#### Approximation Algorithms for Maximum Independent Set of Pseudo-Disks

Discrete Comput Geom
Approximation Algorithms for Maximum Independent Set of Pseudo-Disks
Timothy M. Chan 0 1
Sariel Har-Peled 0 1
0 S. Har-Peled ( ) Department of Computer Science, University of Illinois , 201 N. Goodwin Avenue, Urbana, IL 61801 , USA
1 T.M. Chan School of Computer Science, University of Waterloo, 200 University Ave West , Waterloo, Ontario N2L 3G1 , Canada
We present approximation algorithms for maximum independent set of pseudo-disks in the plane, both in the weighted and unweighted cases. For the unweighted case, we prove that a local-search algorithm yields a PTAS. For the weighted case, we suggest a novel rounding scheme based on an LP relaxation of the problem, which leads to a constant-factor approximation. Most previous algorithms for maximum independent set (in geometric settings) relied on packing arguments that are not applicable in this case. As such, the analysis of both algorithms requires some new combinatorial ideas, which we believe to be of independent interest.
Fréchet distance; Approximation algorithms; Realistic input models
1 Introduction
Let F = {f1, . . . , fn} be a set of n objects in the plane, with weights w1, w2, . . . ,
wn > 0, respectively. In this paper, we are interested in the problem of finding an
independent set of maximum weight. Here a set of objects is independent if no pair
of objects intersect.
A preliminary version of the paper appeared in SoCG 2009 [
17
].
A natural approach to this problem is to build an intersection graph G = (V , E),
where the objects form the vertices, and two objects are connected by an edge if they
intersect, and weights are associated with the vertices. We want the maximum
independent set in G. This is of course an problem, and it is known that no approximation
factor is possible within |V |1−ε for any ε > 0 if NP = ZPP [
27
]. In fact, even if the
maximum degree of the graph is bounded by 3, no PTAS is possible in this case [
11
].
In geometric settings, better results are possible. If the objects are fat (e.g., disks
and squares), PTASs are known. One approach [
15, 22
] relies on a hierarchical
spatial subdivision, such as a quadtree, combined with dynamic programming
techniques [7]; it works even in the weighted case. Another approach [
15
] relies on a
recursive application of a nontrivial generalization of the planar separator theorem
[
30, 38
]; this approach is limited to the unweighted case. If the objects are not fat,
only weaker results are known. For the problem of finding a maximum independent
set of unweighted axis-parallel rectangles, an O(log log n)-approximation algorithm
was recently given by Chalermsook and Chuzhoy [14]. For line segments, a roughly
O(√Opt)-approximation is known [
2
]; recently, Fox and Pach [
24
] have improved
the approximation factor to nε for not only for line segments but curve segments that
intersect a constant number of times.
In this paper we are interested in the problem of finding a large independent set
in a set of weighted or unweighted pseudo-disks. A set of objects is a collection of
pseudo-disks, if the boundary of every pair of them intersects at most twice. This
case is especially intriguing because previous techniques seem powerless: it is
unclear how one can adapt the quadtree approach [
15, 22
] or the generalized separator
approach [15] for pseudo-disks.
Even a constant-factor approximation in the unweighted case is not easy.
Consider the most obvious greedy strategy for disks (or fat objects): select the object
fi ∈ F of the smallest radius, remove all objects that intersect fi from F, and
repeat. This is already sufficient to yield a constant-factor approximation by a simple
packing argument [
21, 32
]. However, even this simplest algorithm breaks down for
pseudo-disks—as pseudo-disks are defined “topologically”, how would one define
the “smallest” pseudo-disk in a collection?
Independent Set via Local Search Nevertheless, we are able to prove that a
different strategy can yield a constant-factor approximation for unweighted pseudo-disks:
local search. In the general settings, local search was used to get (roughly) a Δ/4
approximation to independent set, where Δ is the maximum degree in the graph,
see [26] for a survey. In the geometric settings, Agarwal and Mustafa [2, Lemma 4.2]
had a proof that a local-search algorithm gives a constant-factor approximation for
the special case of pseudo-disks that are rectangles; their proof does not immediately
work for arbitrary pseudo-disks. Our proof provides a generalization of their lemma.
In fact, we are able to do more: we show that local search can actually yield
a PTAS for unweighted pseudo-disks! This gives us by accident a new PTAS for
the special case of disks and squares. Though the local-search algorithm is slower
than the quadtree-based PTAS in these special cases [
15
], it has the advantage that
it only requires the intersection graph as input, not its geometric realization;
previously, an algorithm with this property was only known in further special cases, such
as unit disks [
36
]. Our result uses the planar separator theorem, but in contrast to
the separator-based method in [
15
], a standard version of the theorem suffices and is
needed only in the analysis, not in the algorithm itself.
Planar graphs are special cases of disk intersection graphs, and so our result
applies. Of course, PTASs for planar graphs have been around for quite some time
[
9, 30
], but the fact that a simple local-search algorithm already yields a PTAS for
planar graphs is apparently not well known, if it was known at all.
We can further show that the same local-search algorithm gives a PTAS for
independent set for fat objects in any fixed dimension, reproving known results in [
15, 22
].
This strategy, unfortunately, works only in the unweighted case.
Independent Set via LP It is easy to extract a large independent set from a sparse
unweighted graph. For example, greedily, we can order the vertices from lowest to
highest degree, and pick them one by one into the independent set, if none of its
neighbors was already picked into the independent set. Let dG be the average degree
in G. Then a constant fraction of the vertices have degree O(dG), and the selection
of such a vertex can eliminate O(dG) candidates. Thus, this yields an independent
set of size Ω(n/dG). Alternatively, for better constants, we can order the vertices
by a random permutation and do the same. Clearly, the probability of a vertex v
to be included in the independent set is 1/(d(v) + 1). An easy calculation leads to
Turán’s theorem, which states that any graph G has an independent set of size ≥
n/(dG + 1) [
6
].
