Hypergraphs and VC-dimension.
A hypergraph is a pair where is a set of vertices and is the set of hyperedges of . When is finite, is a finite hypergraph.
A subset is shattered if all its subsets are realized by , meaning . The VC-dimension of , denoted by , is the cardinality of a largest shattered subset of or if arbitrarily large subsets are shattered (which does not happen in finite hypergraphs). This parameter plays a central role in statistical learning, computational geometry, and other areas of computer science and combinatorics [VC71, MATOUSEK, MV17].
Let . An -net for a finite hypergraph is a subset of vertices such that for every hyperedge such that .
Haussler and Welzl [HW87] proved that finite hypergraphs with VC-dimension admit -nets of size , later improved to [KomPaW92]. In the last three decades,
-nets have found applications in diverse areas of computer science, including machine learning[BEHW89], algorithms [Chan18], computational geometry [AFM12] and social choice [ABKKW].
Mustafa and Ray introduced the notion of Mnets [MR17]. For a hypergraph and for a fixed , an -Mnet is a family such that each , each is of size , and, for each such that , for some . They constructed small -Mnets (i.e., such families with small ) for several classes of geometric hypergraphs. These results were extended by Dutta et al. [DGJM19] using polynomial partitioning.
Although Hausssler and Welzl’s proof of the -net theorem is probabilistic, several deterministic constructions of -nets for hypergraphs with finite VC-dimension have been devised [BCM99, Ma95, MC96]. The best result of this kind is Brönniman, Chazelle and Matoušek’s -time algorithm for computing an -net of size [BCM99]. These constructions are used to derandomize applications of
-nets, such as low-dimensional linear programming[Chan18].
In scenarios where the VC-dimension is , the running time of these constructions becomes exponential in . For one such scenario – the hypergraph induced by half-spaces on the discrete cube – Rabani and Shpilka [RS08] presented an efficient explicit construction of an -net, alas of sub-optimal size: for some universal constants , whereas can be obtained by random sampling. Like the aforementioned explicit constructions, the construction of [RS08] is based on derandomization.
1.2 Our problem
We denote by the set of all subsets of cardinality (or “-subsets”) of the set .
Let be a finite hypergraph, a positive integer and . A family of -subsets of is an --net for if for every with there is an such that .
As mentioned already, for this is equivalent to the -net notion, and for this corresponds to the notion of -Mnets. In this paper we study the following problem.
How small are the smallest --nets for ? Can we compute them efficiently?
Instances of the --net problem appear naturally in various contexts in computer science and combinatorics. For example, the following is a basic motivating example for secret sharing [Liu68, Shamir79]: “Eleven scientists are working on a secret project. They wish to lock up the documents in a cabinet so that the cabinet can be opened if and only if six or more of the scientists are present. What is the smallest number of locks needed?”. Consider a variant of this question in which the number of scientists is large. We still insist on the basic security condition – that no less than six scientists can open the cabinet. On the other hand, due to the large number of scientists, we do not require that any six should be able to do so, but rather any sufficiently large group of a certain kind, e.g., at least one tenth of all scientists including a representative of each university involved.
The classical secret sharing methods (see, e.g., [Beimel11]) distribute “keys” to subsets of 6 scientists so that any six scientists will be able to open the cabinet but no five will be able to do that. But as we require only certain groups of scientists to be able to open it, it is possible to distribute shared keys to only some of the 6-subsets. The questions: “What is the minimal number of 6-subsets we can achieve? and how can we choose the 6-subsets of scientists we distribute keys to?” are an instance of the --net problem – with , , and the hyperedges of the hypergraph being all groups of scientists that are required to be able to open the cabinet.
Other contexts in which the --net problem appears (described in sec:applications) include the Turán numbers of hypergraphs, -boundedness of graphs, edge-coloring of hypergraphs and more.
Related work: -Nets and Mnets.
For any , the minimum size of an --net is sandwiched between the corresponding minimum sizes of -nets and of Mnets. Indeed, given an Mnet, one obtains an --net by picking one -subset from each subset, and given an --net, one obtains an -net by taking one vertex from each -subset. The survey [MV17] has most known bounds on these objects.
Notation: we write when the implicit constants depend on parameters and
Hypergraphs of finite VC-dimension have small --nets.
