Meta-Kernelization with Structural ParametersResearch supported by the European Research Council (ERC), project COMPLEX REASON 239962.
Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS’09) and of Fomin et al. (FOCS’10), apply to optimization problems parameterized by solution size. We present meta-kernelization theorems that use a structural parameters of the input and not the solution size. Let be a graph class. We define the 0pt cover number of a graph to be a the smallest number of modules the vertex set can be partitioned into such that each module induces a subgraph that belongs to the class .
We show that each graph problem that can be expressed in Monadic Second Order (MSO) logic has a polynomial kernel with a linear number of vertices when parameterized by the 0ptcover number for any fixed class of bounded rank-width (or equivalently, of bounded clique-width, or bounded Boolean width). Many graph problems such as Independent Dominating Set, -Coloring, and -Domatic Number are covered by this meta-kernelization result.
Our second result applies to MSO expressible optimization problems, such as Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We show that these problems admit a polynomial annotated kernel with a linear number of vertices.
Kernelization is an algorithmic technique that has become the subject of a very active field in parameterized complexity, see, e.g., the references in . Kernelization can be considered as a preprocessing with performance guarantee that reduces an instance of a parameterized problem in polynomial time to a decision-equivalent instance, the kernel, whose size is bounded by a function of the parameter alone ; if the reduced instance is an instance of a different problem, then it is called a bikernel. Once a kernel or bikernel is obtained, the time required to solve the original instance is bounded by a function of the parameter and therefore independent of the input size. Consequently one aims at (bi)kernels that are as small as possible.
Every fixed-parameter tractable problem admits a kernel, but the size of the kernel can have an exponential or even non-elementary dependence on the parameter . Thus research on kernelization is typically concerned with the question of whether a fixed-parameter tractable problem under consideration admits a small, and in particular a polynomial, kernel. For instance, the parameterized Minimum Vertex Cover problem (does a given graph have a vertex cover consisting of vertices?) admits a polynomial kernel containing at most vertices. There are many fixed-parameter tractable problems for which no polynomial kernels are known. Recently, theoretical tools have been developed to provide strong theoretical evidence that certain fixed-parameter tractable problems do not admit polynomial kernels . In particular, these techniques can be applied to a wide range of graph problems parameterized by treewidth and other width parameters such as clique-width, or rank-width. Thus, in order to get polynomial kernels, structural parameters have been suggested that are somewhat weaker than treewidth, including the vertex cover number, max-leaf number, and neighborhood diversity . The general aim is to find a parameter that admits a polynomial kernel while being as general as possible.
We extend this line of research by using results from modular decompositions and rank-width to introduce new structural parameters for which large classes of problems have polynomial kernels. Specifically, we study the rank-width- cover number, which is a special case of a -cover number (see Section 3 for definitions). We establish the following result which is an important prerequisite for our kernelization results.
Hence, for graph problems parameterized by rank-width- cover number, we can always compute the parameter in polynomial time. The proof of Theorem ? relies on a combinatorial property of modules of bounded rank-width that amounts to a variant of partitivity .
Our kernelization results take the shape of algorithmic meta-theorems, stated in terms of the evaluation of formulas of monadic second order logic (MSO) on graphs. Monadic second order logic over graphs extends first order logic by variables that may range over sets of vertices (sometimes referred to as MSO logic). Specifically, for an MSO formula , our first meta-theorem applies to all problems of the following shape, which we simply call MSO model checking problems.
Instance: A graph .
Question: Does hold?
Many 0pthard graph problems can be naturally expressed as MSO model checking problems, for instance Independent Dominating Set, -Coloring, and -Domatic Number.
While MSO model checking problems already capture many important graph problems, there are some well-known optimization problems on graphs that cannot be captured in this way, such as Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. Many such optimization graph problems can be stated in the following way. Let be an MSO formula with one free set variable and .
Instance: A graph and an integer .
Question: Is there a set such that and ?
We call problems of this form MSO optimization problems. MSO optimization problems form a large fragment of the so-called LinEMSO problems . There are dozens of well-known graph problems that can be expressed as MSO optimization problems.
We establish the following result.
In fact, the obtained bikernel is an instance of an annotated variant of the original MSO optimization problem . Hence, Theorem ? provides a polynomial kernel for an annotated version of the original MSO optimization problem.
