Kernelization and approximation of distance-r independent sets on nowhere dense graphs

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Abstract

For a positive integer r, a distance-r independent set in an undirected graph G is a set IV(G) of vertices pairwise at distance greater than r, while a distance-r dominating set is a set DV(G) such that every vertex of the graph is within distance at most r from a vertex from D. We study the duality between the maximum size of a distance-2r independent set and the minimum size of a distance-r dominating set in nowhere dense graph classes, as well as the kernelization complexity of the distance-r independent set problem on these graph classes. Specifically, we prove that the distance-r independent set problem admits an almost linear kernel on every nowhere dense graph class.

Introduction

For a graph G and positive integer r, a distance-r independent set in G is a subset of vertices I whose members are pairwise at distance more than r. On the other hand, a distance-r dominating set in G is a subset of vertices D such that every vertex of G is at distance at most r from some member of D. The cases r=1 correspond to the standard notions of an independent and a dominating set, respectively. In this work we will consider combinatorial questions about distance-r independent and dominating sets, as well as the computational complexity of the corresponding decision problems Distance- r Independent Set and Distance- r Dominating Set: given a graph G and integer k, decide whether G has a distance-r independent set of size at least k, respectively, a distance-r dominating set of size at most k.

In the following, we denote the minimum size of a distance-r dominating set in a graph G by γr(G) and the maximum size of a distance-r independent set by αr(G). Furthermore, if AV(G), we write γr(G,A) for the minimum size of a distance-r dominating set of A, i.e., we only require that each vertex of A is at distance at most r from the dominating set. Similarly, we write αr(G,A) for the maximum size of a distance-r independent subset of A. Observe that for every graph G, vertex subset A, and positive integer r we have α2r(G,A)γr(G,A),because every member of a set that distance-r dominates A can dominate at most one member of a distance-2r independent subset of A. The study of a reverse inequality (in the approximate sense) for certain graph classes is the main combinatorial goal of this work.

Regarding computational complexity, both Independent Set and Dominating Set are NP-hard [29] and this even holds in very restricted settings, e.g., on planar graphs of maximum degree 3 [24], [25]. Even worse, under the assumption that P NP, for every ε>0, the size of a maximum independent set of an n-vertex graph cannot be approximated in polynomial time within a factor better than O(n1ε) [28]. Under the assumption P NP, the domination number of a graph cannot be approximated in polynomial time within a factor better than O(logn) [35]. However, it turns out that in several restricted graph classes the problems can be approximated much better. For instance, for fixed r the distance-r variants of both problems admit a polynomial-time approximation scheme (PTAS) on planar graphs [4] and, more generally, in graph classes with polynomial expansion [27]. We will discuss further approximation results later.

In this paper we are going to study Distance-r Independent Set and Distance- r Dominating Set on nowhere dense graph classes. The notions of nowhere denseness and bounded expansion are the fundamental definitions of the sparsity theory introduced by Nešetřil and Ossona de Mendez [31], [32]. Many familiar classes of sparse graphs, like classes of bounded treewidth, planar graphs, classes of bounded degree, and all classes that exclude a fixed minor or topological minor have bounded expansion and are nowhere dense. In order to facilitate further discussion, we now recall basic definitions.

Nowhere dense classes and classes of bounded expansion are defined by imposing restrictions on the graphs that can be found as bounded depth minors in the class. Formally, for a positive integer r, a graph H with vertex set {v1,,vn} is a depth-r minor of a graph G, written HrG, if there are connected and pairwise vertex disjoint subgraphs H1,,HnG, each of radius at most r, such that if vivjE(H), then there are wiV(Hi) and wjV(Hj) with wiwjE(G). Now, a class C of graphs has bounded expansion if for every positive integer r and every HrG for GC, the edge density |E(H)||V(H)| of H is bounded by some constant d(r). Furthermore, C is nowhere dense if for every positive integer r there exists a constant t(r) such that Kt(r)rG for all GC, where Kt denotes the complete graph on t vertices.

We call C effectively nowhere dense, respectively, of effectively bounded expansion, if the function t(r), respectively d(r), is computable; such effectiveness is enjoyed by essentially all natural classes of sparse graphs. Clearly, every class of bounded expansion is nowhere dense, but the converse is not true. For example the class consisting of all graphs G with girth(G)Δ(G) is nowhere dense, however, it does not have bounded average degree and in particular does not have bounded expansion, see [33].