Our main result is an existence result for small --nets.
theoremmainlarget For every and , every hypergraph on vertices with VC-dimension and dual shatter function admits an --net of size , all elements of which are pairwise disjoint. Here .
(The dual shatter function, described in sec:tuple, is a property of the hypergraph such that we may always take .)
This bound is asymptotically tight when , in the sense that there exist hypergraphs for which any -net, and consequently also any --net, is of size [KomPaW92]. The proof of Theorem 1.3 involves a surprising relation between the --net problem and the existence of spanning trees with a low crossing number, proved by Welzl in 1988 [Welzl88].
Hypergraphs with VC-dimension 1 admit -sized -nets [KomPaW92] and -Mnets [DGJM19]. The latter fact yields the following result, albeit with worse constants. We offer a simple proof.
theoremvcone For every positive integer and , every finite hypergraph on vertices with VC-dimension 1 admits an --net of size at most .
An efficient explicit construction of --nets.
Our second result is a new explicit construction of --nets, for all . The case of (i.e., -nets) is of independent interest, as in this case our construction does not follow the proof strategy of Haussler and Welzl and does not use derandomization (unlike all previously known explicit constructions of -nets). On the other hand, it has a sub-optimal size of , where is the VC-dimension of the underlying hypergraph.
For a higher , we introduce a new parameter of the hypergraph, which we call the -VC-dimension. For hypergraphs of -VC-dimension , we construct --nets of size . We give some first results on the relation between this new parameter and the standard VC-dimension.
Small -2-nets for geometric hypergraphs.
In view of thm:main_ht¿2, which shows that for hypergraphs with a constant VC dimension one can obtain an --net of roughly the same size as the smallest -net, it is natural to ask whether a similar result can be achieved for geometrically-defined hypergraphs that admit an -net of size . We obtain such results for several geometrically-defined hypergraphs in , including the intersection hypergraph of two families of pseudo-disks and the dual hypergraph of a family of regions with linear union complexity. Namely, we show that these hypergraphs have -sized -2-nets provided they have vertices. Interestingly, in some scenarios the minimum size of an -2-net is sensitive to the exact multiplicative constant: there are subhypergraphs on vertices for which any -2-net is of size .
2 Construction of Auxiliary Hypergraphs
2.1 Some preparatory results
Given a hypergraph the trace (also known as projection or restriction) of on is ; shattered subsets are those for which . The shatter function of is
It is bounded by the Sauer–Shelah lemma:
[[VC71, Sau72, She72]] If then . In particular, for one has , where is Euler’s number.
Binary entropy function.
This is . (All logarithms are binary. See fig:entropy.) We will use the following inequality.
For the sake of simplicity, we assume . By the binomial theorem,
& = ∑_i=0^n ni α^i (1-α)^n-i
& ≥∑_i=0^αn ni α^i (1-α)^n-i= ∑_i=0^αn ni (1-α)^n (α1-α)^i
&≥∑_i=0^αn ni (1-α)^n (α1-α)^αn since
&= 2^-n ⋅h(α) ⋅∑_i=0^αn ni.
The binary entropy function restricted to is invertible, and [Calabro, Th. 2.2]:
2.2 A first hypergraph on -subsets
Given a hypergraph and a positive integer , let be the hypergraph where and . That is, its vertices are all -element subsets of and each hyperedge of consists of all such subsets contained in a given hyperedge of .
For , let . Note that .
If is a hypergraph with then .
We assume that , as for , .
To prove the left inequality, let be a shattered subset of vertices in , with . There are sets containing all vertices in and exactly one in . It is easy to see that they form a shattered subset in .
For the right inequality, suppose to the contrary that is a shattered set in with . Let ; clearly . Observe also that . If this were not the case there would exist some such that , which would contradict the fact that is shattered.
We denote ; we have .
Since is shattered in , each is of the form for some . Thus .
On the other hand, , and so by lm:sauer, . It follows from cl:vc_pairs (with ) that . We show that , a contradiction.
Note that . Since is monotone decreasing in the range , we have . As is increasing on , it follows that .
thm:vc_tuples allows us to slightly improve the “trivial” upper bound of on the minimum size of an --net for any hypergraph with constant VC-dimension. Let be a hypergraph on vertices with VC-dimension . For any such that , admits an --net of size .