Now, our intersection graph G may not be sparse. We would like to “sparsify”
it, so that the new intersection graph is sparse and the number of vertices is close
to the size of the optimal solution. Interestingly, we show that this can be done by
solving the LP relaxation of the independent set problem. The relaxation provides us
with a fractional solution, where every object fi has value xi ∈ [
0, 1
] associated with
it. Rounding this fractional solution into a feasible solution is not a trivial task, as
no such scheme exists in the general case. Our basic approach is somewhat similar
to the local-ratio technique [
10
], but, more precisely, it is a variant of the contention
resolution scheme of Chekuri et al. [
18
]. To this end, we prove a technical lemma (see
Lemma 4.1) that shows that the total sum of terms of the form xi xj , over pairs fi fj
that intersect is bounded by the boundary complexity of the union of E objects of F,
where E is the size of the fractional solution. The proof contains a nice application of
the standard Clarkson technique [
19
].
This lemma implies that on average, if we pick fi into our random set of objects,
with probability xi , then the resulting intersection graph would be sparse. This is by
itself sufficient to get a constant-factor approximation for the unweighted case. For
the weighted case, we follow a greedy approach: we examine the objects in a certain
order (based on a quantity we call “resistance”), and choose an object with probability
around xi , on condition that it does not intersect any previously chosen object. We
argue, for our particular order, that each object is indeed chosen with probability
Ω(xi ). This leads to a constant-factor approximation for weighted pseudo-disks.
Interestingly, the rounding scheme we use can be used in more general settings,
when one tries to find an independent set that maximizes a submodular target
function. See Sect. 4.6 for details.
Linear Union Complexity Our LP analysis works more generally for any class of
objects with linear union complexity. We assume that the boundary of the union of
any k of these objects has at most k vertices, for some fixed . For pseudo-disks,
the boundary of the union is made out of at most 6n − 12 arcs, implying = 6 in this
case [
28
].
A family F of simply connected regions bounded by simple closed curves in
general position in the plane is k-admissible (with k even) if for any pair fi , fj ∈ F, we
have: (i) fi \ fj and fj \ fi are connected, and (ii) their boundary intersect at most k
times. Whitesides and Zhao [
40
] showed that the union of such n objects has at most
3kn − 6 arcs; that is, = 3k. So, our LP analysis applies to this class of objects as
well. For more results on union complexity, see the sermon by Agarwal et al. [
5
].
Our local-search PTAS works more generally for unweighted admissible regions
in the plane. For an arbitrary class of unweighted objects with linear union complexity
in the plane, local search still yields a constant-factor approximation.
Rectangles LP relaxation has been used before, notably, in Chalermsook and
Chuzhoy’s recent breakthrough in the case of axis-parallel rectangles [
14
], but their
analysis is quite complicated. Although rectangles do not have linear union
complexity in general, we observe in Sect. 5 that a variant of our approach can yield a
readily accessible proof of a sublogarithmic O(log n/ log log n) approximation factor for
rectangles, even in the weighted case, where previously only logarithmic
approximation is known [
4, 12, 16
] (Chalermsook and Chuzhoy’s result is better but currently
is applicable only to unweighted rectangles).
Discussion Local search and LP relaxation are of course staples in the design of
approximation algorithms, but are not seen as often in computational geometry. Our
main contribution lies in the fusion of these approaches with combinatorial geometric
techniques.
In a sense, one can view our results as complementary to the known results on
approximate geometric set cover by Brönnimann and Goodrich [
13
] and Clarkson
and Varadarajan [
20
]. They consider the problem of finding the minimum number of
objects in F to cover a given point set. Their results imply a constant-factor
approximation for families of objects with linear union complexity, for instance. One version
of their approaches is indeed based on LP relaxation [
23, 31
]. The “dual” hitting
set problem is to find the minimum number of points to pierce a given set of
objects. Brönnimann and Goodrich’s result combined with a recent result of Pyrga and
Ray [37] also implies a constant-factor approximation for pseudo-disks for this
piercing problem. The piercing problem is actually the dual of the independent set problem
(this time, we are referring to linear programming duality). We remark, however, that
the rounding schemes for set cover and piercing are based on different combinatorial
techniques, namely, ε-nets, which are not sufficient to deal with independent set (one
obvious difference is that independent set is a maximization problem).
In Theorem 4.6, we point out a combinatorial consequence of our LP analysis:
for any collection of unweighted pseudo-disks, the ratio of the size of the minimum
piercing set to the size of maximum independent set is at most a constant. (It is easy
to see that the ratio is always at least 1; for disks or fat objects, it is not difficult to
obtain a constant upper bound by packing arguments.) This result is of independent
interest; for example, getting tight bounds on the ratio for axis-parallel rectangles is
a long-standing open problem.
In an interesting independent development, Mustafa and Ray [
35
] have recently
applied the local search paradigm to obtain a PTAS for the geometric set cover
problem for (unweighted) pseudo-disks and admissible regions.
2 Preliminaries
In the following, we have a set F of n objects in the plane, such that the union
complexity of any subset X ⊆ F is bounded by |X|, where is a constant. Here, the
union complexity of X is the number of arcs on the boundary of the union of the
objects of X. Let A(F) denote the arrangement of F, and V(F) denote the set of vertices
of A(F).
In the following, we assume that deciding if two objects intersect takes constant
time.
3 Approximation by Local Search: Unweighted Case
3.1 The Algorithm
In the unweighted case, we may assume that no object is fully contained in another.
We say that a subset L of F is b-locally optimal if L is an independent set and
one cannot obtain a larger independent set from L by deleting at most b objects and
inserting at most b + 1 objects of F.
Our algorithm for the unweighted case simply returns a b-locally optimal solution
for a suitable constant b, by performing a local search. We start with L ← ∅. For every
subset X ⊆ F \ L of size at most b + 1, we verify that X by itself is independent, and,
furthermore, that the set Y ⊆ L of objects intersecting the objects of X is of size
at most |X| − 1. If so, we do L ← (L \ Y ) ∪ X. Every such exchange increases the
size of L by at least one, and as such it can happen at most n times. Naively, there
n
are b+1 subsets X to consider, and for each such subset X it takes O(nb) time to
compute Y . Therefore, the running time is bounded by O(nb+3). (The running time
can be probably improved by being a bit more careful about the implementation.)
3.2 Analysis
We present two alternative ways to analyze this algorithm. The first approach uses
only the fact that the union complexity is low. The second approach is more direct,
and uses the property that the regions are admissible.
