Weighted Geometric Set Multicover via Quasi-uniform Sampling (ESA 2012)

1. Weighted Geometric Set Multicover via Quasi-uniform Sampling(ESA 2012) Kirk Pruhs (U. Pittsburgh) Coauthor: Nikhil Bansal (TU Eindhoven)

2. Motivation for this Research: loglog n Approximation Algorithm for Scheduling Problems [BP10] General class of scheduling problems Reductions Weighted capacitated 2D geometric cover problem Also in Chakabarty, Grant, Konemann IPCO 2010 Fork Reduction Weighted priority geometric cover problem Weighted geometric multicover problem Folklore: loglog n loss Higher dimensional weighted geometric cover problem Weighted geometric cover problem loglog n approximation using Varadarajan’s quasi-uniform sampling technique STOC 10 O(1) approximation using Varadarajan’s quasi-uniform sampling technique STOC 10

3. This Paper/Talk General class of scheduling problems Reductions Weighted capacitated 2D geometric cover problem Also in Chakabarty, Grant, Konemann IPCO 2010 Fork Reduction Weighted priority geometric cover problem Weighted geometric multicover problem O(1) loss Higher dimensional weighted geometric cover problem Weighted geometric cover problem Bottleneck for obtaining O(1) approximation is this side Show how to adapt cover techniques to work for multicover

4. Outline • Randomized rounding and weighted geometric set cover • Varadarajan’s quasi-uniform sampling for weighted geometric set cover • Chan, Grant, Konemann, Sharpe refined quasi-uniform sampling for weighed geometric set cover • Our extension to weighted geometric set multicover • Final comments

5. Weighted Geometric Set MultiCover 7 3 6 • Instance: Geometric objects (here rectangles) r with weights wr, and points p with demands dp • Pick a minimal weight collection of objects such every point p is covered by dp objects • Set Cover = All demands are unit 1 1 2 1 1 3 1 2 2 1 LP: Min rwrxr r : p in r crxr≥dp xrin {0,1}

6. Randomized Rounding For Set Cover • Need to over-sample by log factor to obtain coverage of all points • Doesn’t use geometry • Want to get better than log approximation for geometric instances Weights LP solution 2k 1/k 2k-1 2k-1 1/k 1/k 2k-2 2k-2 2k-2 2k-2 1/k 1/k 1/k 1/k

7. Better Approximation for Geometric Set Cover Union Complexity h(n) of a collection of objects: Take n objects, look at their boundary (vertices,edges, holes). Scales as n h(n) Want approximation ratio o(h(n)). O(n log log n) [Matousek et al 91] O(n log*n exp((n)) [Ezra, Aronov, Sharir 11] O(n) (n2)

8. Round and Force For Unit Weights • Round and force: • Simple randomized rounding • Then force a small number of additional sets to get a cover • Yields better approximation ratios for someunweightedgeometric cover problems 1 1 1 1 1 1 1

9. Why Round and Force Doesn’t Easily Extend to the Weighted Case • Some sets (e.g. the heavy ones below) may be forced with high of a probability, and approximation may be bad Weights LP solution 2k 1/k 2k-1 2k-1 1/k 1/k 2k-2 2k-2 2k-2 2k-2 1/k 1/k 1/k 1/k

10. Outline • Randomized rounding and weighted geometric set cover • Varadarajan’s quasi-uniform sampling for weighted geometric set cover • Chan, Grant, Konemann, Sharpe refined quasi-uniform sampling for weighed geometric set cover • Our extension to weighted geometric set multicover • Final comments

11. Varadarajan’s Quasi-uniform sampling: each object r picked with probability ≤ c xr • Recall xr is probability for picking r according to the LP • Yields c approximation • Two main ideas to achieve quasi-uniform sampling • Sampling order • Successive refinement 2k 2k-1 2k-1 2k-2 2k-2 2k-2 2k-2

12. Sampling Order • Round the objects by decreasing order of the number of points that they cover • (Actually this is done independently for points of different depths) • If not picking an object would leave a point not covered, that set is forced 2k 2k-1 2k-1 2k-2 2k-2 2k-2 2k-2

