On the Approximability of Partial VC Dimension

Abstract

We introduce the problem Partial VC Dimension that asks, given a hypergraph $H=(X,E)$ and integers k and $\ell$, whether one can select a set $C\subseteq X$ of k vertices of H such that the set $\{e\cap C, e\in E\}$ of distinct hyperedge-intersections with C has size at least $\ell$. The sets $e\cap C$ define equivalence classes over E. Partial VC Dimension is a generalization of VC Dimension, which corresponds to the case $\ell =2^k$, and of Distinguishing Transversal, which corresponds to the case $\ell =|E|$ (the latter is also known as Test Cover in the dual hypergraph). We also introduce the associated fixed-cardinality maximization problem Max Partial VC Dimension that aims at maximizing the number of equivalence classes induced by a solution set of k vertices. We study the approximation complexity of Max Partial VC Dimension on general hypergraphs and on more restricted instances, in particular, neighborhood hypergraphs of graphs.

C. Bazgan—Institut Universitaire de France.

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1 Introduction

We study identification problems in discrete structures. Consider a hypergraph (or set system) $H=(X,E)$, where X is the vertex set and E is a collection of hyperedges, that is, subsets of X. Given a subset $C\subseteq X$ of vertices, we say that two hyperedges of E are distinguished (or separated) by C if some element in C belongs to exactly one of the two hyperedges. In this setting, one can tell apart the two distinguished hyperedges simply by comparing their intersections with C. Following this viewpoint, one may say that two hyperedges are related if they have the same intersection with C. This is clearly an equivalence relation, and one may determine the collection of equivalence classes induced by C: each such class corresponds to its own subset of C. Any two hyperedges belonging to distinct equivalence classes are then distinguished by C. We call these classes neighborhood equivalence classes. In general, one naturally seeks to distinguish as many pairs of hyperedges as possible, using a small set C.

It is a well-studied setting to ask for a maximum-size set C such that C induces all possible $2^{|C|}$ equivalence classes. In this case, C is said to be shattered. The maximum size of a shattered set in a hypergraph H is called its Vapnis-Červonenkis dimension (VC dimension for short). This notion, introduced by Vapnis and Červonenkis [40] arose in the context of statistical learning theory as a measure of the structural complexity of the data. It has since been widely used in discrete mathematics; see the references in the thesis [9] for more references. We have the following associated decision problem.

The complexity of VC Dimension was studied in e.g. [17, 21, 34]; it is a complete problem for the complexity class LOGNP defined in [34] (it is therefore a good candidate for an NP-intermediate problem). VC Dimension remains LOGNP-complete for neighborhood hypergraphs of graphs [30] (the neighborhood hypergraph of G has V(G) as its vertex set, and the set of closed neighborhoods of vertices of G as its hyperedge set).

In another setting, one wishes to distinguish all pairs of hyperedges (in other words, each equivalence class must have size 1) while minimizing the size of the solution set C. Following [26], we call the associated decision problem, Distinguishing Transversal.

There exists a rich literature about Distinguishing Transversal. It was studied under different names, such as Test Set in Garey and Johnson’s book [25, SP6]; other names include Test Cover  [18–20], Discriminating Code [16] or Separating System [7, 37].¹ A celebrated theorem of Bondy [8] also implicitely studies this notion. A version of Distinguishing Transversal called Identifying Code was defined for graphs instead of hypergraphs [23, 27]. Similarly as for the well-known relation between the classic graph problem Dominating Set and the hypergraph problem Hitting Set, it is easy to check that an identifying code in graph G is the same as a distinguishing transversal in the neighborhood hypergraph of G.

The goal of this paper is to introduce and study the problem Partial VC Dimension, that generalizes both Distinguishing Transversal and VC Dimension, and defined as follows.

Partial VC Dimension belongs to the category of partial versions of common decision problems, in which, instead of satisfying the problem’s constraint task for all elements (here, all $2^{k}$ equivalence classes), we ask whether we can satisfy a certain number, $\ell$, of these constraints. See for example the papers [22, 29] that study some partial versions of standard decision problems, such as Set Cover or Dominating Set.