The duality between independence and domination numbers on classes of bounded expansion was studied by Dvořák [16], who proved that for such classes, there is a constant-factor multiplicative gap between them. More precisely, Dvořák [16] proved that for every class C of bounded expansion and every positive integer r, there exists a constant c(r) such that every graph GC satisfies α2r(G)γr(G)c(r)α2r(G).A by-product of this combinatorial result is a pair of constant-factor approximation algorithms, for the Distance- r Independent Set and Distance- r Dominating Set problems on any class of bounded expansion. One of the goals of this work is to investigate to what extent the above duality can be lifted to the more general setting of nowhere dense graph classes.

It will be convenient to study the relation between γr(G) and α2r(G) through the lenses of their fractional relaxations. For a graph G and positive integer r, consider the following linear programs; here, Nr(u) denotes the set of vertices at distance at most r from u (including u). γr(G)minvV(G)xvsubject tovNr(u)xv1for all uV(G), andxv0for all uV(G). and α2r(G)maxvV(G)yvsubject tovNr(u)yv1for all uV(G), andyv0for all uV(G). The two above LPs are dual to each other, and requiring the variables to be integral yields the values γr(G) and α2r(G), respectively. Hence we have α2r(G)α2r(G)=γr(G)γr(G).

The relationship between γr(G) and α2r(G) and their fractional relaxations was discussed by Dvořák [15], and in particular, based on the above duality, Dvořák [15] improved the bounds in Equality (1) for classes of bounded expansion and nowhere dense classes.

Consider a ground set U and a set system (family) F consisting of subsets of U. A subset XU is shattered by F if for every subset YX there exists FF such that FX=Y. The Vapnik–Chervonenkis dimension, short VC-dimension, of F is the maximum size of a set shattered by F [10]. We also define the notions of a 2-shattered set and the 2VC-dimension of a set system by restricting subsets YX considered in the above definition only to subsets of size exactly 2. Clearly, the VC-dimension of a set system is upper bounded by its 2VC-dimension.

A fundamental result about VC-dimension is that in set systems of bounded VC-dimension the gap between integral and fractional hitting sets is bounded. A hitting set of a set system F over U is a subset HU that intersects every member of F, while a fractional hitting set is a distribution of weights from [0,1] among elements of U so that every member of F has total weight at least 1. Let τ(F) and τ(F) denote the minimum size, respectively weight, of an integral, respectively fractional, hitting set of F.

Theorem 1 See e.g. [8], [19]

There exists a universal constant C such that for every set system F of VC-dimension at most d, we have τ(F)Cdτ(F)lnτ(F).Moreover, there exists a polynomial-time algorithm that computes a hitting set of F of size bounded as above.

As proved in [1], any nowhere dense class C of graphs is stable (a model theoretic property that describes the complexity of definable set systems); see also [34] for a combinatorial proof of this fact. This, in particular, implies the following assertion: for every positive integer r there exists a constant d(r) such that for every GC the family of distance-r balls Ballsr(G){{v:distG(u,v)r}:uV(G)},treated as a set system over V(G), has VC-dimension at most d(r). Combining this with Theorem 1 shows that γr(G)Cd(r)γr(G)lnγr(G)for every graph GC. However, both [1] and [34] only prove the statement about stability, and consequently do not provide explicit bounds on the constant d(r) in the above inequality.

Observe that VC-dimension is a hereditary measure, i.e., for any subset AU of the universe, the VC-dimension of the system FA{FA:FF} is not larger than the VC-dimension of F. Hence, Theorem 1 applied to the set system FA yields τ(FA)Cdτ(FA)lnτ(FA), and we can make the same conclusion about the system stemming from the r-neighborhoods of graphs from a nowhere dense class C. That is, if AV(G) for GC, then γr(G,A)Cd(r)γr(G,A)lnγr(G,A), and the algorithm provided by Theorem 1 can be applied to compute a vertex subset that distance-r dominates A with this size guarantee.

We first study the VC-dimension of systems of radius-r balls in graphs from nowhere dense graph classes. By following the lines of a recent result of Bousquet and Thomassé [7], we are able to provide explicit bounds for the VC-dimension, in fact even for the 2VC-dimension, of the rth powers of graphs for any nowhere dense class. More precisely, we prove the following theorem.

Theorem 2

Let rN and let G be a graph. If KtrG, then the 2VC-dimension of the set system Ballsr(G) is at most t1.

We immediately derive the following; here, C is the constant provided by Theorem 1.

Corollary 3

Let C be a nowhere dense class of graphs such that Kt(r)rG for all rN. Then for every rN, every GC and every AV(G) we have α2r(G,A)α2r(G,A)=γr(G,A)γr(G,A)Ct(r)γr(G,A)lnγr(G,A).Moreover, there exists a polynomial-time algorithm that computes a distance-r dominating set of A in G of size bounded as above.