3.2.1 Analysis Using Union Complexity
The following lemma by Afshani and Chan [
1
], which was originally intended for
different purposes, will turn out to be useful here (the proof exploits linearity of planar
graphs and the Clarkson technique [
19
]):
Lemma 3.1 Suppose we have n disjoint simply connected regions in the plane and
a collection of disjoint curves, where each curve intersects at most k regions. Call
two curves equivalent if they intersect precisely the same subset of regions. Then the
number of equivalent classes is at most c0nk2 for some constant c0.
Let S be an optimal solution, and let L be a b-locally optimal solution. We will
upper bound |S| in terms of |L|.
Let S>b denote the set of objects in S that intersect at least b + 1 objects of L. Let
S≤b be the set of remaining objects in S.
If fi ∈ S intersects fj ∈ L, then the pair of objects contributes at least two vertices
to the boundary of the union of S ∪ L. Indeed, the objects of S (resp. L) are disjoint
from each other since this is an independent set, and no object is contained inside
another (by assumption). We remind the reader that for any subset X ⊆ F, the union
complexity of the regions of X is ≤ |X|. As such, the union complexity of S>b ∪ L
is ≤ (|S>b| + |L|). Thus,
2(b + 1)|S>b| ≤
|S>b| + |L|
=⇒
|S>b| ≤ 2(b + 1) −
|L|.
On the other hand, by applying Lemma 3.1 with L as the regions and the
boundaries of S≤b as the curves, the objects in S≤b form at most c0b2|L| equivalent classes.
Each equivalent class contains at most b objects: Otherwise we would be able to
remove b objects from L and intersect b + 1 objects in this equivalence class to get
an independent set larger than L. This would contradict the b-local optimality of L.
Thus, |S≤b| ≤ c0b3|L|.
Combining the two inequalities, we get
|S| ≤ |S≤b| + |S>b| ≤
3
c0b + 2(b + 1) −
|L|.
For example, we can set b =
/2 and the approximation factor is O( 3).
Theorem 3.2 Given a set of n unweighted objects in the plane with linear union
complexity, for a sufficiently large constant b, any b-locally optimal independent set
has size Ω(opt), where opt is the size of the maximum independent set of the objects.
3.2.2 Better Analysis for Admissible Regions
A set of regions F is admissible, if for any two regions f, f ∈ F, we see that f \ f
and f \ f are both simply connected (i.e., connected and contains no holes). Note
that we do not care how many times the boundaries of the two regions intersect, and
furthermore, by definition, no region is contained inside another.
Lemma 3.3 Let F be a set of admissible regions, and consider a independent set of
regions I ⊆ F, and a region f ∈ F \ I . Then the core region f \ I = f \ g∈I g is
non-empty and simply connected.
Proof It is easy to verify that for the regions of I to split f into two connected
components, they must intersect, which contradicts their disjointness.
Lemma 3.4 Let X, Y ⊆ F be two independent sets of regions. Then the intersection
graph G of X ∪ Y is planar.
Proof Lemma 3.3 implies the planarity of this graph.
Indeed, for a region f ∈ X, the core f = f \ ∪g∈Y g is non-empty and simply
connected. Place a vertex vf inside this region, and for every object g ∈ Y that
intersects f , create a curve from vf to a point pf,g on the boundary of g that lies inside f .
Clearly, we can create these curves in such a way that they do not intersect each other.
Similarly, for every region g ∈ Y , we place a vertex vg inside g, and connect it
to all the points pf,g placed on its boundary, by curves that are contained in g, and
they are interior disjoint. Clearly, together, these vertices and curves form a planar
drawing of G.
We need the following version of the planar separator theorem. Below, for a set of
vertices U in a graph G, let Γ (U ) denote the set of neighbors of U , and let Γ (U ) =
Γ (U ) ∪ U .
Lemma 3.5 [
25
] There are constants c1, c2 and c3, such that for any planar graph
G = (V , E) with n vertices, and a parameter r , one can find a set of X ⊆ V of
size at most c1n/√r , and a partition of V \ X into n/r sets V1, . . . , Vn/r , satisfying:
(i) |Vi | ≤ c2r , (ii) Γ (Vi ) ∩ Vj = ∅, for i = j , and (iii) |Γ (Vi ) ∩ X| ≤ c3√r .
Let S be the optimal solution and L be a b-locally optimal solution. Consider the
bipartite intersection graph G of S ∪ L. By Lemma 3.4, we can apply Lemma 3.5
to G, for r = b/(c2 + c3). Note that |Γ (Vi )| ≤ c2r + c3√r < b for each i. Let
si = |Vi ∩ S|,
i = |Vi ∩ L|,
and
bi = Γ (Vi ) ∩ X , for each i.
Observe that i + bi ≥ si , for all i. Indeed, otherwise, we can throw away the vertices
of L ∩ Γ (Vi ) from L, and replace them by Vi ∩ S, resulting in a better solution. This
would contradict the local optimality of L. Thus,
|S| ≤
si + |X| ≤
i
i +
i
i
bi + |X|
≤ |L| + c3√r · |S| +r |L| + c1 |S|√+r|L|
≤ |L| + (c1 + c3) |S| + |L| .
√r
It follows that |S| ≤ (1 + O(1/√b))|L|. We can set b to the order of 1/ε2, and we get
the following.
Theorem 3.6 Given a set of n unweighted admissible regions in the plane, any
blocally optimal independent set has size ≥ (1 − O(1/√b)) opt, where opt is the size
of the maximum independent set of the objects. In particular, one can compute an
independent set of size ≥ (1 − ε) opt, in time nO(1/ε2).
3.2.3 Analysis for Fat Objects in Any Fixed Dimension
We show that the same algorithm gives a PTAS for the case when the objects in F
are fat. This result in fact holds in any fixed dimension d. For our purposes, we use
the following definition of fatness: the objects in F are fat if for every axis-aligned
hypercube B of side length r , we can find a constant number c of points such that
every object that intersects B and has diameter at least r contains one of the chosen
points.
Smith and Wormald [
38
] proved a family of geometric separator theorems, one
version of which will be useful for us and is stated below (see also [
15
]):
Lemma 3.7 [
38
] Given a collection of n fat objects in a fixed dimension d with
constant maximum depth, there exists an axis-aligned hypercube B such that at most
2n/3 objects are inside B, at most 2n/3 objects are outside B, and at most O(n1−1/d )
objects intersect the boundary of B.
