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

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

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

Traveling salesman problem 4 MST

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

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

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.

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 H

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

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

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

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 ]

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

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

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)

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

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

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

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

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

Metric TSP 21 R i-1 /M

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

Metric TSP 23

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

Metric TSP 25 2R

Metric TSP 26 2R

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

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

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

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)

Metric TSP 31 2R

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!