Indeed, observe that an -net for is an --net for , and apply the classical -net theorem to .
2.3 A smaller, well-behaved hypergraph on -subsets
A spanning cycle for is a cycle graph on that visits all vertices (exactly once). For , let be the number of edges of with one endpoint in and the other in . The crossing number of with respect to is .
The dual hypergraph of is , where consists of all hyperedges for . Its shatter function is the dual shatter function of , and is denoted by .
If then [Assouad], and hence for every positive , where is a constant depending on . In particular, any hypergraph with finite VC-dimension satisfies the hypotheses of the following theorem.
[[Welzl88, Lemma 3.3 and Theorem 4.2]] Let be a hypergraph on vertices such that for some constants and . Then there exists another constant (depending on and ) and a spanning cycle for with crossing number .
(An additional factor in Welzl’s original result was later removed [Ma99, Sec. 5.4]. Up to constant factors, this theorem is equivalent to the same result for paths or trees.)
Let be a finite hypergraph with . Let be a spanning cycle for whose crossing number is minimal (and thus ). Fix an arbitrary starting point and orientation of . For , let be the -th vertex along . Let (where the subscript stands for low crossing). Observe that its elements are pairwise disjoint. Let be the hypergraph on whose hyperedges are of the form for each .
In order to make uniquely defined, is chosen arbitrarily from all suitable spanning cycles. As is a subhypergraph of , , and thus we also have .
3 Existence of Small --Nets
For , this is simply the -net theorem. For higher , let be such a hypergraph and . Consider the hypergraph defined in sec:tuple. It has vertices and VC-dimension (by Remark 2.3), and thus admits an -net of size . We claim that any such -net is also an --net for .
Indeed, the crossing number of the associated spanning cycle is . Every hyperedge of with fully contains at least elements of , which is as soon as , or equivalenty (noting also that for ) when . One of these -subsets is in .
In general, some fast growth of as a function of is necessary. For example, given any such that , the complete -uniform hypergraph on vertices does not have any --net with fewer than elements. Moreover, there exist geometrically-defined hypergraphs that do not admit --nets of size (see Figure 2 and subsection 5.3.2). On the other hand, in sec:geom we show that certain classes of geometrically-defined hypergraphs have “small” --nets even for “small” values of .
Small -nets, small --nets.
A natural question arising from thm:main_ht¿2 is whether any hypergraph that admits small -nets must also admit --nets of approximately same size. In general, the answer is negative. Take for example a hypergraph whose smallest -net is of size (see [KomPaW92], [PachTa13]), and augment it by adding a vertex that belongs to all hyperedges. Clearly, this second hypergraph has the same VC-dimension and a one-element -net, but any --net is of size .
However, this example is quite artificial. In “natural” scenarios (and for sufficiently large vertex sets) the smallest -nets and --nets might still have approximately same size. In sec:geom we show that this is the case for some geometrically-defined hypergraphs.
Another scenario in which there exist both an -net and an --net of size is when the VC-dimension of the hypergraph is 1. In this case, the existence of an -net of size was proved in [KomPaW92]. The next theorem could be derived from results on Mnets [DGJM19], at the cost of poor multiplicative constants. Here we give a simpler proof for it. * Let be such a hypergraph and . Without loss of generality, . For , there exists an -net that hits each at least times, and . To see this let be an -net of size [KomPaW92]. In the hypergraph induced on the hyperedges hit only times by have cardinality , while the number of vertices is , for a ratio . Take an -net of size for this hypergraph and let . Finally, let the desired --net consist of one -subset from each element of with vertices, of which there are at most by lm:sauer.
4 Deterministic Construction of --Nets
Let be a finite hypergraph with VC-dimension , and fix . In this section we provide an explicit polynomial-time construction of -nets that immediately implies an explicit construction of --nets. The size is far from optimal, but the construction is simpler than previous explicit constructions, as it does not rely on packing numbers nor on pseudo-random choices.
4.1 Deterministic construction of -nets
We start with the following definition:
Let be two subsets of . We say that stabs if for every hyperedge with we have .