When $\ell =|E|$, Partial VC Dimension is precisely the problem Distinguishing Transversal. When $\ell =2^k$, we have the problem VC Dimension. Hence, Partial VC Dimension is NP-hard, even on many restricted classes. Indeed, Distinguishing Transversal is NP-hard [25], even on hypergraphs where each vertex belongs to at most two hyperedges [20], or on neighborhood hypergraphs of graphs that are either: unit disk graphs [32], planar bipartite subcubic [23], graphs that are interval and permutation [24], split graphs [23]. Min Distinguishing Transversal cannot be approximated within a factor of $o(\log n)$ on hypergraphs of order n [20], even on hypergraphs without 4-cycles [10], and on neighborhood hypergraphs of bipartite, co-bipartite or split graphs [23].

When $\ell =2^k$, Partial VC Dimension is equivalent to VC Dimension and unlikely to be NP-hard (unless all problems in NP can be solved in quasi-polynomial time), since $|X|\le 2^k$ and a simple brute-force algorithm has quasi-polynomial running time. Moreover, VC Dimension (and hence Partial VC Dimension) is W[1]-complete when parameterized by k [21].

Recently, the authors in [12] introduced the notion of $(\alpha ,\beta )$ -set systems, that is, hypergraphs where, for any set S of vertices with $|S|\le \alpha$, S induces at most $\beta$ equivalence classes. Using this terminology, if a given hypergraph H is an $(\alpha ,\beta )$-set system, $(H,k,\ell )$ with $k=\alpha$ is a YES-instance of Partial VC Dimension if and only if $\ell \le \beta$.

We will also study the approximation complexity of the following fixed-cardinality maximization problem associated to Partial VC Dimension.

Similar fixed-cardinality versions of classic optimization problems such as Set Cover, Dominating Set or Vertex Cover, derived from the “partial” counterparts of the corresponding decision problems, have gained some attention in the recent years, see for example [13, 14, 29].

Max Partial VC Dimension is clearly NP-hard since Partial VC Dimension is NP-complete; other than that, its approximation complexity is completely unknown since it cannot be directly related to the one of approximating Min Distinguishing Transversal or Max VC Dimension (the minimization and maximization versions of Distinguishing Transversal and VC Dimension, respectively).

Our Results. Our focus is on the approximation complexity of Max Partial VC Dimension. We give positive results in Sect. 3. We first provide polynomial-time approximation algorithms using the VC-dimension for the maximum degree or for the maximum edge-size of the input hypergraph. We apply these to obtain approximation ratios of the form $n^{\delta }$ (for $\delta <1$ a constant) in certain special cases, as well as a better approximation ratio but with exponential time. For neighbourhood hypergraphs of planar graphs, Max Partial VC Dimension admits a PTAS (this is also shown for Min Distinguishing Transversal). In Sect. 4, we give hardness results. We show that any 2-approximation algorithm for Max Partial VC Dimension implies a 2-approximation algorithm for Max VC Dimension. Finally, we show that Max Partial VC Dimension is $\mathsf {APX}$-hard, even for graphs of maximum degree at most 7.

2 Preliminaries

Twin-free Hypergraphs. In a hypergraph H, we call two equal hyperedges twin hyperedges. Similarly, two vertices belonging to the same set of hyperedges are twin vertices.

Clearly, two twin hyperedges will always belong to the same neighborhood equivalence classes. Similarly, for any set T of mutually twin vertices, there is no advantage in selecting more than one of the vertices in T when building a solution set C.

Observation 1

Let $H=(X,E)$ be a hypergraph and let $H'=(X',E')$ be the hypergraph obtained from H by deleting all but one of the hyperedges or vertices from each set of mutual twins. Then, for any set $C\subseteq X$, the equivalence classes induced by C in H are the same as those induced by $C\cap X'$ in $H'$.

Therefore, since it is easy to detect twin hyperedges and vertices in an input hypergraph, in what follows, we will always restrict ourselves to hypergraphs without twins. We call such hypergraphs twin-free.

Degree conditions. In a hypergraph H, the degree of a vertex x is the number of hyperedges it belongs to. The maximum degree of H is the maximum value of the degree of a vertex of H; we denote it by $\varDelta (H)$.

The next theorem gives an upper bound on the number of neighborhood equivalence classes that can be induced when the degrees are bounded.

Theorem 2

([18, 20, 27]). Let $H=(X,E)$ be a hypergraph with maximum degree $\varDelta$ and let C be a subset of X of size k. Then, C cannot induce more than $\frac{k(\varDelta +1)}{2}+1$ neighborhood equivalence classes.

