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

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

Kirk Pruhs (U. Pittsburgh)

Coauthor: Nikhil Bansal (TU Eindhoven)

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

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

- 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

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}

- 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

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)

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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)

- 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

- 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

- 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

- The resulting dp replicas covering point p may all belong to the same original set
- CGKS forcing rule doesn’t obviously extend to multicover

Min rwrxr

r : p in r crxr≥dp

xrin[0, 1]

- 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

- 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

- All points covered by at least L replicas

- 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

- 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

- 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?