Corollary 3 gives an upper bound of O(logγr(G,A)) on the multiplicative gap between γr(G,A) and γr(G,A). For a lower bound, we prove that one cannot expect that this gap can be bounded by a constant on every nowhere dense class; recall that this is the case for classes of bounded expansion [16].

Theorem 4

There exists a nowhere dense class C of graphs with the property that for every rN we have supGCγr(G)γr(G)=+.

Finally, we want to investigate the multiplicative gap between γr(G,A) and α2r(G,A). While the lower bound of Theorem 4 asserts that in some nowhere dense class this gap cannot be bounded by any constant, the upper bound of Corollary 3 does not provide any upper bound in terms of α2r(G,A). To this end, we leverage the kernelization results for Distance- r Dominating Set in nowhere dense classes of [18] to prove the following.

Theorem 5

Let C be a nowhere dense class of graphs. There exists a function fdual:N×RN such that for all GC, AV(G), rN, and ε>0, we have γr(G,A)fdual(r,ε)α2r+1(G,A)1+εfdual(r,ε)α2r(G,A)1+ε.Furthermore, there is a polynomial-time algorithm that given G,A,r,ε as above, computes a distance-r dominating set of A in G of size bounded as above.

Thus, the multiplicative gap is O(α2r(G,A)ε) (and even O(α2r+1(G,A)ε)) for any ε>0.

In the second part of the paper we turn to the parameterized complexity of Distance- r Independent Set on nowhere dense classes. Also from the view of parameterized complexity both Independent Set and Dominating Set are hard: parameterized by the target size k, Independent Set is W[1]-complete and Dominating Set is W[2]-complete [13]. Hence both problems are not likely to be fixed-parameter tractable, i.e., solvable in time f(k)nc on instances of input size n, where f(k) is a computable function, depending only on the value of the parameter k and c is a fixed constant.

Again, it turns out that in several restricted graph classes the problems become easier to handle. As far as classes of sparse graphs are concerned, both Distance- r Dominating Set and Distance- r Independent Set are expressible in first-order logic (for fixed r), and hence fixed-parameter tractable on any nowhere dense class of graphs by the meta-theorem of Grohe et al. [26]. This was earlier proved in the particular case of Distance- r Dominating Set by Dawar and Kreutzer [11].

Once fixed-parameter tractability of a problem on a certain class of graphs is established, we can ask whether we can go even one step further by showing the existence of a polynomial (or even linear) kernel. A kernelization algorithm, or a kernel, is a polynomial-time preprocessing algorithm that given an instance (I,k) of a parameterized problem outputs another instance (I,k), which is equivalent to (I,k), and whose total size |I|+k is bounded by f(k) for some computable function f, called the size of the kernel. If f is a polynomial (respectively, linear) function, then the algorithm is called a polynomial (respectively, linear) kernel. It is known that for decidable problems, the existence of a kernel is equivalent to fixed-parameter tractability, however, in general the function f can be arbitrarily large.

Kernelization of Dominating Set and Distance- r Dominating Set on sparse graphs classes has received a lot of attention in the literature [2], [5], [14], [17], [18], [20], [21], [22], [23], [30]. In particular, Distance- r Dominating Set admits a linear kernel on any class of bounded expansion [14] and an almost linear kernel on any nowhere dense class [18]. The kernelization complexity of Distance- r Independent Set on classes of sparse graphs seems less explored; a linear kernel for the problem is known on any class excluding a fixed apex minor [21].

We prove that for every positive integer r, Distance- r Independent Set admits an almost linear kernel on every nowhere dense class of graphs. In fact, we prove the statement for the slightly more general, annotated variant of the problem, which finds an application e.g. in the model-checking result of Grohe et al. [26].

Theorem 6

Let C be a fixed nowhere dense class of graphs, let r be a fixed positive integer, and let ε>0 be a fixed real. Then there exists a polynomial-time algorithm with the following properties. Given a graph GC, a vertex subset AV(G), and a positive integer k, the algorithm either correctly concludes that αr(G,A)<k, or finds a subset YV(G) of size at most fker(r,ε)k1+ε, for some function fker depending only on C, and a subset BYA such that αr(G,A)kαr(G[Y],B)k.

We remark that in case C is effectively nowhere dense, it is easy to see that the function fker above is computable and the algorithm can be made uniform w.r.t. r and ε: there is one algorithm that takes r and ε also on input, instead of a different algorithm for each choice of r and ε. Furthermore, as in [18], it is easy to follow the lines of the proof to obtain a linear kernel in case C is a class of bounded expansion. That is, the size of the obtained set Y is bounded by O(k), where the constant hidden in the O()-notation depends on C and r.

