Kernelization and approximation of distance- independent sets on nowhere dense graphs☆
Introduction
For a graph and positive integer , a distance-r independent set in is a subset of vertices whose members are pairwise at distance more than . On the other hand, a distance-r dominating set in is a subset of vertices such that every vertex of is at distance at most from some member of . The cases correspond to the standard notions of an independent and a dominating set, respectively. In this work we will consider combinatorial questions about distance- independent and dominating sets, as well as the computational complexity of the corresponding decision problems Distance- Independent Set and Distance- Dominating Set: given a graph and integer , decide whether has a distance- independent set of size at least , respectively, a distance- dominating set of size at most .
In the following, we denote the minimum size of a distance- dominating set in a graph by and the maximum size of a distance- independent set by . Furthermore, if , we write for the minimum size of a distance-r dominating set of A, i.e., we only require that each vertex of is at distance at most from the dominating set. Similarly, we write for the maximum size of a distance- independent subset of . Observe that for every graph , vertex subset , and positive integer we have because every member of a set that distance- dominates can dominate at most one member of a distance- independent subset of . 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 [24], [25]. Even worse, under the assumption that P NP, for every , the size of a maximum independent set of an -vertex graph cannot be approximated in polynomial time within a factor better than [28]. Under the assumption P NP, the domination number of a graph cannot be approximated in polynomial time within a factor better than [35]. However, it turns out that in several restricted graph classes the problems can be approximated much better. For instance, for fixed the distance- 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- Independent Set and Distance- 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 , a graph with vertex set is a depth-r minor of a graph , written , if there are connected and pairwise vertex disjoint subgraphs , each of radius at most , such that if , then there are and with . Now, a class of graphs has bounded expansion if for every positive integer and every for , the edge density of is bounded by some constant . Furthermore, is nowhere dense if for every positive integer there exists a constant such that for all , where denotes the complete graph on vertices.
We call effectively nowhere dense, respectively, of effectively bounded expansion, if the function , respectively , 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 with 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 of bounded expansion and every positive integer , there exists a constant such that every graph satisfies A by-product of this combinatorial result is a pair of constant-factor approximation algorithms, for the Distance- Independent Set and Distance- 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 and through the lenses of their fractional relaxations. For a graph and positive integer , consider the following linear programs; here, denotes the set of vertices at distance at most from (including ). and The two above LPs are dual to each other, and requiring the variables to be integral yields the values and , respectively. Hence we have
The relationship between and 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 and a set system (family) consisting of subsets of . A subset is shattered by if for every subset there exists such that . The Vapnik–Chervonenkis dimension, short VC-dimension, of is the maximum size of a set shattered by [10]. We also define the notions of a 2-shattered set and the 2VC-dimension of a set system by restricting subsets considered in the above definition only to subsets of size exactly . 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 over is a subset that intersects every member of , while a fractional hitting set is a distribution of weights from among elements of so that every member of has total weight at least . Let and denote the minimum size, respectively weight, of an integral, respectively fractional, hitting set of .
Theorem 1 See e.g. [8], [19] There exists a universal constant such that for every set system of VC-dimension at most , we have Moreover, there exists a polynomial-time algorithm that computes a hitting set of of size bounded as above.
As proved in [1], any nowhere dense class 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 there exists a constant such that for every the family of distance- balls treated as a set system over , has VC-dimension at most . Combining this with Theorem 1 shows that for every graph . However, both [1] and [34] only prove the statement about stability, and consequently do not provide explicit bounds on the constant in the above inequality.
Observe that VC-dimension is a hereditary measure, i.e., for any subset of the universe, the VC-dimension of the system is not larger than the VC-dimension of . Hence, Theorem 1 applied to the set system yields , and we can make the same conclusion about the system stemming from the -neighborhoods of graphs from a nowhere dense class . That is, if for , then , and the algorithm provided by Theorem 1 can be applied to compute a vertex subset that distance- dominates with this size guarantee.
We first study the VC-dimension of systems of radius- 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 VC-dimension, of the th powers of graphs for any nowhere dense class. More precisely, we prove the following theorem.
Theorem 2 Let and let be a graph. If , then the VC-dimension of the set system is at most .
We immediately derive the following; here, is the constant provided by Theorem 1.
Corollary 3 Let be a nowhere dense class of graphs such that for all . Then for every , every and every we have Moreover, there exists a polynomial-time algorithm that computes a distance- dominating set of in of size bounded as above.
Corollary 3 gives an upper bound of on the multiplicative gap between and . 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 of graphs with the property that for every we have
Finally, we want to investigate the multiplicative gap between and . 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 . To this end, we leverage the kernelization results for Distance- Dominating Set in nowhere dense classes of [18] to prove the following.
Theorem 5 Let be a nowhere dense class of graphs. There exists a function such that for all , , , and , we have Furthermore, there is a polynomial-time algorithm that given as above, computes a distance- dominating set of in of size bounded as above.
Thus, the multiplicative gap is (and even ) for any .
