Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning

Given a set system (X,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X, \mathcal {R})$$\end{document} such that every pair of sets in R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}$$\end{document} have large symmetric difference, the Shallow Packing Lemma gives an upper bound on |R|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\mathcal {R}|$$\end{document} as a function of the shallow-cell complexity of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}$$\end{document}. In this paper, we first present a matching lower bound. Then we give our main theorem, an application of the Shallow Packing Lemma: given a semialgebraic set system (X,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X, \mathcal {R})$$\end{document} with shallow-cell complexity φ(·,·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (\cdot , \cdot )$$\end{document} and a parameter ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon > 0$$\end{document}, there exists a collection, called an ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document}-Mnet, consisting of O(1ϵφ(O(1ϵ),O(1)))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O\bigl ( \frac{1}{\epsilon } \,\varphi \bigl ( O\bigl (\frac{1}{\epsilon } \bigr ), O(1)\bigr ) \bigr )$$\end{document} subsets of X, each of size Ω(ϵ|X|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ( \epsilon |X| )$$\end{document}, such that any R∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R \in \mathcal {R}$$\end{document} with |R|≥ϵ|X|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|R| \ge \epsilon |X|$$\end{document} contains at least one set in this collection. We observe that as an immediate corollary an alternate proof of the optimal ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document}-net bound follows.

Keywords Epsilon-nets · Haussler's Packing Lemma · Mnets · Shallow-cell complexity · Shallow Packing Lemma 1 Introduction Given a set system (X , R) consisting of a set of base elements X together with a set R of subsets of X , an influential way to capture the complexity of R has been through the concept of VC dimension. First define the projection of R onto any Y ⊆ X to be the system The VC dimension of (X , R), henceforth denoted by VC-dim(R), is the size of any largest subset Y ⊆ X for which |R| Y | = 2 |Y | ; such a set Y is said to be shattered by R.
The usefulness of VC dimension derives from the fact that it is bounded for most natural geometric set systems. For example, consider the case when X is a set of points in R d and the sets in R are defined by containment by half-spaces, namely It is not hard to see that the VC dimension of this set system is d + 1.
For nearly all results on set systems with bounded VC dimension, the key technical property required is the following consequence of bounded VC dimension [29,30]: if (X , R) is a set system with VC-dim(R) = d, then for any Y ⊆ X we have |R| Y | ≤ d i=0 n i ≤ en d d . This is sometimes called the Primal Shatter Lemma.
Most commonly studied set systems derived from geometric configurations can be categorized into two types. When X is a set of points and the sets in R are defined by containment by members of a family of geometric objects O, we say that (X , R) is a primal set system induced by O on X . The second type is when the base set X is a finite subset of O, and R is defined as Then we say that (X , R) is the dual set system induced by O.
Shallow-cell complexity. While most set systems derived from geometry have bounded VC dimension and thus satisfy the Primal Shatter Lemma, they often satisfy a finer property: not only is the size of R| Y polynomially bounded, but for any positive integer r , the number of sets in R| Y of size at most r is bounded by a smaller function. For any positive integer r , define R| Y ,≤r = {R ∈ R| Y : |R| ≤ r }.
For example, let X be a set of n points in R 3 , and R the primal set system induced by half-spaces. Then for any set Y ⊆ X the number of sets in R| Y of size at most r (that is, |R| Y ,≤r |) is O(|Y | · r 2 ) [12]. When r = o(n), this contrasts sharply with the total size of R| Y , which is (|Y | 3 ).
This has motivated the following finer classification of set systems.
Packings. A set system (X , R) is said to be a δ-packing if for every distinct R, S ∈ R, Haussler's proof was later simplified by Chazelle [10], and is an elegant application of the probabilistic method. It has since been applied to several areas ranging from computational geometry and machine learning to Bayesian inference (see [18,20,21]). Haussler [18] also showed that this bound is tight: given positive integers d, n and δ, there exists a δ-packing (X , R) such that |X | = n, VC-dim(R) ≤ d and |R| ≥ n 2e(δ+d) d .
Recent efforts have been devoted to extending the Packing Lemma to these finer classifications of set systems. For k, δ ∈ N + , call (X , R) a k-shallow δ-packing if R is a δ-packing and |S| ≤ k for each S ∈ R. After some earlier bounds [13,14,26], the following lemma was established in [25]. Lemma 1.2 (Shallow Packing Lemma) Let (X , R) be a set system on n elements with shallow-cell complexity ϕ R , and let d 0 , k, δ be positive integers. If VC-dim(R) ≤ d 0 and (X , R) is a k-shallow δ-packing, then Mnets. Given a set system (X , R) and a parameter > 0, an -Mnet is a collection M of subsets of X , each of size ( · |X |), such that any set in R of size at least · |X | completely contains at least one set of M. The following definition formalizes this notion. Definition 1.3 ( -Mnets) Given a set system (X , R) on n elements and parameters and λ in (0, 1), a collection M = {M 1 , . . . , M l } of subsets of X is a λ-heavy -Mnet of size l for R if (i) |M i | ≥ λ n for each i ∈ {1, . . . , l}, and (ii) for any R ∈ R with |R| ≥ n, there exists an index j ∈ {1, . . . , l} such that M j ⊆ R.