13. Setup For Successive Refinement • Make xr L replicas of each object r • Recall xr is LP value for object r • L is large • Each point now covered by ≥ L replicas Weights LP solution 1/k 2k-1 2k-1 1/k 1/k 2k-2 2k-2 2k-2 2k-2 1/k 1/k 1/k 1/k

14. Successive Refinement • Round 1: Sample/retain each replica with probability (log L)/L in sampling order • Equivalent to increasing the probabilities on remaining replicas by L/log L factor • Expect each point to now be covered by log L replicas • If a point is covered < log L replicas, then one of the remaining sets is forced • Otherwise quasi-uniformity might be violated

15. Successive Refinement • Round 2: Sample/retain each remaining replica with probability (loglog L)/log L in sampling order • Expect each point to now be covered by loglog L replicas • If a point is covered < loglog L replicas, then one of the remaining sets is forced

16. Successive Refinement • Round i: Sample/retain each remaining replica with probability (log(i) L)/log(i-1) L in sampling order • Expect each point to be covered by log(i) L replicas • If a point is covered < log(i) L replicas, then one of the remaining sets is forced • Finally, take the last remaining log h(n) replicas • Recall h(n) is union complexity of objects

17. Varadarajan’s Final Result • Theorem: Every object r is selected with probability at most exp(log*(n)) log (h(n)) xr • Quasi-uniform sampling • Corollary: Poly time exp(log*(n)) log (h(n)) approximation algorithm O(k log log k) [Matousek et al 91] O(k log* k exp((k)) [Ezra, Aronov, Sharir 11] O(k) (k2)

18. Outline • Randomized rounding and weighted geometric set cover • Varadarajan’s quasi-uniform sampling for weighted geometric set cover • Chan, Grant, Konemann, Sharpe refined quasi-uniform sampling for weighed geometric set cover • Our extension to weighted geometric set multicover • Final comments

19. Chan, Grant, Konemann, Sharpe (CGKS) • Changes to Varadarajan: • Successive refinement retains each replica with probability ≈ ½ instead of (log L)/L • If a point is covered by a significantly fewer copies than expected, force a set covering that point according to a particular rule guaranteeing that no set can be forced by too many points • Theorem: log (h(n)) quasi-uniform sampling • Shaves off exp(log*(n)) factor and is simpler Varadarajan round Correction Source target CGKS rounds

20. Outline • Randomized rounding and weighted geometric set cover • Varadarajan’s quasi-uniform sampling for weighted geometric set cover • Chan, Grant, Konemann, Sharpe refined quasi-uniform sampling for weighed geometric set cover • Our extension to weighted geometric set multicover • Final comments

21. What doesn’t Varadarajan and CGKS work for multicover? • The resulting dp replicas covering point p may all belong to the same original set • CGKS forcing rule doesn’t obviously extend to multicover

22. Min rwrxr r : p in r crxr≥dp xrin[0, 1] Our Idea • Pick any set that the LP picks with probability > ¼ • Decrease residual cover requirements • Each remaining point p is then covered by at least 4 dp sets • Apply CGKS but also force sets if the number of distinct sets covering a point is much less than expected • Revert to Varadarajan’s method for selecting what sets to force

23. One Slide for Wonks • Invariant: For all rounds, and for all points p: • Σr:pεr min( nr, L/b) ≥ L dp • nr is the number of replicas of object r • L goes down by ≈ ½ each round • b slowly decreases from 4 to 2 • Recall dp is coverage requirement of point p • Consequences of invariant: • All points covered by at least L replicas • same as CGKS • all points p are covered by at least b dp different sets

24. Final Result • Theorem: log (h(n)) quasi-uniform sampling, and hence poly-time log (h(n)) approximation, for weighted geometric set multicover. • Matching bound of CGKS for geometric set cover • Can be extended to some nongeometric network settings, see CGKS and our paper • General extension from set cover to multicover seems unlikely/hard • e.g survivable network design vs. Steiner tree

25. Outline • Randomized rounding and weighted geometric set cover • Varadarajan’s quasi-uniform sampling for weighted geometric set cover • Chan, Grant, Konemann, Sharpe refined quasi-uniform sampling for weighed geometric set cover • Our extension to weighted geometric set multicover • Final comments

26. Open Question • General way to approximate geometric priority cover problems? • Priority cover problems: objects and points each have priorities, and an object can only be covered by objects of higher priority

27. Thanks for listening • Questions?