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

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

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

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

22 1 Instance: Geometric objects (here rectangles) r with weights w r, and points p with demands d p Pick a minimal weight collection of objects such every point p is covered by d p objects Set Cover = All demands are unit Weighted Geometric Set MultiCover LP: Min  r w r x r  r : p in r c r x r ≥ d p x r in {0,1}

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 2k2k 2 k-2 2 k-1 2 k-2 1/k Weights LP solution

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)). Better Approximation for Geometric Set Cover n2)n2) O(n) O(n log log n) [Matousek et al 91] O(n log * n exp(  (n)) [Ezra, Aronov, Sharir 11]

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 some unweighted geometric cover problems

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 2k2k 2 k-2 2 k-1 2 k-2 1/k Weights LP solution

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

2k2k 2 k-2 2 k-1 2 k-2 Varadarajan’s Quasi-uniform sampling: each object r picked with probability ≤ c x r –Recall x r is probability for picking r according to the LP –Yields c approximation Two main ideas to achieve quasi-uniform sampling –Sampling order –Successive refinement

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 2k2k 2 k-2 2 k-1 2 k-2

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

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

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

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

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

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

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 Sourcetarget Varadarajan round Correction CGKS rounds

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

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

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 d p 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 Min  r w r x r  r : p in r c r x r ≥ d p x r in [0, 1]

One Slide for Wonks Invariant: For all rounds, and for all points p: –Σ r:pεr min( n r, L/b) ≥ L d p –n r is the number of replicas of object r –L goes down by ≈ ½ each round –b slowly decreases from 4 to 2 –Recall d p 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 d p different sets

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

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

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

Thanks for listening Questions?