The Sauer-Shelah lemma. The following theorem is known as the Sauer-Shelah Lemma [38, 39] (it is also credited to Perles in [39] and a weaker form was stated by Vapnik and Červonenkis [40]). It is a fundamental tool in the study of the VC dimension.

Theorem 3

(Sauer-Shelah Lemma [38, 39]). Let $H=(X,E)$ be a hypergraph with strictly more than $\sum _{i=0}^{d-1}\left( {\begin{array}{c}|X|\\ i\end{array}}\right)$ distinct hyperedges. Then, S has VC-dimension at least d.

Theorem 3 is known to be tight. Indeed, the system that consists of considering all subsets of $\{1,\ldots ,n\}$ of cardinality at most $d-1$ has VC-dimension equal to $d-1$. Though the original proofs of Theorem 3 were non-constructive, Ajtai [1] gave a constructive proof that yields a (randomized) polynomial-time algorithm, and an easier proof of this type can be found in Miccianio [31].

The following direct corollary of Theorem 3 is observed for example in [10].

Corollary 4

Let $S=(X,E)$ be a hypergraph with VC dimension at most d. Then, for any subset $X'\subseteq X$, there are at most $\sum _{i=0}^{d}\left( {\begin{array}{c}|X'|\\ i\end{array}}\right) \le |X'|^d+1$ equivalence classes induced by $X'$.

Approximation. An algorithm for an optimization problem is a c-approximation algorithm if it returns a solution whose value is always at most a factor of c away from the optimum. The class $\mathsf {APX}$ contains all optimization problems that admit a polynomial-time c-approximation algorithm for some fixed constant c. A polynomial-time approximation scheme (PTAS for short) for an optimization problem is an algorithm that, given any fixed constant $\epsilon >0$, returns in polynomial time (in terms of the instance and for fixed $\epsilon$) a solution that is a factor of $1+\epsilon$ away from the optimum. An optimization problem is $\mathsf {APX}$ -hard if it admits no PTAS (unless P=NP).

Given an optimization problem P, an instance I of P, we denote by $opt_P(I)$ (or opt(I) if there is no ambiguity) the value of an optimal solution for I.

Definition 5

( L -reduction [33]). Let A and B be two optimization problems. Then A is said to be L-reducible to B if there are two constants $\alpha , \beta >0$ and two polynomial time computable functions f, g such that: (i) f maps an instance I of A into an instance $I'$ of B such that $opt_B(I')\le \alpha \cdot opt_A(I)$, (ii) g maps each solution $S'$ of $I'$ into a solution S of I such that $||S|-opt_A(I)|\le \beta \cdot ||S'|-opt_B(I')|$.

L-reductions are useful in order to apply the following theorem.

Theorem 6

[33]. Let A and B be two optimization problems. If A is $\mathsf {APX}$-hard and L-reducible to B, then B is $\mathsf {APX}$-hard.

3 Positive Approximation Results for Max Partial VC Dimension

We start with a greedy polynomial-time procedure that always returns (if it exists), a set $|X'|$ that induces at least $|X'|+1$ equivalence classes.

Lemma 7

Let $H=(X,E)$ be a twin-free hypergraph and let $k\le |X|-1$ be an integer. One can construct, in time $O(k(|X|+|E|))$, a set $C\subseteq X$ of size k that produces at least $\min \{|E|,k+1\}$ neighborhood equivalence classes.

Proof

We produce C in an inductive way. First, let $C_1=\{x\}$ for an arbitrary vertex x of X for which there exists at least one hyperedge of E with $x\notin E$ (if such hyperedge does not exist, then all edges are twin edges; since H is twin-free, $|E|\le 1$ and we are done). Then, for each i with $2\le i\le k$, we build $C_{i}$ from $C_{i-1}$ as follows: select vertex $x_i$ as a vertex in $X\setminus C_{i-1}$ such that $C_{i-1}\cup \{x\}$ maximizes the number of equivalence classes.

We claim that either we already have at least |E| equivalence classes, or $C_{i}$ induces at least one more equivalence class than $C_{i-1}$. Assume for a contradiction that we have strictly less than |E| equivalence classes, but $C_{i}$ has the same number of equivalence classes as $C_{i-1}$. Since we have strictly less than |E| classes, there is an equivalence class consisting of at least two edges, say $e_1$ and $e_2$. But then, since H is twin-free, there is a vertex x that belongs to exactly one of $e_1$ and $e_2$. But $C_{i-1}\cup \{x\}$ would have strictly more equivalence classes than $C_{i}$, a contradiction since $C_{i}$ was maximizing the number of equivalence classes.