We need the following extension of Smith and Wormald’s separator theorem to
multiple clusters (whose proof is similar to the extension of the standard planar
separator theorem in [
25
]):
Lemma 3.8 There are constants c1, c2, c3 and c4, such that for any intersection
graph G = (V , E) of n fat objects in a fixed dimension d with constant maximum
depth, and a parameter r , one can find a set of X ⊆ V of size at most c1n/r1/d , and
a partition of V \ X into n/r sets V1, . . . , Vn/r , satisfying: (i) |Vi | ≤ c2r , (ii) Γ (Vi ) ∩
Vj = ∅, for i = j , and (iii) i |Γ (Vi ) ∩ X| ≤ c3n/r1/d , and (iv) |Γ (Vi ) ∩ X| ≤ c4r .
Proof Assume that all objects are unmarked initially. We describe a recursive
procedure for a given set S of objects. If |S| ≤ c2r , then S is a “leaf” subset and we stop the
recursion. Otherwise, we apply Lemma 3.7. Let S and S be the subset of all objects
inside and outside the separator hypercube B, respectively. Let S be the subset of all
objects intersecting the boundary of B. We mark the objects in S and recursively run
the procedure for the subset S ∪ S and for the subset S ∪ S.
Note that some objects may be marked more than once. Let X be the set of all
objects that have been marked at least once. For each leaf subset Si , generate a subset
Vi of all unmarked objects in Si . Property (i) is obvious. Properties (ii) and (iv) hold,
because the unmarked objects in each leaf subset Si can only intersect objects within
Si and cannot intersect unmarked objects in other Sj ’s.
The total number of marks satisfies the recurrence T (n) = 0 if n ≤ c2r , and
max
T (n) ≤ n1,n2≤2n/3
n1+n2≤n
T n1 + O n1−1/d
+ T n2 + O n1−1/d
+ O n1−1/d
otherwise. The solution is T (n) = O(n/r1/d ). Thus, we have |X| = O(n/r1/d ).
Furthermore, for each object f ∈ X, the number of leaf subsets that f is in is equal to 1
plus the number of marks that f receives. Thus, (iii) follows.
Let S be the optimal solution and L be a b-locally optimal solution. Consider the
bipartite intersection graph G of S ∪ L, which has maximum depth 2. We proceed
as in the proof from Sect. 3.2.2, using Lemma 3.8 instead of Lemma 3.5. Note that
(iii)–(iv) are weaker properties but are sufficient for the same proof to go through.
The only differences are that square roots are replaced by d th roots, and we now set
r = b/(c2 + c4), so that |Γ (Vi )| ≤ c2r + c4r < b. We conclude:
Theorem 3.9 Given a set of n fat objects in a fixed dimension d , any b-locally
optimal independent set has size ≥ (1 − O(1/b1/d )) opt, where opt is the size of the
maximum independent set of the objects. In particular, one can compute an
independent set of size ≥ (1 − ε) opt, in time nO(1/εd ).
4 Approximation by LP Relaxation: Weighted Case
4.1 The Algorithm
We are interested in computing a maximum-weight independent set of the objects in
F = {f1, . . . , fn}, with weights w1, . . . , wn, respectively. To this end, let us solve the
following LP relaxation:
max
n
i=1
wi xi
fi p
0 ≤ xi ≤ 1,
xi ≤ 1
∀p ∈ V(F)
(1)
where V(F) denotes the set of vertices of the arrangement A(F).
In the following, xi will refer to the value assigned to this variable by the solution
of the LP. Similarly, Opt = i wi xi will denote the weight of the relaxed optimal
solution, which is at least the weight opt of the optimal integral solution.
We will assume, for the time being, that no two objects of F fully contain each
other.
For every object fi , let its resistance be the total sum of the values of the objects
that intersect it. Formally, we have
η(fi , F) =
xj .
fj ∈F\{fi }
fi ∩fj =∅
We pick the object in F with minimal resistance, and set it as the first element in
the permutation Π of the objects. We compute the permutation by performing this
“extract-min”, with the variant that objects that are already in the permutation are
ignored. Formally, if we computed the first i objects in the permutation to be Πi =
π1, . . . , πi , then the (i + 1)th object πi+1 is the one realizing
ηi+1 =
min η(f, F \ Πi ).
f ∈F\Πi
(2)
The algorithm starts with an empty candidate set C and an empty independent set
I , and scans the objects according to the permutation in reverse order. At the ith stage,
the algorithm first decides whether to put the object πn−i in C, by flipping a coin that
with probability x(πn−i )/τ comes up heads, where x(πn−i ) is the value assigned by
the LP to the object πn−i and τ is some parameter to be determined shortly. If πn−i
is put into C then we further check whether πn−i intersects any of the objects already
added to the independent set I . If it does not intersect any objects in I , then it adds
πn−i to I and continues to the next iteration.
In the end of the execution, the set I is returned as the desired independent set.
4.2 Analysis
Let F be a set of n objects in the plane, and let u(m) be the maximum union
complexity of m ≤ n objects of F. Furthermore, we assume that the function u(·) is
a monotone increasing function which is well behaved; namely, u(n)/n is a
nondecreasing function, and there exists a constant c, such that u(xr) ≤ cu(r), for any r
and 1 ≤ x ≤ 2. In the following, a vertex p of V(H) is denoted by (p, i, j ), to indicate
that it is the result of the intersection of the ith and j th object.
The key to our analysis lies in the following inequality, which we prove by
adapting the Clarkson technique [
19
].
Lemma 4.1 Let H be any subset of F. Then
E(H) = f ∈H x(f ).
(p,i,j)∈V(H) xi xj = O(u(E(H))), where
Proof Consider a random sample R of H, where an object fi is being picked up with
probability xi /2. Clearly, we find that (p, i, j ) ∈ V(F) appears on the boundary of the
union of the objects of R , if and only if fi and fj are being picked, and none of
the objects that cover p are being chosen into R . In particular, let U(R ) denote the
vertices on the boundary of the union of the objects of R . We have
Pr (p, i, j ) ∈ U R
,
fk p,
k=i,k=j
Pr (p, i, j ) ∈ U R
= E U R
≤ E u R
.
by the inequality k(1 − ak) ≥ 1 − k ak for ak ∈ [
0, 1
], since fk p xk ≤ 1 (as
the LP solution is valid). On the other hand, the number of vertices on the union is
|U(R )| ≤ u(|R |). Thus,
To bound last expression, observe that μ = E[|R |] = E(H)/2. Furthermore, by
Chernoff inequality, Pr[|R | > (t + 1)μ] ≤ 2−t . Thus, E[u(|R |)] ≤ t∞=1 2−t+1u(t μ) =
t=1 2−t+1t O(1)u(μ) = O(u(μ)) = O(u(E(H))), since u(·) is well behaved.