For obtaining the kernel for MSO model checking problems we proceed as follows. First we compute a smallest rank-width- cover of the input graph in polynomial time. Second, we compute for each module a small representative of constant size. Third, we replace each module with a constant size module, which results in the kernel. For the MSO optimization problems we proceed similarly. However, in order to represent a possibly large module with a small module of constant size, we need to keep the information how much a solution projected on a module contributes to the full solution. We provide this information by means of annotations to the kernel.
We would like to point out that a class of graphs has bounded rank-width iff it has bounded clique-width iff it has bounded Boolean-width . Hence, we could have equivalently stated the theorems in terms of clique-width or Boolean width. Furthermore we would like to point out that the theorems hold also for some classes where we do not know whether can be recognized in polynomial time, and where we do not know how to compute the partition in polynomial time. For instance, the theorems hold if is a graph class of bounded clique-width (it is not known whether graphs of clique-width at most can be recognized in polynomial time).
The set of natural numbers (that is, positive integers) will be denoted by . For we write to denote the set .
Graphs. We will use standard graph theoretic terminology and notation (cf. ). A module of a graph is a nonempty set such that for each vertex it holds that either no element of is a neighbor of or every element of is a neighbor of . We say two modules are adjacent if there are vertices and such that and are adjacent. A modular partition of a graph is a partition of its vertex set such that is a module of for each .
Monadic Second-Order Logic on Graphs. We assume that we have an infinite supply of individual variables, denoted by lowercase letters , and an infinite supply of set variables, denoted by uppercase letters . Formulas of monadic second-order logic (MSO) are constructed from atomic formulas , , and using the connectives (negation), (conjunction) and existential quantification over individual variables as well as existential quantification over set variables. Individual variables range over vertices, and set variables range over sets of vertices. The atomic formula expresses adjacency, expresses equality, and expresses that vertex in the set . From this, we define the semantics of monadic second-order logic in the standard way (this logic is sometimes called ).
Free and bound variables
of a formula are defined in the usual way. A sentence is a formula without free variables. We write to indicate that the set of free variables of formula is . If is a graph and we write to denote that holds in if the variables are interpreted by the vertices and the variables are interpreted by the sets (, ).
We review MSOtypes and games roughly following the presentation in . The quantifier rank of an MSO formula is defined as the nesting depth of quantifiers in . For non0ptnegative integers and , let consist of all MSO formulas of quantifier rank at most with free set variables in .
Let and be MSO formulas. We say and are equivalent, written , if for all graphs and , if and only if . Given a set of formulas, let denote the set of equivalence classes of with respect to . The following statement has a straightforward proof using normal forms (see Theorem 7.5 in  for details).
We will assume that for any pair of non0ptnegative integers and the system of representatives of given by Fact ? is fixed.
It follows from Fact ? that up to logical equivalence, every type contains only finitely many formulas. This allows us to represent types using MSO formulas as follows.
Let be a system of representatives of given by Fact ?. Because and are constant, we can consider both the cardinality of and the time required to compute it as constants. Let be the formula defined as , where . We can compute by deciding for each . Since the number of formulas in is a constant, this can be done in polynomial time if can be decided in polynomial time for any fixed .
Let be an arbitrary graph and an 0pttuple of subsets of . We claim that if and only if . Since the forward direction is trivial. For the converse, assume . First suppose . The set is a system of representatives of , so there has to be a such that . But implies by construction of and thus , a contradiction. Now suppose . An analogous argument proves that there has to be a such that and . It follows that , which again yields a contradiction.
Fixed-Parameter Tractability and Kernels. A parameterized problem is a subset of for some finite alphabet . For a problem instance we call the main part and the parameter. A parameterized problem is fixed-parameter tractable (FPT) if a given instance can be solved in time where is an arbitrary computable function of and is a polynomial in the input size .
A bikernelization for a parameterized problem into a parameterized problem is an algorithm that, given , outputs in time polynomial in a pair such that (i) if and only if and (ii) , where is an arbitrary computable function. The reduced instance is the bikernel. If , the reduction is called a kernelization and a kernel. The function is called the size of the (bi)kernel, and if is a polynomial then we say that admits a polynomial (bi)kernel.
It is well known that every fixed-parameter tractable problem admits a generic kernel, but the size of this kernel can have an exponential or even non-elementary dependence on the parameter . Since recently there have been workable tools available for providing strong theoretical evidence that certain parameterized problems do not admit a polynomial kernel .