Hence, setting $C=C_k$ finishes the proof.     $\square$

Proposition 8

Max Partial VC Dimension is $\frac{\min \{2^k,|E|\}}{k+1}$-approximable in polynomial time. For hypergraphs with VC dimension at most d, Max Partial VC Dimension is $k^{d-1}$-approximable. For hypergraphs with maximum degree $\varDelta$, Max Partial VC Dimension is $\frac{\varDelta +1}{2}$-approximable.

Proof

By Lemma 7, we can always compute in polynomial time, a solution with at least $k+1$ neighborhood equivalence classes (if it exists; otherwise, we solve the problem exactly). Since there are at most $\min \{2^k,|E|\}$ possible classes, the first part of the statement follows. Similarly, by Corollary 4, if the hypergraph has VC dimension at most d, there are at most $k^{d}+1$ equivalence classes, and $\frac{k^{d}+1}{k+1}\le k^{d-1}$. Finally, if the maximum degree is at most $\varDelta$, by Theorem 2 there are at most $\frac{k(\varDelta +1)+2}{2}$ possible classes (and when $\varDelta \ge 1$, $\frac{k(\varDelta +1)+2}{2(k+1)}\le \frac{\varDelta +1}{2}$).     $\square$

Corollary 9

For hypergraphs of VC dimension at most d, Max Partial VC Dimension is $|E|^{(d-1)/d}$-approximable.

Proof

By Proposition 8, we have a $\min \{k^{d-1},|E|/k\})$-approximation. If $k^{d-1} < |E|^{(d-1)/d}$ we are done. Otherwise, we have $k^{d-1}\ge |E|^{(d-1)/d}$ and hence $k\ge |E|^{1/d}$, which implies that $\frac{|E|}{k}\le |E|^{(d-1)/d}$.     $\square$

For examples of concrete applications of Corollary 9, hypergraphs with no 4-cycles in its bipartite incidence graph² have VC-dimension at most 3 and hence we have an $|E|^{2/3}$-approximation for this class. Hypergraphs with maximum edge-size d also have VC dimension at most d. Other examples, arising from graphs, are neighborhood hypergraphs of: $K_{d+1}$-minor-free graphs (that have VC dimension at most d [11]); graphs of rankwidth at most r (VC dimension at most $2^{2^{O(r)}}$ [11]); interval graphs (VC dimension at most 2 [10]); permutation graphs (VC dimension at most 3 [10]); line graphs (VC dimension at most 3); unit disk graphs (VC dimension at most 3) [10]; $C_4$-free graphs (VC dimension at most 2); chordal bipartite graphs (VC dimension at most 3 [10]); undirected path graphs (VC dimension at most 3 [10]). Typical graph classes with unbounded VC dimension are bipartite graphs and their complements, or split graphs.

In the case of hypergraphs with no 4-cycles in its bipartite incidence graph (for which Max Partial VC Dimension has an $|E|^{2/3}$-approximation algorithm by Corollary 9), we can also relate Max Partial VC Dimension to Max Partial Double Hitting Set, defined as follows.

Theorem 10

Any $\alpha$-approximation algorithm for Max Partial Double Hitting Set on hypergraphs without 4-cycles in its bipartite incidence graph can be used to obtain a $4\alpha$-approximation algorithm for Max Partial VC Dimension on hypergraphs without 4-cycles in its bipartite incidence graph.

Proof

Let $H=(X,E)$ be a hypergraph without 4-cycles in its bipartite incidence graph, and let $C\subseteq X$ be a subset of vertices. Since H has no 4-cycles in its bipartite incidence graph, note that if some hyperedge contains two vertices of X, then no other hyperedge contains these two vertices. Therefore, the number of equivalence classes induced by C is equal to the number of hyperedges containing at least two elements of C, plus the number of equivalence classes corresponding to a single (or no) element of C. Therefore, the maximum number opt(H) of equivalence classes for a set of size k is at most $opt_{2HS}(H)+k+1$, where $opt_{2HS}(H)$ is the value of an optimal solution for Max Partial Double Hitting Set on H. Observing that $opt_{2HS}(H)\ge \frac{k}{2}$ (since one may always iteratively select pairs of vertices covering a same hyperedge to obtain a valid double hitting set of H), we get that $opt(H)\le 3opt_{2HS}(H) +1 \le 4opt_{2HS}(H)$. Moreover, in polynomial time we can apply the approximation algorithm of Max Partial Double Hitting Set to H to obtain a set C inducing at least $\frac{opt_{2HS}(H)}{\alpha }$ neighborhood equivalence classes. Thus, C induces at least $\frac{opt(H)}{4\alpha }$ neighborhood equivalence classes.     $\square$

