Approximating TSP on Bounded Dimensional Metric Spaces

1. Approximating TSP on Bounded Dimensional Metric Spaces Hubert Chan Joint work with Anupam Gupta

2. Problem setting Traveling Salesperson Problem Given a point setVand a distance functiond : V£V!R+. Find a permutation : [n] !Vthat minimizes: d((n), (1)) + 2 · i·n d((i-1), (i))

3. General This Talk: (1 + )-approx in sub-exp time \>0exp{n} Specific Hardness of Problem General distance function NP-hard to approx within any factor 1.5-approx NP-hard to approx better than 174/173 Metrics Doubling Metrics (1 + )-approx in time exp{ (k/log n)O(k)} (QPTAS) k-Dim Euclidean Metrics (1 + )-approx in time n exp{(k/)O(k)} + O(kn log n) (PTAS) (1 + )-approx in time O(exp{1/2} n) (linear) Planar metrics

4. Roadmap • Problem Definition & Background • Different Notions of Metric Dimensions • Euclidean Dimension • Doubling Dimension • Global Dimension • Techniques for Approximating TSP • Dynamic Program & Hierarchical Decomposition • Use Global Dimension for Better Running Time

5. x = (x1, x2, …, xk) y = (y1, y2, …, yk) Euclidean Dimension Points in k-dimensional space x = (x1, x2, …, xk) Each point has k coordinates. Distance function is the usual Euclidean distance. Drawback: Euclidean dim not applicable to general metrics!

6. Doubling Dimension Generalization of Euclidean Metrics A low-dimensional Euclidean metric has small doubling dimension. Received recent attention in CS community: [Gupta, Krauthgamer, Lee 2003] Hard problems more tractable [Talwar 2004]: Quasi-polynomial time approximation algorithm for TSP, k-median, facility location

7. x R Ball B(x, R) A ballB(x, R)centered atxwith radiusRconsists of points within distance R from x.

8. Doubling Dimension A metric (V,d) has doubling dimension at most k if for any R > 0, every ball of radius 2R is a union of at most 2k balls of radius R. Fact. A k-dim Euclidean metric has doubling dim O(k).

9. R-Net SubsetSinV R > 0 An R-net for S is a subset N of S s.t. • Covering: Every point in S is within distance R of some point in N. • Packing: Points in N are more than distance R away from one another.

10. R R-Nets & Doubling Dimension Useful Property: Given a metric (V, d) with doubling dimension k and any R-net N, any ball of radius 2R contains at most 22k net points in N. • B2Rcovered by 2kBR’s • BRcovered by 2kBR/2’s • EachBR/2 contains at most one net point.

11. R-Nets & Doubling Dimension Useful Property: Given a metric (V, d) with doubling dimension k and any R-net N, any ball of radius R contains at most O(k) net points in N.

12. Limitation of Doubling Dimension Restrictive: bounded growth rate holds everywhere “Every ball covered by 2k balls of half its radius” Real network can behave well on average, but contain locally dense regions. Hence, doubling dim k = (log n). Bounds exponential inkwill not be very useful, e.g.running time for (1 + )-approx for TSP: exp{ (k/ log n)O(k)} >> n!. Database community: fractal dimension [Belussi, Faloutsos ’95] Spatial queries

13. Global Notion of Dimension Trial 1: Global dimension is at mostk, if for allr > 0, x2V |B(x, 2r)| · 2kx 2 V |B(x,r)| Intuition: if for allx2V, |B(x, 2r)| · 2k |B(x,r)|, then the metric has strong doubling dimension at mostk. Good news: Theorem If a metric has doubling dimension at mostk, its global dimension is at mostO(k).

14. (n2) contribution Bad news Any metric can be trivially modified to have small global dim. Add a small “tail” of size n x2V |B(x, 2r)| · 2kx2V |B(x,r)| Definition (Global dim): for all netsN, for allr > 0, x2N |BN(x, 2r)| · 2kx2N |BN(x,r)| The net points hide the “tail”.

15. Roadmap • Problem Definition & Background • Different Notions of Metric Dimensions • Euclidean Dimension • Doubling Dimension • Global Dimension • Techniques for Approximating TSP • Dynamic Program & Hierarchical Decomposition • Use Global Dimension for Better Running Time

16. Easy Instances of TSP Optimal Tour for Tree Metric • Tour enters and leaves subtree through a single point • True for smaller subtrees too. Ideas to approximate TSP: • Decompose the metric recursively into clusters • Assign some points in each cluster as portals • Restrict to tour that enters and leaves a cluster via portals (portal respecting)

17. Dynamic Program • Hierarchical decomposition of metric • Assign portals to each cluster • Find best portal respecting tour y x Each cluster has some DP entries, containing partial tours. Cluster entries filled in bottom-up fashion. Time to compute entries in one cluster: B = No. of portals in child clusters Time = 2O(B log B) Details in a paper by Arnbourg & Proskurowski ’89.

18. Randomized Decomposition & Portal Assignment Conflicting constraints: Di • Too many portals increases running time Di-1 • Too few portals increases cost of tour x y Rand. Hierarchical Decomp. (Bartal) Levelicluster diameter: Di; Di-1·Di/4 Pr[xandyin different leveliclusters] ·d(x,y)/Di ,  = polylog(n) Portal assignment: for level i cluster, pickDi-net ( < 1) Extra distance = O(Di) Highest levelis.t.xandyare separated E[extra distance] = i d(x,y)/Di¢ Di = L d(x,y) SettingL¼ , E[best portal resp. tour] · (1+) OPT

19. R ¸ r Packing in a ball (1) Useful Property: Given a metric (V, d) with doubling dimensionk and any r-net N, any ball of radius R contains at most (R/r)knet points in N.

20. R ¸ r Packing in a ball (2) Useful Property: Given a metric (V, d) with global dimensionk and any r-net N, any ball of radius R contains at most (R/r)k/2n0.5 net points in N.

21. Using Low Dim to Bound Portals B = No. of portals in child clusters Time for each cluster = exp{O(B log B)} With some work, can ensure that child portals form aDi-1-packing,  = /polylog(n). Di Can also ensureDi/Di-1 = constant Doubling dimk, B· (Di/Di-1)k = polylog,k(n) Running time = exp{polylog,k(n)} (QPTAS) Global dimk, B· (Di/Di-1)k/2 n0.5= polylog,k(n) n0.5 Running time = exp{Õ,k(n0.5)} (not sub-exp yet)

22. Hard instance of TSP What’s the problem? Recall: there existsc > 1and instances of TSP that is NP-hard to approximate with ratio better thanc. • Current DP solves for(1+)-aprox locally • Can’t improve, unlessNP µ DTIME(2o(n)) Observe: • Local tour has length a small fraction o(1)of the total tour. • Hence, could useO(1)-approx for local hard instance

23. Patching Pick thresholdB02\ O,k(n) A cluster is dense if it has more than B0child portals. Perform DP bottom-up: • Apply patching for dense clusters. • Construct O(1)-approx tour for child portals. • (1+)-approx for non-dense descendant clusters are kept.

24. Some Technical Issues Bound the extra cost of patching • To use global dimension, ensure the portals from clusters in each level form a packing. • To ensure that each pair (x,y) separated with small probability, need to limit number of levels. • Set Di/Di-1 = (log (n/) loglog n)0.5. After fine tuning all the parameters, the running time:

25. Conclusion Moral: For problems whose dependence on metric is global, metrics with bounded global dimension should have better guarantees than general metrics. Open Problems: • Are there PTAS’s for TSP on metrics with bounded doubling/global dimension? • Any other applications where considering global dimension is useful?

26. Thank you!