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Doubling dimension and the
traveling salesman problem
Yair Bartal
Lee-Ad Gottlieb
Robert Krauthgamer
Hebrew University
Hebrew University
Weizmann Institute
Traveling salesman problem
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Definition: Given a set of cities (points) find a minimum tour that
visits each point
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Classic, well-studied NP-hard problem
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[Karp ‘72; Papadimitriou, Vempala ‘06]
Mentioned in a handbook from 1832!
Common benchmark for optimization methods
Many books devoted to TSP…
Other variants
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Closed tour
Multiple tours
Etc…
Optimal tour
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Traveling salesman problem
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Some additional assumptions on distances
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Symmetric
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d(x,y) = d(y,x)
Metric
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Triangle inequality: d(x,y) + d(y,z) ≤ d(x,z)
Easy 2-approximation via MST
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Since OPT ≥ MST
MST
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Traveling salesman problem
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Some additional assumptions on distances
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Symmetric
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d(x,y) = d(y,x)
Metric
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Triangle inequality: d(x,y) + d(y,z) ≤ d(x,z)
Easy 2-approximation via MST
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Since OPT ≥ MST
Can do better…
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Christophides: MST + Matching ≤ 1½ OPT
MST
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Euclidean TSP
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Sanjeev Arora [A. ‘98] and Joe Mitchell [M. ‘99] :
Euclidean TSP with fixed dimension admits a PTAS
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(1+Ɛ)-approximate tour
In time n(log n)Ɛ-Ỡ(d)
(Easy extension to other norms)
They were awarded the
2010 Gödel Prize
for this discovery
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Euclidean TSP
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To achieve a PTAS, two properties were assumed
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Euclidean space
Fixed dimension
Are both these assumptions required?
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Fixed dimension is necessary
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Hardness: No PTAS for (log n)-dimensions [Trevisan ’00]
Is Euclidean necessary?
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A PTAS for metric space?
Problem: Arbitrary metric space includes high-dimension Euclidean space...
What about metric spaces with low intrinsic dimension?
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Doubling Dimension
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Definition: Ball B(x,r) = all points within distance r from x.
The doubling constant (of a metric M) is the minimum value >0
such that every ball can be covered by  balls of half the radius
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First used by [Assoud ‘83], algorithmically by [Clarkson ‘97].
The doubling dimension is ddim(M)=log(M)
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[Gupta,Krauthgamer,Lee ‘03]
A metric is doubling if its doubling dimension is constant
Packing property of doubling spaces
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A set with diameter D and min. inter-point
distance a, contains at most
(D/a)O(ddim) points
Here ≤7.
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Applications of doubling dimension
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Nearest neighbor search
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Spanner construction, routing
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1
1
2
2
1
1
1
[Har-Peled, Mendel ’06; Bartal, G., Roditty, Kopelowitz, Lewenstein ’11]
[G., Krauthgamer, ’11; Bartal, Recht, Schulman ‘11]
Machine learning
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H
Dimension reduction
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[G., Roditty ’08a, ’08b; Elkin, Solomon ‘12a, ‘12b;
Abraham, Gavoille, Goldberg, Malkhi ‘05]
G
Distance oracles
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[Krauthgamer, Lee ’04; Har-Peled, Mendel ’06; Beygelzimer, Kakade,
Langford ’06; Cole, G. ‘06]
[Bshouty, Yi, Long ‘09; G., Kontorovich, Krauthgamer ’10, ‘12; ]
Extension to nearly-doubling spaces
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[G., Krauthgamer ‘10]
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PTAS for metric TSP?
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Does TSP on doubling metrics admit a PTAS?
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Arora and Mitchell made strong use of Euclidean properties
“Most fascinating problem left open in this area”
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James Lee, tcs math blog, June ‘10
Some attempts
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Quasi-PTAS
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QPTAS for metric with neighborhoods
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[Talwar ‘04] (First description of problem)
[Mitchell ’07; Chan, Elbassioni ‘11]
Subexponential-TAS, under more general growth assumption
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[Chan, Gupta ‘08]
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PTAS for metric TSP?