Unfortunately, the complexity of approximating Max Partial Double Hitting Set seems not to be well-known, even when restricted to hypergraphs with no 4-cycles in its bipartite incidence graph. In fact, the problem Max Densest Subgraph (which, given an input graph, consists of maximizing the number of edges of a subgraph of order k) is precisely Max Partial Double Hitting Set restricted to hypergraphs where each hyperedge has size at most 2 (that is, to graphs), that can be assumed to contain no 4-cycles in its bipartite incidence graph (a 4-cycle would imply the existence of two twin hyperedges). Although Max Densest Subgraph (and hence Max Partial Double Hitting Set for hypergraphs with no 4-cycles in its bipartite incidence graph) is only known to admit no PTAS [28], the best known approximation ratio for it is $O(|E|^{1/4})$ [5].³ We deduce from this result, the following corollary of Theorem 10 for hypergraphs of hyperedge-size bounded by 2. This improves on the $O(|E|^{1/2})$-approximation algorithm given by Corollary 9 for this case.

Corollary 11

Let $\alpha$ be the best approximation ratio in polynomial time for Max Densest Subgraph. Then, Max Partial VC Dimension can be $3\alpha$-approximated in polynomial time on hypergraphs with hyperedges of size at most 2. In particular, there is a polynomial-time $O(|E|^{1/4})$-approximation algorithm for this case.

We will now apply the following result from [4].

Lemma 12

([4]). If an optimization problem is $r_1(k)$-approximable in fpt-time with respect to parameter k for some strictly increasing function $r_1$ depending solely on k, then it is also $r_2(n)$-approximable in fpt-time w.r.t. parameter k for any strictly increasing function $r_2$ depending solely on the instance size n.

Using Proposition 8 showing that Max Partial VC Dimension is $\frac{2^k}{k+1}$-approximable and Lemma 12, we directly obtain the following.

Corollary 13

For any strictly increasing function r, Max Partial VC Dimension parameterized by k is r(n)-approximable in FPT-time.

In the following we establish polynomial time approximation schemes for Min Distinguishing Transversal and Max Partial VC Dimension on planar graphs using the layer decomposition technique introduced by Baker [3].

Given a planar embedding of an input graph, we call the vertices which are on the external face level 1 vertices. By induction, we define level t vertices as the set of vertices which are on the external face after removing the vertices of levels smaller than t [3]. A planar embedding is t-level if it has no vertices of level greater than t. If a planar graph is t-level, it has a t-outerplanar embedding.

Theorem 14

Max Partial VC Dimension on neighborhood hypergraphs of planar graphs admits a PTAS.

Proof

Let G be a planar graph with a t-level planar embedding for some integer t. We aim to achieve an approximation ratio of $1+\varepsilon$. Let $\lambda = \lceil \frac{1}{\varepsilon } \rceil -1$.

Let $G_{i}$ ($0\le i\le \lambda$) be the graph obtained from G by removing the vertices on levels $i\bmod (\lambda +1)$. Thus, graph $G_i$ is the disjoint union of several subgraphs $G_{ij}$ ($0\le j\le p$ with $p=\lceil \frac{t+i}{\lambda +1}\rceil$) where $G_{i0}$ is induced by the vertices on levels $0,\ldots ,i-1$ (note that $G_{00}$ is empty) and $G_{ij}$ with $j\ge 1$ is induced by the vertices on levels $(j-1)(\lambda +1)+i+1, \ldots j(\lambda +1)+i-1$. In other words, each subgraph $G_{ij}$ is the union of at most $\lambda$ consecutive levels and is thus $\lambda$-outerplanar. Hence, $G_{i}$ is also $\lambda$-outerplanar and it has treewidth at most $3\lambda -1$ [6]. Using Courcelle’s theorem⁴, for any integer t and any subgraph $G_{ij}$, we can efficiently determine an optimal set $S_{ij}^t$ of t vertices of $G_{ij}$ that maximizes the number of (nonempty) induced equivalence classes in $G_{ij}$. We then use dynamic programming to construct a solution for $G_i$. Denote by $S_i(q, y)$ a solution corresponding to the maximum feasible number of equivalence classes induced by a set of y vertices of $G_i$ ($0\le y\le k$) among the first q subgraphs $G_{i1}, \ldots , G_{iq}$ ($1\le q\le p$). We have $S_i(q,y) = \max _{0\le x\le y} (S_{iq}^x + S_i(q-1,y-x))$. Let $S_i=S_i(p,k)$.