Let be a hyperedge, , and let . Since the VC-dimension is the set is not shattered. Notice that . We can also assume that , for otherwise is a transversal for of size . Hence there exists at least one non-trivial, proper subset such that . Equivalently, there is a non-trivial partition of into and such that stabs . We say that is of type . Note that could have several types. By the pigeonhole principle, there is a type and a subset such that a fraction of the elements of are stabbed by , hence the following lemma holds: Let be a hyperedge containing vertices of . Then there exists an integer and a subset that stabs subsets of cardinality .
Put . We construct an -net of size as follows. Start with . As long as there is a hyperedge with and , stabbing asserts that some -subset from stabs subsets of with cardinality for an appropriate . Add all elements of this subset to ; we call this a type iteration.
The resulting set is an -net by construction. It is left to show that . As each step of the construction adds at most vertices to it is enough to bound the number of iterations . By the pigeonhole principle, at least of the iterations have the same type, say . After a type- iteration stabs an additional subsets of cardinality none of which were previously stabbed. Since there are subsets of cardinality we have .
We analyze the running time of the above algorithm. We assume that for the algorithm we have a data structure which is the incidence matrix of the hypergraph . Without loss of generality, each hyperedge of may be replaced with a subset of cardinality . This can be done in time due to the fact that .
We consider each . Firstly we check if there is a hyperedge which contains , if not, we continue to the next subset. If yes, we consider each of the proper subsets of . Let be such a subset. We check if is stabbed by . We can do it by going over all hyperedges of . Hence, in total this pre-processing step takes running time. While determining the type of any -subset of and scanning all the hyperedges of the hypergraph, we maintain for any -subset , a list of all the -subsets of that stabs and their number.
Consider some iteration of the algorithm and let be such that and where is the collection of elements found until this iteration. We find a subset of size at most which stabs the most subsets of size .
The running time of each iteration is . Hence in total the running time of the algorithm after the pre-processing step is . Hence the total running of the algorithm described in the previous section is .
Immediate applications to --nets
The construction of -nets in subsec:eps_net gives two straightforward constructions of --nets.
Trivial construction. Use the above algorithm to explicitly construct disjoint -nets of size , and take all -subsets of elements in their union that contain one element from each net. The resulting --net is of size .
Construction via . Use the above algorithm to explicitly construct an -net for the hypergraph , which is an --net for (as was shown in the proof of Theorem 1.3). The resulting --net is of size . (The cycle with a low crossing number required for constructing the hypergraph can be found in polynomial time [Welzl88, Ma99]).
4.2 Deterministic construction of --nets
We present a direct construction of --nets without passing through -nets. For the sake of convenience, we start by for presenting the method for .
The following definition extends the classical notion of VC-dimension.
Let be a positive integer. Also let be a hypergraph, and such that . We say that is -realized by (with respect to ) if for some such that . We say that is -shattered by if every is -realized by (with respect to ). The -VC-dimension of , denoted by , is the maximal size of a vertex set that is -shattered by .
Note that the -VC-dimension is the standard VC-dimension. Moreover, the -VC-dimension is at most the -VC-dimension for any positive integer . We use the following adaptation of def:stab:
Let be a hypergraph. Given two vertex sets , we say that 2-stabs if each hyperedge of that contains also contains at least two vertices from .
For a hypergraph with 2-VC-dimension , one can construct explicitly an --net of size .
Let be a hyperedge and let . Since the 2-VC-dimension is the set is not 2-shattered. Notice that and so and all elements of are 2-realized by with respect to . For our purpose, we can also assume that is 2-realized by (with respect to ), for otherwise is a transversal for of size . This means that there is a partition, say , such that 2-stabs . Let . Note that . We say that is a type partition. We need the following lemma, whose proof is similar to that of stabbing. Let be a hyperedge containing vertices of . Then there exists an integer and a subset with cardinality that 2-stabs subsets of cardinality .
Let be as above and let be fixed. Put . We construct an --net of size as follows. We start with a set . As long as there is a hyperedge with that does not contain any pair , for an appropriate we take an -subset 2-stabbing subsets of with cardinality , and add to all elements of . We call this a type iteration. This is possible by 2stabbing.
The resulting set is an --net by construction. It is left to show that . In each step of the construction we add at most pairs to so it is enough to bound the number of iterations . By the pigeonhole principle, at least of the iterations have the same type, say . There are subsets of cardinality , and in each of the at least type iterations we -stab at least additional subsets of cardinality , so we have so (since ). This completes the proof of thm:direct.