Rank-width The graph invariant rank-width was introduced by Oum and Seymour  with the original intent of investigating the graph invariant clique-width. It later turned out that rank-width itself is a useful parameter, with several advantages over clique-width.
A set function is called symmetric if for all . For a symmetric function on a finite set , a branch-decomposition of is a pair where tree of maximum degree 3 and tT is a bijective function. For an edge of , the connected components of induce a bipartition of the set of leaves of . The width of an edge of a branch-decomposition is . The width of is the maximum width over all edges of . The branch-width of is the minimum width over all branch-decompositions of . If , then we define the branch-width of as . A natural application of this definition is the branch-width of a graph, as introduced by Robertson and Seymour [?], where , and the connectivity function of .
There is, however, another interesting application of the aforementioned general notions, in which we consider the vertex set of a graph as the ground set.
For a graph and , let A_G denote the -submatrix of the adjacency matrix over the two-element field , i.e., the entry , and , of A_G is if and only if is an edge of . Thecut-rank function of a graph is defined as follows: For a bipartition of the vertex set , equals the rank of A_G over . A rank-decomposition and rank-width of a graph is the branch-decomposition and branch-width of the cut-rank function of on , respectively.
Let be the trivial single-vertex graph, and let be a graph class such that . We define a 0ptcover of as a modular partition of such that the induced subgraph belongs to the class for each . Accordingly, the 0ptcover number of is the size of a smallest 0ptcover of .
Of special interest to us are the classes of graphs of rank-width at most . We call the 0ptcover number also the rank-width- cover number. If is the class of complete and edgeless graphs, then the 0ptcover number equals the neighborhood diversity , and clearly . Figure 1 shows the relationship between the rank-width- cover number and some other graph invariants.
We state some further properties of rank-width- covers.
( ?) The neighborhood diversity of a graph is also a rank-width- cover. The neighborhood diversity is known to be upper-bounded by . ( ?) follows immediately from the definition of rank-width- covers.
3.1Finding the Cover
Next we state several properties of modules of graphs. These will be used to obtain a polynomial algorithm for finding smallest rank-width- covers.
The symmetric difference of sets is . Sets overlap if but neither nor .
If or the result is immediate. Suppose and overlap and let , and . It follows from Theorem ? that these sets are modules of . Let , and . We show that . By assumption, both and have rank-width at most . Since rank-width is preserved by taking induced subgraphs, the graphs , , and also have rank-width at most . Let , , and be witnessing rank decompositions of , and , respectively.
We construct a rank decomposition of as follows. Let be the leaf (note that is bijective) of such that . Moreover, let and be the leaves of and such that and , respectively. We obtain from by adding disjoint copies of and and then identifying with the copies of and . Since , and are subcubic, so is .
We define the mapping t is a leaf of by
where maps nodes in to their copies in . The mappings , and are bijections and is injective, so is injective. By construction, the image of under is the set of leaves of , so is a bijection. Thus is a rank decomposition of .
We prove that the width of is at most . Given a rank decomposition and an edge , the connected components of induce a bipartition of the leaves of . We set . Take any edge of . There is a natural bijection from the edges in to the edges of . Accordingly, we distinguish three cases for :
. Let . Without loss of generality assume that . Then by construction of , we have . Pick any and . Since is a module of with but we have . As a consequence, can be obtained from by copying the column corresponding to . This does not increase the rank of the matrix.
. This case is symmetric to case ?, with and taking the roles of and , respectively.
. Let . Without loss of generality assume that . Then . Let and . Since is a module and but , we must have , so one can simply copy the column corresponding to . Now consider . Suppose . Since but , we must have because is a module. Then since and we must have because is a module. A symmetric argument proves that implies . It follows that . So again can be obtained from by copying columns, and thus the two matrices have the same rank.
Since is bijective, this proves that the rank of any bipartite adjacency matrix induced by removing an edge is bounded by . We conclude that the width of is at most and thus .
Let be a graph and . For every , the singleton is a module of , so is reflexive. Symmetry of is trivial. For transitivity, let such that and . Then there are modules of such that , , and . By Lemma ? is a module of with . In combination with that implies . This concludes the proof that is an equivalence relation.
Now let and let . For each there is a module of with and . By Lemma ?, is a module of and . Clearly, . On the other hand, implies by definition of , so . That is, .