∞
Lemma 4.2 For any i, the resistance of the ith object πi (as defined by Eq. (2)) is
ηi = O( u(EE((FF))) ).
Proof Fix an i, and let K = F \ {π1, . . . , πi−1}. By Lemma 4.1,
xj η(fj , K) = 2
xi xj = O u E(K)
fj ∈K
=⇒
fj ∈K
xj
E(K)
(p,i,j)∈V(K)
η(fj , K) = O
u(E(K))
E(K)
= O
u(E(F))
E(F)
by the monotonicity of u(n)/n, where E(K) =
minfj ∈K η(fj , K) = O( u(EE((FF))) ).
fj ∈K xj . It follows that
Lemma 4.3 For a sufficiently large constant c, setting τ = c u(EE((FF))) , the algorithm in
Sect. 4.1 outputs in expectation an independent set of weight Ω((n/u(n))Opt).
Proof Indeed, the j th object in the permutation is added to C with probability xj /τ .
Let K be the set of all objects of F that were already considered and intersect fj .
Clearly, E(K) is exactly the resistance ηj of fj . Furthermore by picking c large
enough, we have E(K) = ηj ≤ τ /2 by Lemma 4.2. This implies that
Pr fj ∈ I | fj ∈ C = Pr K ∩ C = ∅ | fj ∈ C = Pr[K ∩ C = ∅] =
fj ∈K
As such, the expected value of the independent set output is
yj wj =
j
j
xj
2τ wj = Ω
Opt
τ
= Ω
n
u(n)
Opt ,
as E(F) ≤ n.
4.3 Remarks
Variant In the conference version of this paper [
17
], we proposed a different variant
of the algorithm, where instead of ordering the objects by increasing resistance, we
order the objects by decreasing weights. An advantage of the resistance-based
algorithm is that it is oblivious to (i.e., does not look at) the input weights. This feature
is shared, for example, by Varadarajan’s recent algorithm for weighted geometric set
cover via “quasi-random sampling” [
39
]. Another advantage of the resistance-based
algorithm is its extendibility to other settings; see Sect. 4.6.
Derandomization The variance of the expected weight of the returned independent
set I could be high, but fortunately the algorithm can be derandomized by the
standard method of conditional probabilities/expectations [
34
]. To this end, observe that
the above analysis provide us with a constructive way to estimate the weight of the
generated solution. It is now straightforward to decide for each region whether to
include it or not inside the generated solution, using conditional probabilities. Indeed,
for each object we compute the expected weight of the solution if it is include in the
solution, and if it is not included in the solution, and pick the one that has higher
value.
Coping with Object Containment We have assumed that no object is fully
contained in another, but this assumption can be removed by adding the constraint
fi ⊂fj xj ≤ 1 for each i to the LP. Then, for any subset H of F, we have
fi ,fj ∈H
fi ⊂fj
xi xj ≤ E(H),
and so Lemma 4.1 still holds. The rest of the analysis then holds verbatim.
Time to Solve the LP This LP is a packing LP with O(n2) inequalities, and n
variables. As such, it can be (1 + ε)-approximated in O(n3 + ε−2n2 log n) = O(n3) by a
randomized algorithm that succeeds with high probability [
29
]. For our purposes, it
is sufficient to set ε to be a sufficient small constant, say ε = 10−4.
We have thus proved:
Theorem 4.4 Given a set of n weighted objects in the plane with union complexity
O(u(n)), one can compute an independent set of total weight Ω((n/u(n)) opt), where
opt is the maximum weight over all independent sets of the objects. The running time
of the randomized algorithm is O(n3), and polynomial for the deterministic version.
The running time of the deterministic algorithm of Theorem 4.4 is dominated by
the time it takes to deterministically solve (approximately) the LP. One can use the
ellipsoid algorithm to this end, but faster algorithm are known, see [
29
] and references
therein.
Corollary 4.5 Given a set of n weighted pseudo-disks in the plane, one can compute,
in O(n3) time, a constant-factor approximation to the maximum-weight independent
set of pseudo-disks.
Theorem 4.4 can be applied to cases where the union complexity is low. Even in
the case of fat objects, where PTASs are known [
15, 22
], the above approach is still
interesting as it can be extended to more general settings, as noted in Sect. 4.6.
4.4 A Combinatorial Result: Piercing Number
In the unweighted case, we obtain the following result as a byproduct:
Theorem 4.6 Given a set of n pseudo-disks in the plane, let opt be the size of the
maximum independent set and let opt be the size of the minimum set of points that
pierce all the pseudo-disks. Then opt = Ω(opt ).
Proof By the preceding analysis, we have opt = Ω(Opt), i.e., the integrality gap of
our LP is a constant. (Here, all the weights are equal to 1.)
For piercing, the LP relaxation is
min
n
p∈V(S)
yp
Let Opt be the value of this LP. Known analysis [
23, 31
] implies that the
integrality gap of this LP is constant if there exist ε-nets of linear size for a corresponding
class of hypergraphs formed by objects in F and points in V(S). Pyrga and Ray [37,
Theorem 12] obtained such an existence proof for this (“primal”) hypergraph for
pseudo-disks. Thus, opt = O(Opt ).
To conclude, observe that the two LPs are precisely the dual of each other, and so
Opt = Opt .
4.5 A Discrete Version of the Independent Set Problem
We now show that our algorithm can be extended to solve a variant of the independent
set problem where we are given not only a set F of n weighted objects but also a set P
of m points. The goal is to select a maximum-weight subset S ⊆ F such that each point
p ∈ P is contained in at most one object of S. (The original problem corresponds to
the case where P is the entire plane.) Unlike in the original independent set problem,
it is not clear if local search yields good approximation here, even in the unweighted
case.
We can use the same LP as in Sect. 4.1 to solve this problem, except that we now
have a constraint for each p ∈ P instead of each p ∈ V(F). In the rest of the algorithm
and analysis, we just reinterpret “fi ∩ fj = ∅” to mean “fi ∩ fj ∩ P = ∅”.