Previous Work
Mnets (short for combinatorial Macbeath regions) were introduced by Mustafa and Ray [26] as the combinatorial analogue of Macbeath regions [5,6] for set systems.
Using several different techniques, they gave the following upper bounds on the size of -Mnets.

Theorem 1.4
There exists an absolute constant λ > 0 such that there exist λ-heavy -Mnets of size

Our Contributions
In this paper we present progress on several aspects of shallow packings (the precise statements and their proofs are presented in the indicated sections): Section 2: We show that the Shallow Packing Lemma is tight, up to a constant factor, for the most common case of shallow-cell complexity: Section 3: We restrict ourselves to semialgebraic set systems (a natural property defined in Sect. 3). For such set systems with shallow-cell complexity ϕ(·, ·), we show the existence of -Mnets of size . The proof also implies an efficient algorithm to construct such a collection. Section 4: We give several implications of our -Mnet theorem for geometric set systems. In particular, we obtain -Mnets for a larger class of geometric set systems than previously known, and also improve some of the earlier bounds of [26]. Our -Mnet theorem also gives another proof of the optimal bound on -nets as a function of the shallow-cell complexity of the set system, resolving the main open question of Mustafa and Ray [26]. Finally, we show that substantially improving our -Mnet bound is not possible. Appendix A: We prove a generalization of the Packing Lemma that implies several previous statements related to the Packing Lemma.

Optimality of the Shallow Packing Lemma
While Haussler [18] gave a matching lower bound to his Packing Lemma, the optimality of the Shallow Packing Lemma (Lemma 1.2) was an open question in previous work [13,14,25,26]. We show that it is tight, up to a constant factor, for the case where Proof We build a set system with the desired shallow-cell complexity and then show that it contains a large shallow packing. Without loss of generality we assume that n is an integer multiple of d. The ground set X will be a subset of N × N.
For each 1 ≤ i ≤ d 1 , set X i = {i} × 1, . . . , n d . This can be viewed as simply considering d 1 disjoint copies of 1, . . . , n d . Define the following set system P i on each X i : Intuitively, consider a balanced binary tree T i on X i , with its leaves labeled (i, 1), (i, 2), . . . , i, n d . For each node v ∈ T i , the family P i contains a set consisting of the leaves of the subtree rooted at v. Here α is the height of the node (so 2 α is the number of elements in the corresponding subset), while β identifies one of the nodes of that height (among the 2 log( n d )−α = 2 −α · n d choices). See also Fig. 1.
Proof For any Y ⊆ X i , the sets in P i | Y are in a one-to-one correspondence with the nodes of T i whose left and right subtrees, if they exist, both contain leaves labeled by Y . It is easy to see that if the nodes of T i corresponding to Y form a connected sub-tree, then these nodes define a new binary tree whose leaves are still labeled by Y , and thus their number is at most 2|Y | − 1. Otherwise, the statement holds by induction on the number of connected components of Y in T i .
which can be seen as prefix sets of the sequence (i, 1), . . . , i, n d .
Finally, the required base set will be The set system R 0 is defined on X by taking d-wise unions of the sets in P i 's and Q i 's: We first bound the shallow-cell complexity of R 0 , and then construct a subset of R 0 which forms a large packing.
Thus by Claims 2.2 and 2.3, we have the required bound: It remains to show that some subset of R 0 is a large k-shallow δ-packing. For the given parameters k, δ and for all 1 ≤ i ≤ d 1 and The intuition here is that we pick only the nodes in our binary trees T i which have height at least log 2 δ (and thus a symmetric difference of at least δ elements).
Similarly in Q j we only pick every δ-th set. All these sets have size at most k d . This is straightforward for Q (k,δ) i ; on the other hand, a set in P (k,δ) i defined by the pair (α, β) has size 2 α ≤ k d . All those sets also are integer intervals of the form {μδ + 1, . . . , νδ} for some μ, ν ∈ N and thus pairwise δ-separated (for the P (k,δ) i , notice that 2 α is a multiple of δ). Hence is a δ-packing which is k-shallow. We bound its size:

Remark 2.5
This lower-bound proof is constructive; the packing that we built can be realized in a number of simple ways, for example with points on a square grid and sets induced by some specific (2d)-gons, i.e., a semialgebraic set system with constant description complexity.

Mnets for Semialgebraic Set Systems
Building on the work of Mustafa and Ray [26], we present a general upper bound on the size of the smallest -Mnets for semialgebraic set systems. The proof uses two new ingredients: the Shallow Packing Lemma (Lemma 1.2) and the polynomial partitioning method of Guth and Katz [17], specifically a multi-level refinement due to Matoušek and Patáková [23].
Semialgebraic sets are subsets of R d obtained by taking Boolean operations such as unions, intersections, and complements of sets of the form {x ∈ R d : Denote by d, ,s the family of all semialgebraic sets in R d obtained by taking Boolean operations on at most s polynomial inequalities, each of degree at most . In this paper d, and s are all regarded as constants and therefore the sets in d, ,s have constant description complexity (see [7]). For a set X of points in R d and a set system R on X , we say that (X , R) is a semialgebraic set system generated by d, ,s if for each S ∈ R there exists a γ ∈ d, ,s such that S = X ∩ γ .
The constant in the asymptotic notation depends on d, , and s. In particular, if ϕ R (·, ·) is a non-decreasing function in the first argument, then (X , R) has a λ d, ,s -heavy -Mnet of size where the constant in the asymptotic notation depends on d, , and s.
Before presenting the proof of Theorem 3.1, we first give a brief overview of a technical tool that is used in the proof.

Preliminaries: Polynomial Partitioning
For two subsets γ and ω of R d , we say that γ crosses In other words, the set P can be partitioned into a small number of parts defined by a set S k of semialgebraic regions, such that every set γ ∈ d, ,s either contains, or is disjoint from, "most" of these regions. Theorem 3.2 extends the Guth-Katz [17] polynomial partitioning theorem, a partition of R d by an algebraic variety which is balanced with respect to the set P. Here partitioning is applied not once but recursively on varieties of decreasing dimension. This allows us to dispense with assumptions of genericity.

