Improved Approximation Algorithms for Geometric Set Cover

The set cover problem

• Given `bb(U)`, a set; here, a set of points
• Given collection `bb(S)` of subsets of `bb(U)`, here called "objects"
• Given `bb(M) subset bb(U)`, find the smallest subset `C` of `bb(S)` that covers `bb(M)`

For example: siting of basestations

• Objects are footprints of basestations, `bb(M)` is market area
• Footprint is area visible to basestation
• Choose from among discrete set of such footprints
• Often modeled as circular disks in the plane
• An idealization, esp. for directional antennas

Simplified cases of basestation siting

• Results here apply to a few simplifications:
• Fat triangles
• Visibility regions at points in a 1.5d terrain `approx` "guarding"

(A few) old set cover results

If `c := |bb(C)^{:**:} |`, `n := |bb(S)|`, can get within factor:

• `O(log n)`, with the greedy algorithm;
• `O(log c)`, for polytope approximation in `RR^m` [C94] ([L87][W88])
• `O(log c)`, when the VC dimension is bounded [BG]
• `O(1)`, polytope approximation in 3d [BG]
  • Or `bb(M) subset RR^3`, `bb(S)` is a collection of halfspaces
  • Or `bb(M) subset RR^2`, `bb(S)` is a collection of disks
  • Use connection with `epsilon`-nets

New results: hypothesis

• If complement of `cup(R)` has roughly linear combinatorial complexity
• Random `R subset bb(S)`
• Decomposable into expected `f(r) = o(r log r)` simple regions, `r:=|R|`
• For example, trapezoidization of complement of linear-complexity union

New results: conclusion

• Then approximation ratio is `f(c)//c = o(log c)`:
• Fat triangles `f(r) = O(r log log r)` [MPSSW], yields `O(log log c)` approx.
• Similar-sized fat objects with `t` boundary crossings `f(r) = lambda_(t+2)(r)` [Efrat]
• Regions bounded by Jordan arcs with `le 3` intersections `(alpha (n))` approx.[EGHPPSSS]
• In some cases `f(r)=O(r)` and ratio is `O(1)`:
• Similar-sized fat triangles [MPSSW], fat wedges [ERS]
• Arrangements of pseudo-disks [KLPS]
• 1.5d terrain guarding (matching [BMKM05]) *
• Unit cubes in `RR^3` [BSTY] *

Outline

• `epsilon`-nets
  • An easy set cover problem
• Small `epsilon`-nets `=>` small covers
• Size of cover is within a constant factor of size of net
• As in [BG], [ERS]
• Small complement decompositions `=>` small `epsilon`-nets
• (Generalizing "shallow cuttings" [M])
• Particular case of [CF]
• 1.5d terrain guarding `=>` small complement decomposition
• Unit cubes: use fatness to create clusters, need only decompose boundary

`epsilon`-nets

• `epsilon>0` a given parameter
• Cover for easy-to-cover points `x in bb(M)_epsilon`
• `x in bb(M)_epsilon` if more than `epsilon {:| bb(S) |:}` objects contain x
• To construct: pick `r` objects from `bb(S)` at random
• Probability that:
• Given `x in bb(M)_epsilon` not covered is ` le (1-epsilon)^r`
• Some `x` not covered is `le |bb(M)_epsilon|(1-epsilon)^r`
• All `x` covered `ge 1 - |bb(M)_epsilon| e^(-r epsilon)`
• `r = O(log |bb(M)_epsilon|)//epsilon` is big enough
• [HW86]

`epsilon`-nets: equivalent points

• `log |bb(M)_epsilon|` could be large
• Maybe not in geometric settings
• Say that `x approx y` for `x,y in {: bbb()_epsilon :}` if `x in S iff y in S`, for all `S in bb(S)`
• Need only consider equivalence classes `bb(M)_epsilon/approx`
• For example, if VC dimension is `d'`, have `O(r^{d'})` classes for `R`
• Implies sample of size `O(d' \log (1//\epsilon))//epsilon` is a cover of `bb(M)_epsilon`

`epsilon`-nets: a different approach

• Sometimes `bb(U) setminus cup(R)` is "simple"
• For triangles, can make trapezoidal decomposition
• A trapezoid T is in decomposition iff:
  • `d=O(1)` triangles "defining" `T` are in `R`
  • None of the `|T|` triangles meeting `T` are in `R`
• `R` induces a collection of trapezoids `F_0(R)`,

  `cup(F_0(R)) = bb(U) setminus cup(R)`

`epsilon`-nets using `F_0(R)`

• If `x in bb(M)_epsilon` and `x` in trapezoid `T`,
• Then `|T| ge epsilon |bb(S)|`
• (`|T|=` number of sets meeting `T`)
• There are `O(r^d)` such `T` possible, since `d` triangles define `T`
• For random subset of size `r`,
• Probability that some such `T` is not met by `R` is `O(r^d)(1-epsilon)^r`
• Similar to before, `r=(d \log(1//\epsilon))//epsilon` is big enough
• [C86]