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Does TSP on doubling metrics admit a PTAS?
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Yes!
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(1+Ɛ)-approximate tour
2
In time:
n2O(ddim) 2Ɛ-Ỡ(ddim) 2O(ddim ) log½n
Euclidean: n (log n)Ɛ-Ỡ(d)
We’ll jump right in to the construction
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Metric partition
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Starting point – a quadtree
like hierarchy [Talwar ‘04,
Bartal ‘96]
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Metric partition
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Starting point – a quadtree
like hierarchy [Talwar ‘04,
Bartal ‘96]
Random radius
Ri = [2i, 2·2i]
Arbitrary center
point, ordering
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Metric partition
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Starting point – a quadtree
like hierarchy [Talwar ‘04,
Bartal ‘96]
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Metric partition
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Starting point – a quadtree
like hierarchy [Talwar ‘04,
Bartal ‘96]
Random radius
Ri-1 = [2i-1, 2·2i-1]
Arbitrary center
point
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Caveat: logn hiearchical levels suffice
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Ignore tiny distances < 1/n2
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Metric TSP
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Definition: A tour is (m,r)-light on a hierarchy if it enters all cells
(clusters)
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At most r times
Only via m portals
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Portals are 2i-1/M –net points
m = MO(ddim)
2i-1/M
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Metric TSP
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Theorem [Arora ‘98,Talwar ‘04]: Given a partition
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The best (m,r)-light tour on the partition can be computed exactly
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mr O(ddim) nlogn time
Via simple dynamic programming
Join tours for small clusters
into tour for larger cluster
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Metric TSP
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Our contribution
Theorem: Given an optimal tour T, there exists a partition with
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(m,r)-light tour T’
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M = ddim logn/Ɛ
m = MO(ddim) = (logn/Ɛ)Ỡ(ddim)
r = Ɛ-O(d) loglogn
Length of T’ is within (1+Ɛ) factor of the length of T
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Metric TSP
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Our contribution
Theorem: Given an optimal tour T, there exists a partition with
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(m,r)-light tour T’
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M = ddim logn/Ɛ
m = MO(ddim) = (logn/Ɛ)Ỡ(ddim)
r = Ɛ-O(d) loglogn
Length of T’ is within (1+Ɛ) factor of the length of T
If the partition were known, then T’ could be found in time
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mr O(ddim) n logn = n 2Ɛ-Ỡ(ddim) loglog2n
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Metric TSP
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Our contribution
Theorem: Given an optimal tour T, there exists a partition with
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(m,r)-light tour T’
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Length of T’ is within (1+Ɛ) factor of the length of T
If the partition were known, then T’ could be found in time
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M = ddim logn/Ɛ
m = MO(ddim) = (logn/Ɛ)Ỡ(ddim)
r = Ɛ-O(d) loglogn
mr O(ddim) n logn = n 2Ɛ-Ỡ(ddim) loglog2n
It remains only to prove the Theorem, and to show how to find
the partition
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Metric TSP
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Modify a tour to be (m,r)-light
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Part I: Focus on m (i.e. net points)
[Arora ‘98, Talwar ‘04]
Move cut edges to be incident on net points
Ri-1/M
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Metric TSP
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Modify a tour to be (m,r)-light
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Part I: Focus on m (i.e. net points)
[Arora ‘98, Talwar ‘04]
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Move cut edges to be incident on net points
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Expected cost to edge
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At the (i-1)-level, radius Ri-1~2i-1
Expected cost
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~ (Ri-1/M)(ddim/Ri-1) = ddim/M = Ɛ/logn
(assuming edge length 1)
Ri-1/M
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Expected cost to edge over all levels:
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logn * Ɛ/logn = Ɛ
(1+Ɛ)-approximate tour
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Metric TSP
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Modify a tour to be (m,r)-light
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Part II: Focus on r (i.e. number of crossing edges)
Reduce number of crossings
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Metric TSP
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Modify a tour to be (m,r)-light
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Part II: Focus on r (i.e. number of crossing edges)