Proof of Theorem 3.1
Proof We will prove the existence of a λ-heavy -Mnet, where we set with the same constants C 1 , C 2 , C 3 and C d, ,s as in Theorem 3.2.
If ≤ 8(16·d·C d, ,s ) C 1 d n , then the collection of singleton sets {{ p} : p ∈ X } is an -Mnet for (X , R) of size n, which is at most 8(16 · d · C d, ,s ) C 1 d · 1 . Furthermore, it is λ-heavy since each set has size 1, which is at least n 8(16·d·C d, ,s ) C 1 d ≥ λ n. Therefore we may restrict ourselves to the case when For i = 0, . . . , log 1 , let R i ⊆ R be a maximum-cardinality 2 i−1 n -packing, with the additional constraint that each set in R i has cardinality in [2 i n, 2 i+1 n). From the Shallow Packing Lemma, we have For every such index i let m i = |R i |, and say R i = {R i1 , . . . , R im i }. For each j = 1, . . . , m i , consider the multilevel polynomial partitioning of R i j as given by Theorem 3.2, for a parameter r (independent of i and j) to be fixed later. We write It remains to show that M is the required λ-heavy -Mnet for an appropriate value of r . Namely, (i) the promised upper bound on |M| holds, (ii) each set in M has size at least λ n, and (iii) for any R ∈ R with |R| ≥ n, there exists a set M ∈ M such that M ⊆ R.
We set r = (16dC d, ,s ) d . Inequality (2) implies that To see (i), observe that Thus we have To see (ii), observe that each set added to M satisfies by (1). To see (iii), let R ∈ R be any set such that |R| ≥ n, and let i ∈ 1, . . . , log 1 be the index such that |R| ∈ [2 i n, 2 i+1 n). Since R i is a maximal (2 i−1 n)-packing, there exists an index j such that R i j ∈ R i and |R R i j | ≤ 2 i−1 n. Using the fact that |R i j | ∈ [2 i n, 2 i+1 n), we get It now suffices to show that R ∩ R i j must contain a set i jkl such that i jkl ∈ M i , i.e., such that | i jkl | ≥ 2 i n 8C 3 dr C 2 . Assume otherwise. Consider the contribution of the sets i jkl to R ∩ R i j , Then we have (a) The total number of points contained in R ∩ R i j from all sets i jkl such that | i jkl | < 2 i n 8C 3 dr C 2 , summed over all indices k and l, is at most Thus we have This contradicts inequality (5), and completes the proof.

Remark 3.3
The main open question in [26] was the following interesting pattern: set systems that had -nets of size O 1 log f 1 , for some function f : R + → R + , had -Mnets of size O 1 f 1 . Theorem 3.1 provides the explanation-it turns out that f 1 is in fact the function ϕ R O 1 , O(1) (the constant in the asymptotic notation depends on d), and it is this function that dictates the bounds on both -nets and -Mnets.

Corollaries of Theorem 3.1
Nets. First we show that the -net theorem of Chan et al. [9] follows immediately, though for the special case of unweighted semialgebraic set systems.

Remark 4.2
The constant in the asymptotic notation of the -net size above depends linearly on 1/λ d, ,s .
Observe that the primal set system (X O , S W ) where sets of S W are induced by semialgebraic regions in O W is by construction the same as (X , R). This construction shows that dual set system of semialgebraic regions can be represented as primal set system of semialgebraic regions. The shallow-cell complexity of the dual set system induced by objects with union complexity κ(·) is ϕ(m, r ) = O κ(m) m · r 2 (e.g., see [27]) which together with Theorem 3.1 implies the stated bound. The remaining bounds follow from the facts that κ(m) for fat triangles with approximately same size [24] is O(m), for α-fat triangles [15] is O(m log * m) (where the constant of proportionality depends only on α), and for locally γ -fat objects [1] is O(m2 log * m ), where the constant of proportionality in the linear term depends only on γ . 2. Let (X , R) be the set system induced on a set X of n points in R 2 by the family of axis-parallel rectangles. Aronov et al. [2] showed that there exists another set system R on X with ϕ R (m, r ) = O(r · log m), such that for any R ∈ R, there exists an R ∈ R such that R ⊆ R and |R | ≥ |R| 2 . Thus a 2 -Mnet for R is an , and the primal set system bound for hyperplanes follows. The bound for the dual set system then follows by point-hyperplane duality. [26] for lines, strips and cones in the plane were weaker by polylogarithmic factors: they were O 1 2 log 2 1 , O 1 2 log 3 1 2 and O 1 2 log 4 1 respectively.

Remark 4.5 Earlier bounds
Mustafa and Ray [26] already gave a number of non-linear lower bounds (in terms of 1 ), such as 1 2 for primal systems induced by points and lines in the plane and − (d+1)/3 for 1 2 -heavy -Mnets for primal set systems induced by points and half-spaces in R d . We give a tight lower bound for the set system induced by points and hyperplanes in R d .
The standard Veronese lifting [22] maps P and C to a set P of n points in R d and a family of hyperplanes H in R d satisfying the following conditions: This implies that any (2λ)-heavy -Mnet for the primal set system defined by H on P must have at least |H| = 1 d sets.