Small `epsilon`-nets `=>` small covers

• Two approaches:
• Using "doubling" of objects [L87][W88][C92][BG94]
• Using LP relaxation [ERS], also observed by Kasturi
• Size bounds give `O(log c)` approximation
• Sometimes, can make `epsilon`-net smaller, even `O(1//epsilon)`
• Implies constant-factor approximations

Small `epsilon`-nets `=>` small covers: LP

• From [ERS]
• Tweak set system until every point is in `bb(M)_(1//2c)`
• Existence of solution of size `c` implies weights `w_S in [0..1]` with

  `{:(sum_S w_S le c), (sum_{S ni x} w_S ge 1 text(for all) x in bb(M)):}`

• Make `bb(S)'`, a multiset, with `|~ w_S n//c ~|` copies of object `S`
• At least `n//c` sets contain each point, `2n` sets total

Small `epsilon`-nets `=>` small covers: doubling

• [L87][W88][C92][BG94]
• Tweak set system until all points in `bb(M)` covered:
• Find `(1//2c)`-net `E` of current `bb(S)'`; initially `bb(S)' := bb(S)`;
• For any `T in F_0(E)`, some `S in bb(C)` meets it;
• Double the multiplicity of all `S in bb(S')` that meet `T`;
• After `c` rounds:
• Factor increase of total multiplicity of `bb(S) le (1+1/{:2c:})^c approx sqrt{e}`;
• Increase of total mult. of `C ge 2`;
• Eventually `E` must be a cover

Small `F_0(R)` `=>` small `epsilon`-nets: alteration

• Use the alteration/mutation/repair method
• As in [CF], specialized to small `F_0(R)`
• Take random sample `R` of size `r:=1//epsilon`;

• Will find `epsilon`-net of size `O(f(r))`, `f(r)=E|F_0(R)|`
• Compute `F_0(R)`;
• "Patch up" regions in `F_0(R)`: find `epsilon_T`-nets for region `T`
• `epsilon`-net is union of `R` and `epsilon_T`-nets for each `T in F_0(R)`

Small `F_0(R)` `=>` small `epsilon`-nets: analysis

• How much patching up to do?
• Start from: with high prob., `R` is an `O((log r)//r)`-net;
• Need `epsilon_T` with `epsilon_T |T| = n//r`, or `epsilon_T= n//rj`, where `j={:|T|:}`
• To repair `T`, need `(jr//n) log (jr//n)` sample size, no more than `(jr//n)^2`

Small `F_0(R)` `=>` small `epsilon`-nets:

final analysis

• From [CS88], `E sum_{T in F_0(R)} {:|T|:}^k = O(f(r)n^k//r^k)`
• Or: expected proportion of `T` with `|T| ge i n//r` is `le e^{-i}`
• Take `k=2`, expected `sum_{T in F_0(R)} ({:|T|r:}//n)^2 = O(f(r))`

Terrain guarding: visibility

• Have terrain `P`, an `x`-monotone polygonal path
• Set `G` of point guards on `P`
• `g in G` sees a point `p` if:
• line segment `bar(gp)` does not meet region below terrain

More visibility

• `x(g) lt x(p)`, then `g` sees `p` from the left
• `g` is the leftmost guard that sees `p` from the left, then `g` owns `p`
• No lower point at `x(p)` is seen by any guard, then `g` owns `x(p)` at `p`

Why ownership?

• Any guarded point is also owned
• A set owns every point iff it sees every point
• Ownership is combinatorially simpler

Order claim

• Generalizing [BMKM] slightly: for
• Guards `a`, `b`, points `p`, `p'`
• `x(a) lt x(b) le x(p) le x(p')`
• `a` owns `x(p)` at `p`, `b` sees `p'`
• Have:
• `a` also sees `p'`
• `b` must be below `bar(ap)`, `p` must be below `bar(bp')`
• So `b` cannot own `p'`
• No ownership pattern `a cdots b cdots a cdots b`

Davenport-Schinzel-orama

• Order claim implies ownership is `DS(m,2)` for `m` guards
• At most `2m-1` (left) ownership intervals
• Shows `|F_0(R)| lt 2r-1` for appropriate configuration system
• Implies small `epsilon`-nets for left visibility
• Combine with small `epsilon`-net for right visibility
• From small `epsilon`-nets, small covers

Concluding remarks

• Some other geometric set cover problems have `O(1)` approx.; any relation?
• Balls covering points in 3d?
• Thanks!