Lemma 4.1 is now replaced by the following.
Lemma 4.7 Let H be any subset of F. Then
fi ∩fj ∩P=∅ xi xj = O(u(E(H))).
fi ,fj ∈H
Proof Consider a random sample R of H, where an object fi is being picked up with
probability xi /2. Let VD(R) be the cells in the vertical decomposition of the
complement of the union U(R). For a cell Δ ∈ VD(R), let xΔ = fi ∈H,fi ∩int(Δ)=∅ xi
be the total energy of the objects of H that intersects the interior of Δ. A
minor modification of the analysis of Clarkson [
19
] implies that, for any constant c,
E[ Δ∈VD(R)(xΔ)c] = O(E[u(E(H))]).
A point of p ∈ P is active for R, if it is outside the union of objects of R. Let P be
the set of active points in P. We have
fi ∩fj ∩P =∅
1
2
xi xj ≤ E
xi xj ≤ E
fi ∩fj ∩P=∅
The above proof is inspired by a proof from [
3
]. There is an alternative argument
based on shallow cuttings, but the known proof for the existence of such cuttings
requires a more complicated sampling analysis [
33
].
The rest of the analysis then goes through unchanged. We therefore obtain an
O(1)-approximation algorithm for the discrete independent set problem for
unweighted or weighted pseudo-disks in the plane.
4.6 Contention Resolution and Submodular Functions
The algorithm of Theorem 4.4 can be interpreted as a contention resolution scheme;
see Chekuri et al. [
18
] for details. The basic idea is that, given a feasible fractional
solution x ∈ [
0, 1
]n, a contention resolution scheme scales down every coordinate of
x (by some constant b) such that, given a random sample C of the objects according
to x (i.e., the ith object fi is picked with probability bxi ), the contention resolution
scheme computes (in our case) an independent set I such that Pr[fi ∈ I | fi ∈ C] ≥ c,
for some positive constant c. The proof of Lemma 4.3 implies exactly this property
in our case.
As such, we can apply the results of Chekuri et al. [
18
] to our settings. In
particular, they show that one can obtain constant approximation to the optimal solution,
when considering independence constraints and submodular target function.
Intuitively, submodularity captures the diminishing-returns nature of many optimization
problems. Formally, a function g : 2F → R is submodular if g(X ∪ Y ) + g(X ∩ Y ) ≤
g(X) + g(Y ), for any X, Y ⊆ F.
As a concrete example, consider a situation where each object in F represents a
coverage area by a single antenna. If a point is contained inside such an object, it
is fully serviced. However, even if it is not contained in a object, it might get some
reduced coverage from the closest object in the chosen set. In particular, let ν(r)
be some coverage function which is the amount of coverage a point gets if it is at
distance r from the closest object in the current set I . We assume here ν(·) is a
monotone decreasing function. Because of interference between antennas we require
that the regions these antennas represent do not intersect (i.e., the set of antennas
chosen needs to be an independent set).
Lemma 4.8 Let P be a set of points, and let F be a set of objects in the plane. Let ν(·)
be a monotone decreasing function. For a subset H ⊆ F, consider the target function
p∈P
where dp(H) is the distance of p to its nearest neighbor in H. Then the function αP(H)
is submodular.
Proof The proof is not hard and is included for the sake of completeness. For a point
p ∈ P, it is sufficient to prove that the function ν(dp(H)) is submodular, as αP(H) is
just the sum of these functions, and a sum of submodular functions is submodular.
To prove the latter, it is sufficient to prove that for any sets X ⊆ Y ⊆ F, and an object f ∈ F \ Y ,
ν dp X ∪ {f }
− ν dp(X) ≥ ν dp Y ∪ {f }
− ν dp(Y ) .
To this end, let x and y be the closest objects to p in X and Y , respectively. Similarly,
let x , y , f be the distance of p to x, y and f , respectively. The above then becomes
ν min( x , f ) − ν x ≥ ν min( y , f ) − ν y .
(3)
Observe that as X ⊆ Y , we have x ≥ y , so ν( y ) ≥ ν( x ) as ν is monotone
decreasing. Now, one of the following holds:
• If f ≤ y ≤ x then Eq. (3) becomes ν( f ) − ν( x ) ≥ ν( f ) − ν( y ), which holds.
• If y ≤ f ≤ x then Eq. (3) becomes ν( f ) − ν( x ) ≥ ν( y ) − ν( y ), which is
equivalent to ν( f ) ≥ ν( x ). This in turn holds by the decreasing monotonicity
of ν.
• If y ≤ x ≤ f then Eq. (3) becomes 0 = ν( x ) − ν( x ) ≥ ν( y ) − ν( y ) = 0.
We conclude that αP(·) is submodular.
To solve our problem, using the framework of Chekuri et al. [
18
], we need the
following:
(A) The target function is indeed submodular and can be computed efficiently. This
is Lemma 4.8.
(B) State an LP that solves the fractional problem (and its polytope contains the
optimal integral solution). This is just the original LP, see Eq. (1).
(C) Observe that our rounding (i.e., contention resolution) scheme is still applicable
in this case. This follows by Lemma 4.3.
One can now plug this into the algorithm of Chekuri et al. [
18
] and get an
Ω(α)approximation algorithm, where α is the rounding scheme gap. The algorithm of
Chekuri et al. [
18
] uses a continuous optimization to find the maximum of a
multilinear extension of the target function inside the feasible polytope, and then uses this
fractional value with the rounding scheme to get the desired approximation.
We thus get the following.
Problem 4.9 Let P be a set of n points in the plane, and let F be a set of m objects
in the plane. Let ν(r) be a monotone decreasing function, which returns the amount
of coverage a point gets if it is at distance r from one of the regions of F. Consider
the scoring function that for an independent set H ⊆ F returns the total coverage it
provides; that is,
We refer to the problem of computing the independent set maximizing this function as
the partial coverage problem.
Theorem 4.10 Given a set of n points in the plane and a set m of unweighted
objects in the plane with union complexity O(u(n)), one can compute, in polynomial
time, an independent set. Furthermore, this independent set provides an
Ω(n/u(n))approximation to the optimal solution of the partial coverage problem.
Observe that the above algorithm applies for any pricing function that is
submodular. In particular, one can easily encode into this function weights for the ranges, or
other similar considerations.