Conclusion
We conclude with some remarks and open questions.
Lower bound for the Shallow Packing Lemma. The lower bound construction given in the proof of Theorem 2.1, showing the optimality of the Shallow Packing Lemma (Lemma 1.2), is constructive. Also observe that it can be realized in a number of simple ways, for example with points on a square grid and sets induced by some specific (2d)gons, i.e., a semialgebraic set system with constant description complexity.
Closing the gap between lower and upper bounds. Except for a few cases, there are gaps between lower and upper bounds on the size of Mnets for various primal and dual set systems.
Applications of Mnets. Corollary 4.1 shows that the existence of small -nets follows immediately from the more general structure of Mnets. Macbeath regions for convex bodies have recently found algorithmic applications such as volume estimation of convex bodies [3,4]. We believe that Mnets will also find important applications and connections to various aspects of set systems with bounded VC dimension.
Computing Mnets. In the real RAM model of computation one can compute exactly with arbitrary real numbers and each arithmetic operation takes unit time. Matoušek and Patáková [23] gave an algorithmic counterpart of Theorem 3.2, showing that the sets * , i j , S i j from Theorem 3.2 can be computed in time O(nr C ) in the real RAM model, where C is a constant depending only on d. Using this result and the construction in the proof of Theorem 3.1, we obtain a randomized algorithm with time complexity poly n, 1 Combining Claim A.4 and the above lower bound on E [W ], we get This implies |R| ≤ 2l · E [|R| A |].

Proof of Theorem A.1 Let
A ⊆ X be a random sample of size s := 8l(l−1)dn where we used the fact that |(R\R 1 )| A | ≤ |A|·ϕ(|A|, t), where t = max S∈R\R 1 |S| < 4l ks n . Now the bound follows from Lemma A.3.
Finally we give the proof of Claim A.5.

Proof of Claim A.5
Consider a set Q ∈ R| A , and let R Q be the sets of R whose projection is Q. Once the choice of a has been made, Q will be split into two sets, those sets containing that choice of a -say there are b 1 of these, and those sets not containing a, say a number b 2 . From the definition of weights, the expected contribution of sets of R Q to edge weight will be E [min{b 1 , b 2 }] ≥ E[b 1 b 2 ] b 1 +b 2 . The above inequality follows from the fact min{b 1 , b 2 } ≥ b 1 b 2 b 1 +b 2 . Observe that b 1 b 2 is the number of ordered pairs (S 1 , S 2 ) ∈ R Q × R Q with a ∈ S 1 and a / ∈ S 2 . Therefore for each fixed pair of sets (S 1 , S 2 ) ∈ R Q ×R Q , the probability that the randomly chosen last element a ∈ S 1 \S 2 is |S 1 \S 2 | n−s−1 . Therefore the contribution of (S 1 , S 2 ) in R Q to b 1 b 2 is |S 1 \S 2 | n−s−1 . Noting that b = b 1 + b 2 is fixed independent of the choice of a, summing up over all pairs of sets in R Q , we get the expected contribution of the sets in R Q to the edge weight to be at least For all l sets S 1 , . . . , S l ∈ R Q , we have 2≤ j≤l S 1 S j = (S 1 ∪ · · · ∪ S l ) \ (S 1 ∩ · · · ∩ S l ) .
So for every l tuple there exists one pair (S 1 , S j ) with |S 1 S j | ≥ δ l−1 . Define the graph G R Q := (R Q , E Q ), where {S 1 , S 2 } ∈ E if |S 1 S 2 | ≥ δ l−1 . As R Q is an l-wise δ-packing we do not have independent sets of size l in G R Q . From Turán's theorem, see [28], we have 2l . Therefore The last inequality follows from the facts |E Q | ≥ b(b − l) 2l and |R Q | = b.
Summing up over all sets of R| A ,