Q)))χeQ
is feasible for the Ti -join LP, and
LemmaLet F be a randomly sampled tree Fi , and T the correspondingvertices Ti . Let Q ∈ Cτ be a τ -narrow cut. Then
Pr[|δ(Q) ∩ F| = 1] ≥ 2− x∗(δ(Q))
Pr[|Q ∩ T | odd] ≤ x∗(δ(Q))− 1.
Lemma ∑Q∈Cτ
(2− x∗(δ(Q)))ceQ ≤ E [c(F st)].
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David P. Williamson s-t path TSP
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David P. Williamson s-t path TSP
Vygen’s Improvement
Vygen (2015) gives a 1.599-approximation algorithm.
Key idea: Modify the initial convex combination of trees intoanother one that avoids certain bad properties.
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David P. Williamson s-t path TSP
Vygen’s Improvement
Vygen (2015) gives a 1.599-approximation algorithm.
Key idea: Modify the initial convex combination of trees intoanother one that avoids certain bad properties.
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David P. Williamson s-t path TSP
Integrality Gap
The performance of Best-of-Many Christofides’ cannot do betterthan the integrality gap of the LP relaxation.
The integrality gap is
µ ≡ sup OPTOPTLP
over all instances of the problem.
Note that we have shown µ ≤ 85 , since we find a tour of cost at
most 85OPTLP .
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David P. Williamson s-t path TSP
Integrality Gap
The performance of Best-of-Many Christofides’ cannot do betterthan the integrality gap of the LP relaxation.
The integrality gap is
µ ≡ sup OPTOPTLP
over all instances of the problem.
Note that we have shown µ ≤ 85 , since we find a tour of cost at
most 85OPTLP .
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David P. Williamson s-t path TSP
Integrality Gap
The performance of Best-of-Many Christofides’ cannot do betterthan the integrality gap of the LP relaxation.
The integrality gap is
µ ≡ sup OPTOPTLP
over all instances of the problem.
Note that we have shown µ ≤ 85 , since we find a tour of cost at
most 85OPTLP .
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David P. Williamson s-t path TSP
Integrality Gap
We can show a lower bound on the integrality gap using aninstance of graph TSP: input is a graph G = (V ,E ), cost ce fore = (i , j) is number of edges in a shortest i-j path in G .
s t
k
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David P. Williamson s-t path TSP
Integrality Gap
We can show a lower bound on the integrality gap using aninstance of graph TSP: input is a graph G = (V ,E ), cost ce fore = (i , j) is number of edges in a shortest i-j path in G .
s t
k
OPTLP ≈ 2k
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David P. Williamson s-t path TSP
Integrality Gap
We can show a lower bound on the integrality gap using aninstance of graph TSP: input is a graph G = (V ,E ), cost ce fore = (i , j) is number of edges in a shortest i-j path in G .
s t
k
OPT ≈ 3k
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David P. Williamson s-t path TSP
Integrality Gap
We can show a lower bound on the integrality gap using aninstance of graph TSP: input is a graph G = (V ,E ), cost ce fore = (i , j) is number of edges in a shortest i-j path in G .
s t
k
OPTOPTLP
→ 32 as k →∞
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David P. Williamson s-t path TSP
Graph Instances
Sebő and Vygen (2014) show that for graph TSP instances of s-tpath TSP, can get a 3
2 -approximation algorithm (i.e. the algorithmproduces a solution of cost at most 3
2OPTLP), so the integralitygap is tight for these instances.
We’ll present a simplified version of this result due to Gao (2013).
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David P. Williamson s-t path TSP
Graph Instances
Given the input graph G = (V ,E ) and an optimal solution, canreplace any edge (i , j) in the optimal solution with the i-j path inG since these have the same cost.
So finding an optimal solution is equivalent to finding a multiset Fof edges such that (V ,F ) is connected, degF (s) and degF (t) areodd, degF (v) is even for all v ∈ V − {s, t}, and |F | is minimum.
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David P. Williamson s-t path TSP
LP Relaxation
Min∑e∈E
xe
subject to: x(δ(S)) ≥{
1, |S ∩ {s, t}| = 1,2, |S ∩ {s, t}| 6= 1,
xe ≥ 0, ∀e ∈ E .