Let be the set of equivalence classes of . It is immediate from Proposition ? that is a rank-width- cover of . Let be a partition of with . By the pigeonhole principle, there have to be vertices and indices , such that but and , where . Thus , so there is no module of such that and . In particular, is not a module or . So is not a rank-width- cover of .
By Lemma ? we can compute the unique minimal (with respect to set inclusion) module containing and in time . Since rank-width is preserved for induced subgraphs, there is a module containing and with if and only if . By Theorem ? this can be decided in time .
Let be a constant. Given a graph , we can compute the set of equivalence classes of by testing whether for each pair of vertices . By Proposition ?, this can be done in polynomial time, and by Corollary ?, is a smallest rank-width- cover of .
4Kernels for MSO Model Checking
In this section, we show that every MSO model checking problem admits a polynomial kernel when parameterized by the 0ptcover number of the input graph, where is some recursively enumerable class of graphs satisfying the following properties:
contains the single0ptvertex graph, and a 0ptcover of a graph with minimum cardinality can be computed in polynomial time.
There is an algorithm that decides whether in time polynomial in for any fixed MSO sentence and any graph .
Let be a graph and . Let be an 0pttuple of vertices of , and let be an 0pt tuple of sets of vertices of . We write to refer to the elementwise intersection of with . Similarly, we let , denote the subsequence of elements from contained in . If is a modular partition of and we will abuse notation and write and if there is no ambiguity about what partition the index belongs to.
For , we write and . By Theorem ?, Condition ? of Definition ? is equivalent to . That is, for each , duplicator has a winning strategy in the 0ptround MSO game played on and starting from . We construct a strategy witnessing by aggregating duplicator’s moves from these games in the following way:
Suppose spoiler makes a set move and assume without loss of generality that . For , let , and let be duplicator’s response to according to . Then duplicator responds with .
Suppose spoiler makes a point move and again assume without loss of generality that . Then for some . Duplicator responds with according to .
Assume duplicator plays according to this strategy and consider a play of the 0ptround MSO game on and starting from . Let and be the point moves and and be the set moves, so that and the moves made in the same round have the same index. We claim that defines a partial isomorphism between and .
Let and let such that and . Suppose . Since duplicator plays according to a winning strategy in the game on and , the restriction defines a partial isomorphism between and . It follows that if and only if and if and only if . Now suppose . Then and also since and by choice of duplicator’s strategy. By congruence, and are adjacent in if and only if and are adjacent in , so we must have if and only if .
Let and let such that . By construction of duplicator’s strategy, we have . Note that if then if and only if for arbitrary sets and . Combined with the fact that defines a partial isomorphism between and , this observation implies that is contained in any of the sets from if and only if is contained in the sets from with the same indices.
By Lemma ? we can compute a formula capturing the type of in polynomial time. Given , a graph satisfying can be effectively computed as follows. We start enumerating and check for each graph whether . If this is the case, we stop and output . Since this procedure must terminate eventually. Fixing and the order in which graphs are enumerated, the number of graphs we have to check depends only on . By Fact ? the number of rank 0pttypes is finite for each , so we can think of the total number of checks as bounded by a constant. Moreover the time spent on each check depends only on and the size of the graph . Because the number of graphs enumerated is bounded by a constant, we can think of the latter as bounded by a constant as well. Thus the algorithm computing a model of runs in constant time.
For each , we compute a graph of constant size with the same MSO rank- type as . By Lemma ?, this can be done in polynomial time. Now let be the graph obtained from the disjoint union of the graphs for as follows. For , let denote the set of vertices from the copy of . If and are adjacent in for and , we insert an edge for every and . Then is a modular partition of , and for and , modules and are adjacent in if and only if and are adjacent in . It is readily verified that and are 0ptcongruent.
Let be a graph with 0ptcover number , and let be a smallest 0ptcover given by ?. Let be the quantifier rank of . By Lemma ? and ?, we can in polynomial time compute a graph and a modular partition of such that and are 0ptcongruent and for each , is bounded by a constant. It follows from Lemma ? that . In particular, if and only if . Moreover, we have , so is a polynomial kernel.
Immediate from Theorems ?, ?, and ? in combination with Proposition ?.
5Kernels for MSO Optimization
By definition, MSO formulas can only directly capture decision problems such as -colorability, but many problems of interest are formulated as optimization problems. The usual way of transforming decision problems into optimization problems does not work here, since the MSO language cannot handle arbitrary numbers.
