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Doubling dimension and the traveling salesman problem Yair BartalHebrew University Lee-Ad GottliebHebrew University Robert KrauthgamerWeizmann Institute TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

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Traveling salesman problem Definition: Given a set of cities (points) find a minimum tour that visits each point Classic, well-studied NP-hard problem [Karp ‘72; Papadimitriou, Vempala ‘06] Mentioned in a handbook from 1832! Common benchmark for optimization methods Many books devoted to TSP… Other variants Closed tour Multiple tours Etc… 2 Optimal tour

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Traveling salesman problem Some additional assumptions on distances Symmetric d(x,y) = d(y,x) Metric Triangle inequality: d(x,y) + d(y,z) ≤ d(x,z) Easy 2-approximation via MST Since OPT ≥ MST 3 MST

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Traveling salesman problem 4 MST

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Sanjeev Arora [A. ‘98] and Joe Mitchell [M. ‘99] : Euclidean TSP with fixed dimension admits a PTAS (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 5 Euclidean TSP 5

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6 To achieve a PTAS, two properties were assumed Euclidean space Fixed dimension Are both these assumptions required? Fixed dimension is necessary Hardness: No PTAS for (log n)-dimensions [Trevisan ’00] Is Euclidean necessary? A PTAS for metric space? Problem: Arbitrary metric space includes high-dimension Euclidean space... What about metric spaces with low intrinsic dimension? 6

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Doubling Dimension Definition: Ball B(x,r) = all points within distance r from x. The doubling constant (of a metric M) is the minimum value such that every ball can be covered by balls of half the radius First used by [Assoud ‘83], algorithmically by [Clarkson ‘97]. The doubling dimension is ddim(M)=log (M) [Gupta,Krauthgamer,Lee ‘03] A metric is doubling if its doubling dimension is constant Packing property of doubling spaces A set with diameter D and min. inter-point distance a, contains at most (D/a) O(ddim) points 7 Here ≤7.

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Applications of doubling dimension Nearest neighbor search [Krauthgamer, Lee ’04; Har-Peled, Mendel ’06; Beygelzimer, Kakade, Langford ’06; Cole, G. ‘06] Spanner construction, routing [G., Roditty ’08a, ’08b; Elkin, Solomon ‘12a, ‘12b; Abraham, Gavoille, Goldberg, Malkhi ‘05] Distance oracles [Har-Peled, Mendel ’06; Bartal, G., Roditty, Kopelowitz, Lewenstein ’11] Dimension reduction [G., Krauthgamer, ’11; Bartal, Recht, Schulman ‘11] Machine learning [Bshouty, Yi, Long ‘09; G., Kontorovich, Krauthgamer ’10, ‘12; ] Extension to nearly-doubling spaces [G., Krauthgamer ‘10] 8 G 2 1 1 H 2 1 1 1

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9 PTAS for metric TSP? Does TSP on doubling metrics admit a PTAS? Arora and Mitchell made strong use of Euclidean properties “Most fascinating problem left open in this area” James Lee, tcs math blog, June ‘10 Some attempts Quasi-PTAS [Talwar ‘04] (First description of problem) QPTAS for metric with neighborhoods [Mitchell ’07; Chan, Elbassioni ‘11] Subexponential-TAS, under more general growth assumption [Chan, Gupta ‘08] 9

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10 PTAS for metric TSP? Does TSP on doubling metrics admit a PTAS? Yes! (1+ Ɛ )-approximate tour In time:n 2 O(ddim) 2 Ɛ -Ỡ(ddim) 2 O(ddim 2 ) log ½ n Euclidean:n (log n) Ɛ -Ỡ(d) We’ll jump right in to the construction 10

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Metric partition 11 Starting point – a quadtree like hierarchy [Talwar ‘04, Bartal ‘96]

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Starting point – a quadtree like hierarchy [Talwar ‘04, Bartal ‘96] Metric partition 12 Arbitrary center point, ordering Random radius R i = [2 i, 2·2 i ]