It is not difficult to see that for classes closed under taking subgraphs, this result cannot be extended further. More precisely, similarly as in [14] for the case of Distance- r Dominating Set, we provide the following lower bound for completeness.

Theorem 7

Let C be a class of graphs that is closed under taking subgraphs and that is not nowhere dense. Then there exists an integer r such that Distance-r Independent Set is W[1]-hard on C.

Our proof of Theorem 6 uses a similar approach as [14], [18] for the kernelization of Distance- r Dominating Set. We aim to iteratively remove vertices from A that are irrelevant for distance-r independent sets in the following sense. A vertex vA is irrelevant if the following assertion holds: provided A contains a distance-r independent subset of size k, then also A{v} contains a distance-r independent subset of size k.

In order to find such an irrelevant vertex, we start by computing a good approximation of a distance-r2 dominating set D of A. If we do not find a sufficiently small such set, we can reject the instance, as by Theorem 5 this implies that there does not exist a large distance-r independent set in A. We now classify the remaining vertices of A with respect to their interaction with the set D and argue that if A is large we may find an irrelevant vertex.

We repeat this construction until A becomes small enough (almost linear in k) and return the resulting set as the set B. It now suffices to add a small set of vertices and edges so that short distances between the elements of B are exactly preserved. The result will be the output of the kernelization algorithm.

We assume familiarity with graph theory and refer to [12] for undefined notation. We provide basic facts about nowhere dense graph classes in Section 2 and refer to [33] for a broader discussion of the area. We present our results on the VC-dimension of power graphs in Section 3 and the construction of a nowhere dense class witnessing the non-constant gap between distance-r domination and distance-2r independence in Section 4. Finally, we present the kernelization algorithm for Distance- r Independent Set on nowhere dense graph classes in Section 5.

Section snippets

Preliminaries

We shall need some basic notions and tools for kernelization in nowhere dense classes used by Eickmeyer et al. [18]. For consistency and completeness of this paper, we have included these preliminaries also here, and they are largely taken verbatim from [18].

2VC-dimension of nowhere dense classes

In this section we prove Theorem 2, which we repeat for convenience. We remark again that our proof follows the lines of the work of Bousquet and Thomassé [7], who proved that the 2VC-dimension of the set system of all balls (of all radii) in a graph that excludes Kt as a minor is bounded by t1. As noted in [7], this result was in turn based on the case of planar graphs considered by Chepoi et al. [9].

Theorem 2

Let rN and let G be a graph. If KtrG, then the 2VC-dimension of the set system Ballsr(G)

Gap between γr and γr.

We first prove Theorem 4, which we repeat for convenience.

Theorem 4

There exists a nowhere dense class C of graphs with the property that for every rN we have supGCγr(G)γr(G)=+.

We will first prove the following auxiliary lemma, which essentially encompasses the statement for r=1. Here Δ(G) denotes the maximum degree of a vertex in G, whereas girth(G) is the minimum length of a cycle in G.

Lemma 12

For every sufficiently large dN there exists a graph Gd, say with n vertices, satisfying the following properties.

Kernelization

We now come to the proof of Theorem 6, which we repeat for convenience.

Theorem 6

Let C be a fixed nowhere dense class of graphs, let r be a fixed positive integer, and let ε>0 be a fixed real. Then there exists a polynomial-time algorithm with the following properties. Given a graph GC, a vertex subset AV(G), and a positive integer k, the algorithm either correctly concludes that αr(G,A)<k, or finds a subset YV(G) of size at most fker(r,ε)k1+ε, for some function fker depending only on C, and a subset

Hardness on somewhere dense classes

We finally prove Theorem 7, which we repeat for convenience.

Theorem 7

Let C be a class of graphs that is closed under taking subgraphs and that is not nowhere dense. Then there exists an integer r such that Distance-r Independent Set is W[1]-hard on C.

We shall use the following well-known characterization of somewhere dense graph classes; recall that G(r) denotes the exact r-subdivision of a graph G.

Lemma 18

[31]

Let C be somewhere dense graph class that is closed under taking subgraphs. Then there exists rN such that

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    Our work is supported by the National Science Centre of Poland via POLONEZ grant agreement UMO-2015/19/P/ST6/03998, which has received funding from the European Union’s Horizon 2020 research and innovation programme (Marie Skłodowska-Curie Grant Agreement No. 665778).

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