The only significant difference between the constructions of subsec:eps_net and of subsec:construction is the factor that depends on the size of the resulting net. Hence, the complexity of the algorithm in this section is bounded by , where is the -VC-dimension of .
4.2.1 Extension of the Direct Construction of --Nets to --Nets
Now we show how our deterministic construction of -2-nets can be extended to --nets for other values of . The following argument is a direct adaptation of the argument from subsec:construction.
Let be a hypergraph. Given two disjoint vertex sets , we say that -stabs if each hyperedge that contains must contain at least vertices from .
Let be a hypergraph with -VC-dimension . Then one can construct explicitly an --net for of size .
Let be a hyperedge and let be a subset of with cardinality . Since the -VC-dimension is then the set cannot be -shattered. Notice that and so each with is -realized by . For our purpose, we can also assume that is -realized by (with respect to ), for otherwise the set of all -subsets in is a transversal for of size . This means that there exists a subset of size between and that is not -realized by (with respect to ). Equivalently, there is a partition, say such that -stabs . Let . Note that . We say that is a type partition. Note that there could be more than one type partition for the same set . We need the following lemma: Let be a hyperedge containing vertices of . Then there exists an integer and a subset with cardinality that -stabs subsets of cardinality .
For each subset in there exists a partition with one of the above stabbing types. By the pigeonhole principle at least of these subsets have the same type, say . Each such subset of type is charged by a -piercing subset of cardinality . Then by the pigeonhole principle there is a subset of cardinality that is charged at least times. This means that 2-stabs subsets of cardinality , as asserted.
Let be as above and let be fixed. Put . We construct an --net of size as follows. We start with a set . As long as there is a hyperedge with that does not include any -subset of , for an appropriate we take an -subset -stabbing subsets of with cardinality , and add to all elements of . We call this a type iteration. This is possible by 2stabbingt.
Obviously the resulting set is an --net by construction. It is left to show that . In each step of the construction we add at most subsets to so it is enough to bound the number of iterations. Denote this number by . By the pigeonhole principle, at least of the iterations have the same type, say . There are subsets of cardinality and in each of the at least type iterations we -stab at least additional subsets of cardinality . We have that so (since ). This completes the proof.
4.3 -VC-dimension versus classical VC-dimension
What can be said about the relation between VC-dimension and our newly introduced -VC-dimension, for ? By definition, . As shown below ideas from Dudley’s unpublished lecture notes [Dudley99, Th. 4.37] yield . This is sharp for some small hypergraphs, such as that with vertex set and hyperedges , , , and , which has VC-dimension 1 but 2-VC-dimension 3.
Let be a hypergraph then . Assume that be -shattered. We can show that for every either or is shattered. This yields the desired result by taking of cardinality .
If is not shattered, then there exists . For any , there is a set , , such that , because is -shattered. Since we must have . But then this implies , that is, is shattered.
For general , we conjecture that . The reasoning below gives roughly .
Let be a hypergraph of finite VC-dimension with a largest -shattered subset of vertices . As is -shattered, we have . This yields
with the last inequality following from lm:sauer. When , applying cl:vc_pairs gives
From this inequality we obtain: For , the -VC-dimension of a hypergraph of VC-dimension is at least , at most (where , and, as , at most .
An interesting geometric example is the hypergraph whose vertex set is a finite subset of and whose hyperedges are induced by half-spaces. It is well-known that .
More generally, we have for all . Indeed, by Tverberg’s theorem (see, e.g., [MATOUSEK]), every set of points in admits a partition into pairwise disjoint and non-empty sets such that the intersection of their convex hulls is non-empty. No half-space can -realize since any half-space that contains must contain at least one point from each , that is, at least points of .
Therefore, for this hypergraph and , the direct construction yields an --net of size , while the trivial construction (described at the end of Section 4.1) yields only a weaker upper bound of . With good bounds on , the construction via (see again Section 4.1) might provide even smaller --nets. In the plane (namely, where ), it follows from [GITS19] that , and so the upper bounds obtained using the direct construction and using are the same – .
5 Geometric -2-Nets
For a fixed , any hypergraph with VC-dimension and