In the second part of the paper we turn to the parameterized complexity of Distance- 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 , Independent Set is -complete and Dominating Set is -complete [13]. Hence both problems are not likely to be fixed-parameter tractable, i.e., solvable in time on instances of input size , where is a computable function, depending only on the value of the parameter and 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- Dominating Set and Distance- Independent Set are expressible in first-order logic (for fixed ), 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- 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 of a parameterized problem outputs another instance , which is equivalent to , and whose total size is bounded by for some computable function , called the size of the kernel. If 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 can be arbitrarily large.
Kernelization of Dominating Set and Distance- 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- 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- 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 , Distance- 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 be a fixed nowhere dense class of graphs, let be a fixed positive integer, and let be a fixed real. Then there exists a polynomial-time algorithm with the following properties. Given a graph , a vertex subset , and a positive integer , the algorithm either correctly concludes that , or finds a subset of size at most , for some function depending only on , and a subset such that .
We remark that in case is effectively nowhere dense, it is easy to see that the function above is computable and the algorithm can be made uniform w.r.t. and : there is one algorithm that takes and also on input, instead of a different algorithm for each choice of and . Furthermore, as in [18], it is easy to follow the lines of the proof to obtain a linear kernel in case is a class of bounded expansion. That is, the size of the obtained set is bounded by , where the constant hidden in the -notation depends on and .
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- Dominating Set, we provide the following lower bound for completeness.
Theorem 7 Let be a class of graphs that is closed under taking subgraphs and that is not nowhere dense. Then there exists an integer such that Distance- Independent Set is -hard on .
Our proof of Theorem 6 uses a similar approach as [14], [18] for the kernelization of Distance- Dominating Set. We aim to iteratively remove vertices from that are irrelevant for distance- independent sets in the following sense. A vertex is irrelevant if the following assertion holds: provided contains a distance- independent subset of size , then also contains a distance- independent subset of size .
In order to find such an irrelevant vertex, we start by computing a good approximation of a distance- dominating set of . 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- independent set in . We now classify the remaining vertices of with respect to their interaction with the set and argue that if is large we may find an irrelevant vertex.
We repeat this construction until becomes small enough (almost linear in ) and return the resulting set as the set . It now suffices to add a small set of vertices and edges so that short distances between the elements of 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- domination and distance- independence in Section 4. Finally, we present the kernelization algorithm for Distance- 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 as a minor is bounded by . 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 and let be a graph. If , then the VC-dimension of the set system
Gap between and .
We first prove Theorem 4, which we repeat for convenience.
Theorem 4 There exists a nowhere dense class of graphs with the property that for every we have
We will first prove the following auxiliary lemma, which essentially encompasses the statement for . Here denotes the maximum degree of a vertex in , whereas is the minimum length of a cycle in .
Lemma 12 For every sufficiently large there exists a graph , say with vertices, satisfying the following properties.
Kernelization
We now come to the proof of Theorem 6, which we repeat for convenience.
Theorem 6 Let be a fixed nowhere dense class of graphs, let be a fixed positive integer, and let be a fixed real. Then there exists a polynomial-time algorithm with the following properties. Given a graph , a vertex subset , and a positive integer , the algorithm either correctly concludes that , or finds a subset of size at most , for some function depending only on , and a subset
Hardness on somewhere dense classes
We finally prove Theorem 7, which we repeat for convenience.
Theorem 7 Let be a class of graphs that is closed under taking subgraphs and that is not nowhere dense. Then there exists an integer such that Distance- Independent Set is -hard on .
We shall use the following well-known characterization of somewhere dense graph classes; recall that denotes the exact -subdivision of a graph .
Lemma 18 Let be somewhere dense graph class that is closed under taking subgraphs. Then there exists such that[31]
References (38)
- et al.
Interpreting nowhere dense graph classes as a classical notion of model theory
European J. Combin.
(2014) A probabilistic proof of an asymptotic formula for the number of labelled regular graphs
European J. Combin.
(1980)- et al.
VC-dimension and Erdős-Pósa property
Discrete Math.
(2015) - et al.
Fixed-parameter tractability and completeness II: On completeness for W[1]
Theoret. Comput. Sci.
(1995) Constant-factor approximation of the domination number in sparse graphs
European J. Combin.
(2013)- et al.
Hitting sets when the VC-dimension is small
Inform. Process. Lett.
(2005) - et al.
Some simplified NP-complete graph problems
Theoret. Comput. Sci.
(1976) - et al.
On nowhere dense graphs
European J. Combin.
(2011) - et al.
Grad and classes with bounded expansion I. Decompositions
European J. Combin.
(2008) The asymptotic distribution of short cycles in random regular graphs
J. Combin. Theory Ser. B
(1981)
Colouring graphs with bounded generalized colouring number
Discrete Math.
Polynomial-time data reduction for dominating set
J. ACM
High degree graphs contain large-star factors
Approximation algorithms for NP-complete problems on planar graphs
J. ACM
(Meta) kernelization
J. ACM
Almost optimal set covers in finite VC-dimension
Discrete Comput. Geom.
Covering planar graphs with a fixed number of balls
Discrete Comput. Geom.
Theory of uniform convergence of frequencies of events to their probabilities and problems of search for an optimal solution from empirical data
Autom. Remote Control
Domination problems in nowhere-dense classes
Cited by (0)
- ☆
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). .