Among $S_0,\ldots , S_{\lambda }$, we choose the best solution, that we denote by S. We now prove that S is an $(1+\varepsilon )$-approximation of the optimal value opt(G) for Max Partial VC Dimension on G. Let $S_{opt}$ be an optimal solution of G. Then, there is at least one integer r such that at most $1/(\lambda +1)$ of the equivalent classes induced by $S_{opt}$ in G are lost when we remove vertices on the levels congruent to $r\bmod (\lambda + 1)$.

Thus, $val(S)\ge val(S_r)\ge opt(G)-\frac{opt(G)}{\lambda +1}=\frac{\lambda }{\lambda +1}opt(G)\ge (1-\frac{1}{\varepsilon })opt(G)$, which completes the proof.

The overall running time of the algorithm is $\lambda$ times the running time for graphs of treewidth at most $3\lambda -1$, that is, $O(\lambda n)$.     $\square$

As a side result, using the same technique, we provide the following theorem about Min Distinguishing Transversal, which is an improvement over the 7-approximation algorithm that follows from [36] (in which it is proved that any YES-instance satisfies $\ell \le 7k$) and solves an open problem from [23]. Due to space constraints, its proof is omitted.

Theorem 15

Min Distinguishing Transversal on neighborhood hypergraphs of planar graphs (equivalently, Min Identifying Code on planar graphs) admits a PTAS.

4 Hardness of Approximation Results for Max Partial VC Dimension

We define Max VC Dimension as the maximization version of VC Dimension.

Not much is known about the complexity of Max VC Dimension: it is trivially $\log _2 |E|$-approximable by returning a single vertex; a lower bound on the running time of a potential PTAS has been proved [17]. It is mentioned as an outstanding open problem in [15]. In the following we establish a connection between the approximability of Max VC Dimension and Max Partial VC Dimension.

Theorem 16

Any 2-approximation algorithm for Max Partial VC Dimension can be transformed into a randomized 2-approximation algorithm for Max VC Dimension with polynomial overhead in the running time.

Proof

Let H be a hypergraph on n vertices that is an instance for Max VC Dimension, and suppose we have a c-approximation algorithm $\mathscr {A}$ for Max Partial VC Dimension.

We run $\mathscr {A}$ with $k=1,\ldots ,\log _2 |X|$, and let $k_0$ be the largest value of k such that the algorithm outputs a solution with at least $\frac{2^k}{c}$ neighborhood equivalence classes. Since $\mathscr {A}$ is a c-approximation algorithm, we know that the optimum for Max Partial VC Dimension for any $k>k_0$ is strictly less than $2^k$. This implies that the VC-dimension of S is at most $k_0$.

Now, let X be the solution set of size $k_0$ computed by $\mathscr {A}$, and let $H_X$ be the sub-hypergraph of H induced by X. By our assumption, this hypergraph has at least $\frac{2^{k_0}}{c}$ distinct edges. We can now apply the Sauer-Shelah Lemma (Theorem 3).

We have $c=2$, and we apply the lemma with $|X|=k_0$ and $d=\frac{k_0}{2}+1$; it follows that the VC dimension of $H_X$ (and hence, of H) is at least $\frac{k_0}{2}+1$. By the constructive proof of Theorem 3, a shattered set Y of this size can be computed in (randomized) polynomial time [1, 31]. Set Y is a 2-approximation, since we saw in the previous paragraph that the VC dimension of H is at most $k_0$.     $\square$

We note that the previous proof does not seem to apply for any other constant than 2, because the Sauer-Shelah Lemma would not apply. Though the approximation complexity of Max VC Dimension is not known, our result shows that Max Partial VC Dimension is at least as hard to approximate.