Reduce number of crossings
Patchings [Arora ‘98]: Reroute edges back into cluster
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Cost: ~ length of tour on the patched endpoints
~ MST of these points
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MST in doubling spaces
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Bound the weight of MST in doubling space
[Talwar ‘04]: For any r-point set S
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MST(S) = Rr1-1/ddim « Rr
Per point cost = R/r1/ddim
2R
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Metric TSP
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Modify a tour to be (m,r)-light – Part II
Focus on r (i.e. number of crossing edges)
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Reduce number of crossings
Expected cost to edge
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At the (i-1)-level, radius Ri-1~2i-1
Probability edge is patched ≤ Probability edge is cut
(Ri-1/r1/ddim)(ddim/Ri-1) = ddim/r1/ddim
2R
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Metric TSP
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Modify a tour to be (m,r)-light – Part II
Focus on r (i.e. number of crossing edges)
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Reduce number of crossings
Expected cost to edge
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At the (i-1)-level, radius Ri-1~2i-1
Probability edge is patched ≤ Probability edge is cut
(Ri-1/r1/ddim)(ddim/Ri-1) = ddim/r1/ddim
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As before, want this term to be equal to Ɛ/logn
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So we get a contribution of Ɛ over logn levels
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Could take r = (ddim logn/Ɛ)ddim
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But dynamic program runs in time mr – QPTAS! [Talwar ‘04]
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Challenge: smaller value for r
2R
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Metric TSP
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Key observation:
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Space can be decomposed into sparse neighborhoods
Consider an (i-1)-level ball
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If the local tour weight inside is at least Ri-1/Ɛ
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“Dense” ball
Ball can be removed, each subproblem solved
separately
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Metric TSP
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Key observation:
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Space can be decomposed into sparse neighborhoods
Consider an (i-1)-level ball
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If the local tour weight inside is at least Ri-1/Ɛ
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“Dense” ball
Ball can be removed, each subproblem solved
separately
Cost to join tours: only Ri-1
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Metric TSP
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Sparse decomposition:
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Search hierarchy bottom-up for “dense” balls.
Remove “dense” ball
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Ball is composed of sparse subballs
So it’s barely dense
Recurse on remaining point set
How do we know the local weight of the tour in a ball?
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Can be estimated using the local MST
Modulo some caveats, error terms…
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OPT Ʌ B(u,R) = O(MST(S))
B(u,3R) Ʌ OPT = Ω(MST(S)) -Ɛ-O(ddim) R
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Metric TSP
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Suppose a tour is q-sparse with respect to hierarchy
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Cluster formed by cuts from many levels
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Every R-ball contains weight Rq (for all R=2i)
Expectation: Random R-ball cuts weight Rq/R = q
Expectation: q cuts per level
If r = q 2loglogn
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Expectation: (i-1)-level patching includes
cuts from 2loglogn higher levels
Charge patching to edges in top loglogn
levels
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Ri-1/M
Cut Pr: (ddim/Ri+loglogn) = (ddim/Ri-1 logn)
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Metric TSP
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Modify a tour to be (m,r)-light – Part II
Focus on r (i.e. number of crossing edges)
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Expected cost to edge
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Reduce number of crossings
At the (i-1)-level, radius Ri-1~2i-1
Probability edge is patched ≤ Probability edge is cut
(Ri-1/r1/ddim)(ddim/Ri-1logn) = ddim/logn r1/ddim
2R
As before, want this term to be equal to Ɛ/logn
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Can take r = (ddim/Ɛ)ddim
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PTAS!
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Metric TSP
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Outstanding problem:
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Previous analysis assumed ball cuts only q edges
True in expectation… Not good enough
Solution: try many hierachies
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choose logn random radii for each ball
Then some hierarchy has balls which cut only q edges
Drives up runtime of dynamic program
Thank you!
Ri-1/M
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