Nevertheless, there is a known solution. Arnborg, Lagergren, and Seese  (while studying graphs of bounded tree-width), and later Courcelle, Makowsky, and Rotics  (for graphs of bounded clique-width), specifically extended the expressive power of MSO logic to define so-called optimization problems, and consequently showed the existence of efficient (parameterized) algorithms for such problems in the respective cases.
The MSO optimization problems (problems of the form ^_) considered here are a streamlined and simplified version of the formalism introduced in . Specifically, we consider only a single free variable , and ask for a satisfying assignment of with minimum or maximum cardinality. To achieve our results, we need a recursively enumerable graph class that satisfies ? and ? along with the following property:
Let be a fixed MSO formula. Given a graph , a set of minimum (maximum) cardinality such that can be found in polynomial time, if one exists.
Our approach will be similar to the MSO kernelization algorithm, with one key difference: when replacing the subgraph induced by a module, the cardinalities of subsets of a given 0pttype may change, so we need to keep track of their cardinalities in the original subgraph.
To do this, we introduce an annotated version of ^_. Given a graph , an annotation is a set of triples with . For every set we define
We call the pair an annotated graph. If the integer is represented in binary, we can represent a triple in space . Consequently, we may assume that the size of the encoding of an annotated graph is polynomial in .
Each MSO formula and gives rise to an annotated MSO-optimization problem.
Instance: A graph with an annotation and an integer .
Question: Is there a set such that and ?
Notice that any instance of ^_ is also an instance of ^_ with the trivial annotation . The main result of this section is a bikernelization algorithm which transforms any instance of ^_ into an instance of ^_; this kind of bikernel is called an annotated kernel .
The results below are stated and proved for minimization problems ^_ only. This is without loss of generality – the proofs for maximization problems are symmetric.
Suppose there exists an 0pttuple of sets of vertices of , and a formula such that but for every 0pttuple of sets of vertices of we have . Let . Clearly, and but , a contradiction.
Let be the quantifier rank of . By Lemma ? and ?, we can in polynomial time compute a graph and a modular partition of such that and are 0ptcongruent, is bounded by a constant, and for each . To compute the annotation , we proceed as follows. For each , we go through all subsets . By Lemma ?, we can compute a formula such that for any graph and we have if and only if . Since has constant size for every , this can be done within a constant time bound. By Lemma ? and because and are 0ptcongruent, there has to be a such that . Using the algorithm given by ?, we can compute a minimum0ptcardinality subset with this property in polynomial time. We then add the triple to . In total, the number of subsets processed is in . From this observation we get the desired bounds on the total runtime, , and the encoding size of .
We claim that ^_ if and only if ^_. Suppose there is a set of vertices such that and . Since is a partition of , we have , where . For each , let be a subset of minimum cardinality such that . By Lemma ? and 0ptcongruence of and , there is for each such that . By construction, contains a triple . Observe that and implies . Let . Then by 0pt congruence of and and Lemma ?, we must have . In particular, . Furthermore,
For the converse, let such that and , let denote for . By construction, there is a set for each such that and . Let . Then by congruence and Lemma ? we get and thus . Moreover, .
Let be an instance of ^_. By ? a smallest 0ptcover of can be computed in polynomial time. Let be an annotated graph computed from and according to Lemma ?. Let and suppose . Then we can solve in time for some constant that only depends on and . To do this, we go through all subsets of and test whether . If that is the case, we check whether . By Fact ? this check can be carried out in time for suitable constants and depending only on and . Thus we can find a such that the entire procedure runs in time whenever is large enough. If we find a solution we return a trivial yes0pt instance; otherwise, a trivial no0ptinstance (of ^_). Now suppose . Then and so the encoding size of is polynomial in . Thus is a polynomial bikernel.
Immediate from Theorems ?, ?, and ? when combined with Proposition ?.
Recently Bodlaender et al.  and Fomin et al.  established meta-kernelization theorems that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems apply to optimization problems parameterized by solution size. Our results are, along with very recent results parameterized by the modulator to constant0pttreedepth , the first meta-kernelization theorems that use a structural parameter of the input and not the solution size. In particular, we would like to emphasize that our Theorem ? applies to a large class of optimization problems where the solution size can be arbitrarily large.
It is also worth noting that our structural parameter, the rank-width- cover number, provides a trade-off between the maximum rank-width of modules (the constant ) and the maximum number of modules (the parameter ). Different problem inputs might be better suited for smaller and larger