Before proving our next result, we first need an intermediate result for Max Partial Vertex Cover (also known as Max k -Vertex Cover [14]), which is defined as follows.

Proposition 17

([35]). Max Partial Vertex Cover is $\mathsf {APX}$-hard, even for cubic graphs.

Theorem 18

Max Partial VC Dimension is $\mathsf {APX}$-hard, even for graphs of maximum degree 7.

Proof

We will give an L-reduction from Max Partial Vertex Cover (which is $\mathsf {APX}$-hard, by Proposition 17) to Max Partial VC Dimension. The result will then follow from Theorem 6. Given an instance $I=(G, k)$ of Max Partial Vertex Cover with $G=(V,E)$ a cubic graph, we construct an instance $I'=(G', k')$ of Max Partial VC Dimension with $G'=(V',E')$ of maximum degree 7 in the following way. For each vertex $v \in V$, we create a gadget $P_v$ with twelve vertices where four among these twelve vertices are special: they form the set $F_v = \{f_v^1,f_v^2,f_v^3,f_v^4\}$. The other vertices are adjacent to the subsets $\{f_v^4\}$, $\{f_v^2,f_v^3\}$, $\{f_v^1,f_v^3\}$, $\{f_v^2,f_v^4\}$, $\{f_v^1,f_v^4\}$, $\{f_v^1,f_v^3,f_v^4\}$, $\{f_v^1,f_v^2,f_v^4\}$,$\{f_v^1,f_v^2,f_v^3,f_v^4\}$, respectively. We also add edges between $f_v^1$ and $f_v^2$, between $f_v^2$ and $f_v^3$ and between $f_v^3$ and $f_v^4$. Since G is cubic, for each vertex v of G, there are three edges $e_1$, $e_2$ and $e_3$ incident with v. For each edge $e_i$ ($1\le i\le 3$), the endpoint v is replaced by $f_v^i$. Moreover, each of these original edges of G is replaced in $G'$ by two edges by subdividing it once (see Fig. 1 for an illustration). We call the vertices resulting from the subdivision process, edge-vertices. Finally, we set $k'=4k$.

Fig. 1.

a) Vertex-gadget $P_v$ and b) illustration of the reduction.

From any optimal solution S with $|S|=k$ covering opt(I) edges of G, we construct a set $C = \{ f_v^j : 1 \leqslant j \leqslant 4, v \in S \}$ of size 4k. By construction, C induces 12 equivalence classes in each vertex gadget. Moreover, for each covered edge $e=xy$ in G, the corresponding edge-vertex $v_e$ in $G'$ forms a class of size 1 (which corresponds to one or two neighbor vertices $f_x^i$ and $f_y^j$ of $v_e$ in C). Finally, all vertices in $G'$ corresponding to edges not covered by S in G, as well as all vertices in vertex gadgets corresponding to vertices not in S, belong to the same equivalence class (corresponding to the empty set). Thus, C induces in $G'$ $12k+opt(I)+1$ equivalence classes, and hence we have

$$\begin{aligned} opt(I')\ge 12k+opt(I)+1. \end{aligned}$$

(1)

Conversely, given a solution $C'$ of $I'$ with $|C'|=4k$, we transform it into a solution for I as follows. First, we show that $C'$ can be transformed into another solution $C''$ such that (1) $C''$ only contains vertices of the form $f_v^i$, (2) each vertex-gadget contains either zero or four vertices of $C''$, and (3) $C''$ does not induce less equivalence classes than $C'$. To prove this, we proceed step by step by locally altering $C'$ whenever (1) and (2) are not satisfied, while ensuring (3).

Suppose first that some vertex-gadget $P_v$ of $G'$ contains at least four vertices of $C'$. Then, the number of equivalence classes involving some vertex of $V(P_v)\cap C'$ is at most twelve within $P_v$ (since there are only twelve vertices in $P_v$), and at most three outside $P_v$ (since there are only three vertices not in $P_v$ adjacent to vertices in $P_v$). Therefore, we can replace $V(P_v)\cap C'$ by the four special vertices of the set $F_v$ in $P_v$; this choice also induces twelve equivalence classes within $P_v$, and does not decrease the number of induced classes.