Let x∗ be an optimal LP solution.
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David P. Williamson s-t path TSP
Narrow Cuts
As before, focus on narrow cuts S such that x∗(δ(S)) < 2 (i.e. aτ -narrow cut for τ = 1). Recall:
Theorem (An, Kleinberg, Shmoys (2012))If S1, S2 are narrow cuts, S1 6= S2, then either S1 ⊂ S2 or S2 ⊂ S1.
So the narrow cuts look like s ∈ S1 ⊂ S2 ⊂ · · · ⊂ Sk ⊂ V .
s . . . t
Let S0 ≡ ∅, Sk+1 ≡ V , Li ≡ Si − Si−1.
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David P. Williamson s-t path TSP
Key Idea
Find a tree spanning Li in the support of x∗ for each i . Connecteach of these via a single edge from Li to Li+1. Let F be theresulting tree, T the vertices in F whose parity needs changing.
Then |F | = n − 1 and |δ(Si) ∩ F | = 1 for each narrow cut Si .
s L1 L2 . . . Lk t
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David P. Williamson s-t path TSP
Key Lemma
Recall:
LemmaLet S be an odd set. If |S ∩ {s, t}| = 1, then |F ∩ δ(S)| is even.
Min∑e∈E
cexe
subject to: x(δ(S)) ≥ 1, ∀S ⊆ V , |S ∩ T | oddxe ≥ 0, ∀e ∈ E .
Lemmay = 1
2x∗ is feasible for the the T -join LP.
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David P. Williamson s-t path TSP
Gao (2013)
Theorem (Gao (2013))For spanning tree F constructed by the algorithm, let J be aminimum-cost T -join. Then c(F ∪ J) ≤ 3
2OPTLP .
Min∑e∈E
xe
subject to: x(δ(S)) ≥{
1, |S ∩ {s, t}| = 1,2, |S ∩ {s, t}| 6= 1,
xe ≥ 0, ∀e ∈ E .
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David P. Williamson s-t path TSP
Last LemmaLet E (x∗) = {e ∈ E : x∗e > 0} be the support of LP solution x∗,H = (V ,E (x∗)) the support graph of x∗, H(S) the graph inducedby a set S of vertices.
Lemma (Gao (2013))
For 1 ≤ p ≤ q ≤ k + 1, H(⋃
p≤i≤q Li)is connected.
s L1 L2 . . . Lk t
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David P. Williamson s-t path TSP
The Big Question
Is there a 32 -approx. alg. for s-t path TSP for general costs?
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David P. Williamson s-t path TSP
One Idea
Idea: Construct a spanning tree F just as in Gao’s algorithm forthe graph case. Then again y = 1
2x∗ is feasible for the T -join LP,
and the overall cost of F plus the T -join is at mostc(F ) + 1
2∑
e∈E cex∗e .
Problem: Not clear how to bound the cost of F . Gao (2014) hasan example showing that F can have cost greater than OPTLP .
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David P. Williamson s-t path TSP
One Idea
Idea: Construct a spanning tree F just as in Gao’s algorithm forthe graph case. Then again y = 1
2x∗ is feasible for the T -join LP,
and the overall cost of F plus the T -join is at mostc(F ) + 1
2∑
e∈E cex∗e .
Problem: Not clear how to bound the cost of F . Gao (2014) hasan example showing that F can have cost greater than OPTLP .
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David P. Williamson s-t path TSP
The Bigger Question
Best-of-Many Christofides’ is provably better than Christofides’ fors-t path TSP. What about the standard TSP?
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David P. Williamson s-t path TSP
An empirical answer
Did some computational work with Cornell CS undergraduate KyleGenova to see whether Best-of-Many Christofides is any betterthan standard Christofides in practice. Paper to appear inupcoming ESA.
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David P. Williamson s-t path TSP
The algorithms
We implement algorithms to do the following:• Run the standard Christofides’ algorithm (Christofides 1976);• Construct explicit convex combination via column generation(An 2012);
• Construct explicit convex combination via splitting off (Frank2011, Nagamochi, Ibaraki 1997);
• Add sampling scheme SwapRound to both of above; givesnegative correlation properties (Chekuri, Vondrák, Zenklusen2010);
• Compute and sample from maximum entropy distribution(Asadpour, Goemans, Madry, Oveis Gharan, Saberi 2010).