5 Weighted Rectangles
5.1 The Algorithm
For the (original) independent set problem in the case of weighted axis-aligned
rectangles, we can solve the same LP, where the set V(F) contains both intersection points
and corners of the given rectangles.
Define two subgraphs G1 and G2 of the intersection graph: if the boundaries of fi
and fj intersect zero or two times, put fi fj in G1; if the boundaries intersect four
times instead, put fi fj in G2.
We first extract an independent set I of G1 using the algorithm of Theorem 4.4.
It is well known (e.g., see [
8
]) that G2 forms a perfect graph (specifically, a
comparability graph), so find a Δ-coloring of the rectangles of I in G2, where Δ denotes the
maximum clique size, i.e., the maximum depth in A(R). Let I be the color subclass
of I of the largest total weight. Clearly, the objects in I are independent, and we
output this set.
5.2 Analysis
As in Lemma 4.1, let H be any subset of F. Observe that if fi fj ∈ G1, then fi contains
a corner of fj or vice versa. Letting Vj denote the corners of fj , we have
Applying the same analysis as before, we conclude that the expected total weight
of I (i.e., the independent set of G1) computed by the algorithm is of size Ω(Opt).
To analyze I , we need a new lemma which bounds the maximum depth of R:
Lemma 5.1 Δ = O(log n/ log log n) with probability at least 1 − 1/n.
Proof Fix a parameter t > 1. Fix a point p ∈ V(F). The depth of p in A(R), denoted
by depth(p, R), is a sum of independent 0-1 random variables with overall mean
μ = p∈fi xi ≤ 1. By the Chernoff bound [34, p. 68],
Pr depth(p, R) > (1 + δ)μ <
eδ
(1 + δ)1+δ
μ
for any δ > 0 (possibly large). By setting δ so that t = (1 + δ)μ, this
probability becomes less than (e/t )t . Since |V(F)| = O(n2), the probability that Δ > t
is at most O((e/t )t n2), which is at most 1/n by setting the value of t to be
Θ(log n/ log log n).
By construction of I , we know that
fi ∈I
1
wi ≥ Δ
where 1A denotes the indicator variable for event A. With t = Θ(log n/ log log n), we
conclude that
E
fi ∈I
wi ≥ Ω(log log n/ log n) E
wi − Opt/n
fi ∈I
≥ Ω(log log n/ log n) · Opt.
5.3 Remarks
Derandomization This algorithm can also be derandomized by the method of
conditional expectations. The trick is to consider the following random variable:
1
Z := t
wi − Opt ·
(1 + δp)depth(p,R)−t ,
fi ∈I
p∈V(F)
where δp is the δ from the proof of Lemma 5.1 and t is the same as before. This
variable Z lower-bounds fi ∈I wi (the bound is trivially true if Δ > t , since Z
would be negative). Our analysis still implies that E[Z] ≥ Ω(log log n/ log n) · Opt
(since the standard proof of the Chernoff bound [34, pp. 68–69] actually shows that
Pr depth(p, R) > (1 + δ)μ <
E[(1 + δ)depth(p,R)]
(1 + δ)(1+δ)μ
eδ
≤ (1 + δ)1+δ
μ
,
which for δ = δp and μ ≤ 1 implies E[(1 + δp)depth(p,R)−t ] < (e/t )t ). The advantage
of working with Z is that we can calculate E[Z] exactly in polynomial time, even
when conditioned to the events that some objects are known to be in or not in R (since
depth(p, R) is a sum of independent 0-1 random variables, making (1 + δp)depth(p,R)
a product of independent random variables).
We have thus proved:
Theorem 5.2 Given a set of n weighted axis-aligned boxes in the plane, one can
compute in polynomial time an independent set of total weight Ω (log log n/ log n) · opt,
where opt is the maximum weight over all independent sets of the objects.
Higher Dimensions By a standard divide-and-conquer method [
4
], we get an
approximation factor of O(logd−1 n/ log log n) for weighted axis-aligned boxes in any
constant dimension d .
Acknowledgements We thank Esther Ezra for discussions on the discrete version of the independent
set problem considered in Sect. 4.3. The somewhat cleaner presentation in the paper, compared to the
preliminary version [
17
], was suggested by Chandra Chekuri. The results of Sect. 4.6 were inspired by
discussions with Chandra Chekuri. We also thank the anonymous referees for their comments.
1. Afshani , P. , Chan , T.M. : Dynamic connectivity for axis-parallel rectangles . In: Proc. 14th European Sympos. Algorithms. Lecture Notes Comput. Sci. , vol. 4168 , pp. 16 - 27 ( 2006 )
2. Agarwal , P.K. , Mustafa , N.H. : Independent set of intersection graphs of convex objects in 2D . Comput. Geom., Theory Appl . 34 ( 2 ), 83 - 95 ( 2006 )
3. Agarwal , P.K. , Aronov , B. , Chan , T.M. , Sharir , M. : On levels in arrangements of lines, segments, planes, and triangles . Discrete Comput. Geom . 19 , 315 - 331 ( 1998 )
4. Agarwal , P.K., van Kreveld , M. , Suri , S. : Label placement by maximum independent set in rectangles . Comput. Geom., Theory Appl . 11 , 209 - 218 ( 1998 )
5. Agarwal , P.K. , Pach , J. , Sharir , M. : State of the union-of geometric objects . In: Goodman, J.E. , Pach , J. , Pollack , R . (eds.) Surveys in Discrete and Computational Geometry Twenty Years Later . Contemporary Mathematics , vol. 453 , pp. 9 - 48 . AMS, Providence ( 2008 )
6. Alon , N. , Spencer , J.H. : The Probabilistic Method, 2nd edn . Wiley-Interscience, New York ( 2000 )
7. Arora , S. : Polynomial time approximation schemes for Euclidean TSP and other geometric problems . J. Assoc. Comput. Mach . 45 ( 5 ), 753 - 782 ( 1998 )