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Metric partition 13 Starting point – a quadtree like hierarchy [Talwar ‘04, Bartal ‘96]

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Metric partition 14 Random radius R i-1 = [2 i-1, 2·2 i-1 ] Arbitrary center point Starting point – a quadtree like hierarchy [Talwar ‘04, Bartal ‘96] Caveat: logn hiearchical levels suffice Ignore tiny distances < 1/n 2

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Metric TSP 15 2 i-1 /M Definition: A tour is (m,r)-light on a hierarchy if it enters all cells (clusters) At most r times Only via m portals Portals are 2 i-1 /M –net points m = M O(ddim)

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16 Metric TSP Theorem [Arora ‘98,Talwar ‘04]: Given a partition The best (m,r)-light tour on the partition can be computed exactly m r O(ddim) nlogn time Via simple dynamic programming Join tours for small clusters into tour for larger cluster

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Metric TSP Our contribution Theorem: Given an optimal tour T, there exists a partition with (m,r)-light tour T’ M = ddim logn/ Ɛ m = M O(ddim) = (logn/ Ɛ ) Ỡ(ddim) r = Ɛ -O(d) loglogn Length of T’ is within (1+ Ɛ ) factor of the length of T 17

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Metric TSP Our contribution Theorem: Given an optimal tour T, there exists a partition with (m,r)-light tour T’ M = ddim logn/ Ɛ m = M O(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 m r O(ddim) n logn = n 2 Ɛ -Ỡ(ddim) loglog 2 n 18

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Metric TSP Our contribution Theorem: Given an optimal tour T, there exists a partition with (m,r)-light tour T’ M = ddim logn/ Ɛ m = M O(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 m r O(ddim) n logn = n 2 Ɛ -Ỡ(ddim) loglog 2 n It remains only to prove the Theorem, and to show how to find the partition 19

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Metric TSP 20 Modify a tour to be (m,r)-light Part I: Focus on m (i.e. net points) [Arora ‘98, Talwar ‘04] Move cut edges to be incident on net points R i-1 /M

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Metric TSP 21 R i-1 /M

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Metric TSP 22 Modify a tour to be (m,r)-light Part II: Focus on r (i.e. number of crossing edges) Reduce number of crossings

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Metric TSP 23

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MST in doubling spaces Bound the weight of MST in doubling space [Talwar ‘04]: For any r-point set S MST(S) = Rr 1-1/ddim « Rr Per point cost = R/r 1/ddim 24 2R

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Metric TSP 25 2R

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Metric TSP 26 2R

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Metric TSP Key observation: Space can be decomposed into sparse neighborhoods Consider an (i-1)-level ball If the local tour weight inside is at least R i-1 / Ɛ “Dense” ball Ball can be removed, each subproblem solved separately 27

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Metric TSP Key observation: Space can be decomposed into sparse neighborhoods Consider an (i-1)-level ball If the local tour weight inside is at least R i-1 / Ɛ “Dense” ball Ball can be removed, each subproblem solved separately Cost to join tours: only R i-1 28

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Metric TSP Sparse decomposition: Search hierarchy bottom-up for “dense” balls. Remove “dense” ball 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? Can be estimated using the local MST Modulo some caveats, error terms… OPT B(u,R) = O(MST(S)) B(u,3R) OPT = Ω(MST(S)) - Ɛ -O(ddim) R 29

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Metric TSP 30 R i-1 /M Suppose a tour is q-sparse with respect to hierarchy Every R-ball contains weight Rq (for all R=2 i ) Expectation: Random R-ball cuts weight Rq/R = q Cluster formed by cuts from many levels Expectation: q cuts per level If r = q 2loglogn Expectation: (i-1)-level patching includes cuts from 2loglogn higher levels Charge patching to edges in top loglogn levels Cut Pr: (ddim/R i+loglogn ) = (ddim/R i-1 logn)

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Metric TSP 31 2R

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Metric TSP 32 R i-1 /M Outstanding problem: Previous analysis assumed ball cuts only q edges True in expectation… Not good enough Solution: try many hierachies 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!

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