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David P. Williamson s-t path TSP
The experiments
The algorithms were implemented in C++, run on a machine witha 4.00Ghz Intel i7-875-K processor with 8GB DDR3 memory.
We run these algorithms on several types of instances:• 59 Euclidean TSPLIB (Reinelt 1991) instances up to 2103vertices (avg. 524);
• 5 non-Euclidean TSPLIB instances (gr120, si175, si535,pa561, si1032);
• 39 Euclidean VLSI instances (Rohe) up to 3694 vertices (avg.1473);
• 9 graph TSP instances (Kunegis 2013) up to 1615 vertices(avg. 363).
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David P. Williamson s-t path TSP
The results
Std ColGen ColGen+SRBest Ave Best Ave
TSPLIB (E) 9.56% 4.03% 6.44% 3.45% 6.24%VLSI 9.73% 7.00% 8.51% 6.40% 8.33%TSPLIB (N) 5.40% 2.73% 4.41% 2.22% 4.08%Graph 12.43% 0.57% 1.37% 0.39% 1.29%
MaxEnt Split Split+SRBest Ave Best Ave Best Ave
TSPLIB (E) 3.19% 6.12% 5.23% 6.27% 3.60% 6.02%VLSI 5.47% 7.61% 6.60% 7.64% 5.48% 7.52%TSPLIB (N) 2.12% 3.99% 2.92% 3.77% 1.99% 3.82%Graph 0.31% 1.23% 0.88% 1.77% 0.33% 1.20%
Costs given as percentages in excess of optimal.
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David P. Williamson s-t path TSP
The results
Standard Christofides MST (Rohe VLSI instance XQF131)
Splitting off + SwapRound
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David P. Williamson s-t path TSP
The resultsBoMC yields more vertices in the tree of degree two.
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David P. Williamson s-t path TSP
The results
So while the tree costs more (as percentage of optimal tour)...
Std BOMTSPLIB (E) 87.47% 98.57%VLSI 89.85% 98.84%TSPLIB (N) 92.97% 99.36%Graph 79.10% 98.23%
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David P. Williamson s-t path TSP
The results
...the matching costs much less.
Std CG CG+SR MaxE Split Sp+SRTSPLIB (E) 31.25% 11.43% 11.03% 10.75% 10.65% 10.41%VLSI 29.98% 14.30% 14.11% 12.76% 12.78% 12.70%TSPLIB (N) 24.15% 9.67% 9.36% 8.75% 8.77% 8.56%Graph 39.31% 5.20% 4.84% 4.66% 4.34% 4.49%
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David P. Williamson s-t path TSP
Conclusion
Q: Are there empirical reasons to think BoMC might be provablybetter than Christofides’ algorithm?
A: Yes.
Maximum entropy sampling, or splitting off with SwapRound seemlike the best candidates.
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David P. Williamson s-t path TSP
Conclusion
Q: Are there empirical reasons to think BoMC might be provablybetter than Christofides’ algorithm?A: Yes.
Maximum entropy sampling, or splitting off with SwapRound seemlike the best candidates.
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David P. Williamson s-t path TSP
ConclusionHowever, we have to be careful, as the following, very recent,example of Schalekamp and van Zuylen shows.
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David P. Williamson s-t path TSP
Conclusions
So it seems that randomization, or at least, careful construction ofthe convex combination is needed.
Vygen (2015) also uses careful construction to improve s-t pathTSP from 1.6 to 1.5999.
If we want to use the best sample from Max Entropy orSwapRound, then might need to prove some tail bounds.
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David P. Williamson s-t path TSP
Conclusions
So it seems that randomization, or at least, careful construction ofthe convex combination is needed.
Vygen (2015) also uses careful construction to improve s-t pathTSP from 1.6 to 1.5999.
If we want to use the best sample from Max Entropy orSwapRound, then might need to prove some tail bounds.
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David P. Williamson s-t path TSP
Conclusions
So it seems that randomization, or at least, careful construction ofthe convex combination is needed.
Vygen (2015) also uses careful construction to improve s-t pathTSP from 1.6 to 1.5999.
If we want to use the best sample from Max Entropy orSwapRound, then might need to prove some tail bounds.