8. Asplund , E. , Grübaum , B. : On a coloring problem . Math. Scand. 8 , 181 - 188 ( 1960 )
9. Baker , B.S. : Approximation algorithms for NP-complete problems on planar graphs . J. Assoc. Comput. Mach . 41 , 153 - 180 ( 1994 )
10. Bar-Yehuda , R. , Bendel , K. , Freund , A. , Rawitz , D. : Local ratio: A unified framework for approximation algorithms . In memoriam: Shimon Even 1935-2004. ACM Comput. Surv . 36 , 422 - 463 ( 2004 )
11. Berman , P. , Fujito , T. : On approximation properties of the independent set problem for low degree graphs . Theor. Comput. Sci . 32 ( 2 ), 115 - 132 ( 1999 )
12. Berman , P. , DasGupta , B. , Muthukrishnan , S. , Ramaswami , S. : Efficient approximation algorithms for tiling and packing problems with rectangles . J. Algorithms 41 , 443 - 470 ( 2001 )
13. Brönnimann , H. , Goodrich , M.T.: Almost optimal set covers in finite VC-dimension . Discrete Comput. Geom . 14 , 263 - 279 ( 1995 )
14. Chalermsook , P. , Chuzhoy , J.: Maximum independent set of rectangles . In: Proc. 20th ACM-SIAM Sympos. Discrete Algorithms , pp. 892 - 901 ( 2009 )
15. Chan , T.M. : Polynomial-time approximation schemes for packing and piercing fat objects . J. Algorithms 46 ( 2 ), 178 - 189 ( 2003 )
16. Chan , T.M.: A note on maximum independent sets in rectangle intersection graphs . Inf. Process. Lett . 89 , 19 - 23 ( 2004 )
17. Chan , T.M. , Har-Peled , S. : Approximation algorithms for maximum independent set of pseudo-disks . In: Proc. 25th Annu. ACM Sympos. Comput. Geom. , pp. 333 - 340 ( 2009 ). cs.uiuc.edu/~sariel/papers/08/w_indep
18. Chekuri , C. , Vondrák , J. , Zenklusen , R.: Submodular function maximization via the multilinear relaxation and contention resolution schemes . In: Proc. 43rd Annu. ACM Sympos. Theory Comput . ( 2011 , to appear)
19. Clarkson , K.L. , Shor , P.W. : Applications of random sampling in computational geometry, II. Discrete Comput . Geom. 4 , 387 - 421 ( 1989 )
20. Clarkson , K.L. , Varadarajan , K.R. : Improved approximation algorithms for geometric set cover . Discrete Comput. Geom . 37 ( 1 ), 43 - 58 ( 2007 )
21. Efrat , A. , Katz , M.J. , Nielsen , F. , Sharir , M. : Dynamic data structures for fat objects and their applications . Comput. Geom., Theory Appl . 15 , 215 - 227 ( 2000 )
22. Erlebach , T. , Jansen , K. , Seidel , E.: Polynomial-time approximation schemes for geometric intersection graphs . SIAM J. Comput . 34 ( 6 ), 1302 - 1323 ( 2005 )
23. Even , G. , Rawitz , D. , Shahar , S. : Hitting sets when the VC-dimension is small . Inf. Process. Lett . 95 ( 2 ), 358 - 362 ( 2005 )
24. Fox , J. , Pach , J.: Computing the independence number of intersection graphs . In: Proc. 22nd ACMSIAM Sympos. Discrete Algorithms , pp. 1161 - 1165 ( 2011 )
25. Frederickson , G.N. : Fast algorithms for shortest paths in planar graphs, with applications . SIAM J. Comput . 16 ( 6 ), 1004 - 1022 ( 1987 )
26. Halldórsson , M.M.: Approximations of independent sets in graphs . In: The 2nd Intl. Work. Approx. Algs. Combin. Opt. Problems , pp. 1 - 13 ( 1998 )
27. Hastad , J.: Clique is hard to approximate within n1−ε . Acta Math . 182 , 105 - 142 ( 1996 )
28. Kedem , K. , Livne , R. , Pach , J. , Sharir , M. : On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles . Discrete Comput. Geom. 1 , 59 - 71 ( 1986 )
29. Koufogiannakis , C. , Young , N.E. : Beating simplex for fractional packing and covering linear programs . In: Proc. 48th Annu. IEEE Sympos. Found. Comput. Sci , pp. 494 - 506 ( 2007 )
30. Lipton , R.J. , Tarjan , R.E.: A separator theorem for planar graphs . SIAM J. Appl. Math . 36 , 177 - 189 ( 1979 )
31. Long , P.M.: Using the pseudo-dimension to analyze approximation algorithms for integer programming . In: Proc. 7th Workshop Algorithms Data Struct. Lecture Notes Comput. Sci. , vol. 2125 , pp. 26 - 37 ( 2001 )
32. Marathe , M.V. , Breu , H. , Hunt , H.B. III , Ravi , S.S. , Rosenkrantz , D.J.: Simple heuristics for unit disk graphs . Networks 25 , 59 - 68 ( 1995 )
33. Matoušek , J.: Reporting points in halfspaces . Comput. Geom., Theory Appl . 2 ( 3 ), 169 - 186 ( 1992 )
34. Motwani , R. , Raghavan , P. : Randomized Algorithms. Cambridge University Press, New York ( 1995 )
35. Mustafa , N.H. , Ray , S.: PTAS for geometric hitting set problems via local search . In: Proc. 25th Annu. ACM Sympos. Comput. Geom. , pp. 17 - 22 ( 2009 )
36. Nieberg , T. , Hurink , J. , Kern , W.: A robust PTAS for maximum weight independent set in unit disk graphs . In: Proc. 30th Int. Workshop Graph-Theoretic Concepts in Comput. Sci. Lecture Notes Comput. Sci. , vol. 3353 , pp. 214 - 221 ( 2005 )
37. Pyrga , E. , Ray , S. : New existence proofs for ε-nets . In: Proc. 24th ACM Sympos. Comput. Geom. , pp. 199 - 207 ( 2008 )
38. Smith , W.D. , Wormald , N.C. : Geometric separator theorems and applications . In: Proc. 39th Annu. IEEE Sympos. Found. Comput. Sci. , pp. 232 - 243 ( 1998 )
39. Varadarajan , K. : Weighted geometric set cover via quasi-uniform sampling . In: Proc. 42nd Annu. ACM Sympos. Theory Comput. , pp. 641 - 648 ( 2010 )
40. Whitesides , S. , Zhao , R. : k-admissible collections of Jordan curves and offsets of circular arc figures . Technical Report SOCS 90 .08, McGill Univ. , Montreal, Quebec ( 1990 )