Next, we show that it is always best to select the four special vertices of $F_v$ from some vertex-gadget (rather than having several vertex-gadgets containing less than four solution vertices each). To the contrary, assume that there are two vertex-gadgets $P_u$ and $P_v$ containing respectively a and b vertices of $C'$, where $1 \le b \le a \le 3$. Then, we remove an arbitrary vertex from $C'\cap V(P_v)$; moreover we replace $C'\cap V(P_u)$ with the subset $\{f_u^i, 1\le i\le a+1\}$, and similarly we replace $C'\cap V(P_v)$ with the subset $\{f_v^i, 1\le i\le b-1\}$. Before this alteration, the solution vertices within $V(P_u)\cup V(P_v)$ could contribute to at most $2^{a} + 2^b-2$ equivalence classes. After the modification, one can check that this quantity is at least $2^{a+1} + 2^{b-1}-2$ classes. Observing that $2^{a+1} + 2^{b-1}-2 \ge 2^{a} + 2^b-2$ since $2^a - 2^{b-1} \ge 0$ yields our claim. Hence, by this argument, we conclude that all vertex-gadgets (except possibly at most one) contain either zero or four vertices from the solution set $C'$.

Suppose that there exists one vertex-gadget $P_v$ with i solution vertices, $1\le i\le 3$. We show that we may add $4-i$ solution vertices to it so that $C'\cap V(P_v)=F_v$. Consider the set of edge-vertices belonging to $C'$. Since we had $|C'|=4k$ and all but one vertex-gadget contain exactly four solution vertices, there are at least $4-i$ edge-vertices in the current solution set. Then, we remove an arbitrary set of $4-i$ edge-vertices from $C'$ and instead, we replace the set $V(P_v)\cap C'$ by the set $F_v$ of special vertices of $P_v$. We now claim that this does not decrease the number of classes induced by $C'$. Indeed, any edge-vertex, since it has degree 2, may contribute to at most three equivalence classes, and the i solution vertices in $P_v$ can contribute to at most $2^{i}$ classes. Summing up, in the old solution set, these four vertices contribute to at most $3(4-i)+2^{i}$ classes, which is less than 12 since $1\le i\le 3$. In the new solution, these four vertices contribute to at least 12 classes, which proves our above claim.

We now know that there are 4i edge-vertices in $C'$, for some $i\le k$. All other solution vertices are special vertices in some vertex-gadgets. By similar arguments as in the previous paragraph, we may select any four of them and replace them with some set $F_v$ of special vertices of some vertex-gadget $P_v$. Before this modification, these four solution vertices may have contributed to at most $3\cdot 4=12$ classes, while the new four solution vertices now contribute to at least 12 classes.

Applying the above arguments, we have proved the existence of the required set $C''$ that satisfies conditions (1)–(3).

Therefore, we may now assume that the solution $C''$ contains no edge-vertices, and for each vertex-gadget $P_v$, $C''\cap V(P_v)\in \{\emptyset , F_v\}$. We define as solution S for I the set of vertices v of G for which $P_v$ contains four vertices of $C''$. Then, $val(S)=val(C')-12k-1$. Considering an optimal solution $C'$ for $I'$, we have $opt(I) \ge opt(I')-12k-1$. Using (1), we conclude that $opt(I') = opt(I)+12k+1 \le opt(I) + 24 opt(I) + 1$ since $k \leqslant 2opt(I)$ and thus $opt(I') \le 26 opt(I)$.

Moreover, we have $opt(I) - val(S) = opt(I') -12k - 1 - (val(C')-12k-1) = opt(I')-val(C')$.

Thus, our reduction is an L-reduction with $\alpha =26$ and $\beta = 1$.     $\square$

Proposition 8 and Theorem 18 give the following corollary:

Corollary 19

Max Partial VC Dimension is $\mathsf {APX}$-complete for bounded degree graphs.

5 Conclusion

In this paper, we defined and studied generalization of Distinguishing Transversal and VC Dimension. The probably most intriguing open question seems to be the approximation complexity of Max Partial VC Dimension. In particular, does the problem admit a constant-factor approximation algorithm? As a first step, one could determine whether such an approximation algorithm exists in superpolynomial time, or on special subclasses such as neighbourhood hypergraphs of specific graphs. We have seen that there exist polynomial-time approximation algorithms with a sublinear ratio for special cases; does one exist in the general case?