On the Weisfeiler-Leman Dimension of Fractional Packing

10/24/2019 ∙ by V. Arvind, et al. ∙ 0

The k-dimensional Weisfeiler-Leman procedure (k-WL), which colors k-tuples of vertices in rounds based on the neighborhood structure in the graph, has proven to be immensely fruitful in the algorithmic study of Graph Isomorphism. More generally, it is of fundamental importance in understanding and exploiting symmetries in graphs in various settings. Two graphs are k-WL-equivalent if the k-dimensional Weisfeiler-Leman procedure produces the same final coloring on both graphs. 1-WL-equivalence is known as fractional isomorphism of graphs, and the k-WL-equivalence relation becomes finer as k increases. We investigate to what extent standard graph parameters are preserved by k-WL-equivalence, focusing on fractional graph packing numbers.

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1 Introduction

The -dimensional version of the Weisfeiler-Leman procedure is the classical color refinement applied to an input graph : each vertex of is initially colored by its degree. The procedure refines the color of each vertex in rounds, using the multiset of colors of vertices in the neighborhood of the vertex . In the -dimensional version (described in [37]), all vertex pairs

are classified by a similar procedure of coloring them in rounds. The extension of this procedure to a classification of all

-tuples of is due to Babai (see historical overview in [4, 7]) and is known as the -dimensional Weisfeiler-Leman procedure, abbreviated as . Graphs and are said to be -equivalent (denoted ) if they are indistinguishable by .

The WL invariance of graph parameters.

Let be a graph parameter. By definition, whenever and are isomorphic (denoted ). We say that is a -invariant graph parameter if the equality is implied even by the weaker condition . The smallest such will be called the Weisfeiler-Leman (WL) dimension of .

If no such exists, we say that the WL dimension of is unbounded. Knowing that a parameter has unbounded WL dimension is important because this implies that cannot be computed by any algorithm formalizable in fixed-point logic with counting (FPC), a robust framework for study encoding-invariant (or “choiceless”) computations; see the survey [11].

The focus of our paper is on graph parameters with bounded WL dimension. If is the indicator function of a graph property , then -invariance of precisely means that is definable in the infinitary -variable counting logic . While minimizing the number of variables is a recurring theme in descriptive complexity; see, e.g. [24, 18], our interest in the study of -invariance has an additional motivation: If we know that a graph parameter is -invariant, this gives us information not only about but also about .

Indeed, -invariance admits the following interpretation. We say that a (not necessarily numerical) graph invariant subsumes a graph invariant if whenever , that is, whenever distinguishes two graphs, then also does this. Let denote the graph invariant computed by on input . As easily seen, a parameter is -invariant if and only if is subsumed by . Which graph parameters are subsumed by is an interesting question even for dimensions and , in view of the importance of (color refinement) and (the original Weisfeiler-Leman) in isomorphism testing [4, 5] and, more recently, also in other application areas [28, 30]

. It is known, for example, that the largest eigenvalue of the adjacency matrix has WL dimension 1 (see

[35]), and the whole spectrum of a graph has WL dimension 2 (see [12, 19]). Kiefer and Neuen [25] recently proved that subsumes, in a certain strong sense, the decomposition of a graph into 3-connected components.

Fractional graph parameters.

In this paper, we mainly consider fractional

graph parameters. Algorithmically, a well-known approach to tackling intractable optimization problems is to consider an appropriate linear programming (LP) relaxation. Many standard integer-valued graph parameters have fractional real-valued analogues, obtained by LP-relaxation of the corresponding 0-1 linear program; see, e.g., the monograph

[35]. The fractional counterpart of a graph parameter is denoted by . While is often hard to compute, provides a polynomial-time computable approximation, and is sometimes even a good approximation of .

The WL dimension of a natural fractional parameter is a priori bounded, where natural means that is determined by an LP which is logically interpretable in terms of an input graph . A striking results of Anderson, Dawar, Holm [1] says that the optimum value of an interpretable LP is expressible in FPC. It follows from the known immersion of FPC into the finite-variable infinitary counting logic [31] , that each such is -invariant for some . While this general theorem is applicable to many graph parameters of interest, it is not clear if the bound on it yields will be explicit or will be even close to optimal.

We are interested in explicit and, possibly, exact bounds for the WL dimension. The simplest question here would be to pinpoint which fractional parameters are -invariant. This natural question, using fractional isomorphisms [35], can be recast as follows: Which fractional graph parameters are invariant under fractional isomorphism? It appears that this question has not received adequate attention in the literature. The only earlier result we could find is the -invariance of the fractional domination number shown in the Ph.D. thesis of Rubalcaba [34].

We show that the fractional matching number is also preserved by fractional isomorphism. Indeed, the matching number is an instance of the -packing number of a graph, corresponding to . Here, we use the standard notation for the complete graphs, for the path graphs, and for the cycle graphs on vertices. In general, is the maximum number of vertex-disjoint subgraphs of such that is isomorphic to the fixed pattern graph . While the matching number is computable in polynomial time, computing is NP-hard whenever has a connected component with at least 3 vertices [27], in particular, for . Note that -packing is the optimization version of the archetypal NP-complete problem Partition Into Triangles [20, Problem GT11]. We show that the fractional -packing number , like , is -invariant, whereas the WL dimension of the fractional triangle packing is 2.

In fact, we present a general treatment of fractional -packing numbers . We begin in Section 2 with introducing a concept of equivalence between two linear programs and ensuring that equivalent and have equal optimum values. Next, in Section 3, we consider the standard fractional versions of the generic Set Packing and Hitting Set problems, generalizing both -Packing and Dominating Set. We observe that the LP relaxations of Set Packing (or Hitting Set) are equivalent whenever the incidence graphs of the input set systems are -equivalent. This general fact readily implies Rubalcaba’s result [34] on the -invariance of the fractional domination number and also shows that, if the pattern graph has vertices, then the fractional -packing number is -invariant for some . This bound for comes from a logical definition of the instance of Set Packing corresponding to -Packing in terms of an input graph (see Section 3.3). Though the bound is quite decent, it does not need to be optimal. We elaborate on a more precise bound, where we need to use additional combinatorial arguments even in the case of fractional matching.

Integrality gap via invariance ratio.

We also briefly discuss approximate invariance of some standard integral graph parameters. The -invariance of the fractional matching number has two consequences. The first follows from [35, Theorem 2.1.3] that if is bipartite. Hence, over bipartite graphs the integral parameter is also -invariant; cf. [6, 3].

Another consequence concerns all graphs and is based on the result by Choi, Kim, and O [9] that

(1)

for every . As is -invariant, it follows that

(2)

for any pair of nonempty -equivalent graphs and . The bound in Equation (2) is tight, as seen for the -equivalent graphs and . Consequently, (1) is also tight.

The maximum is known as the integrality gap. Analogously, we define the -invariance ratio for the parameter as the quotient maximized over all -equivalent graph pairs . The integrality gap is important for a computationally hard graph parameter , as it bounds how well the polynomial-time computable parameter approximates . In this context, we now consider the fractional domination number .

As shown by Rubalcaba [34], is -invariant. Chappell, Gimbel, and Hartman [8] have shown that the integrality gap for the domination number is logarithmic, i.e., for vertex graphs . Indeed, the LP-based algorithm for computing is essentially optimal for this problem, as is hard-to-approximate within a sublogarithmic factor assuming  [33]. The results of [34] and [8] imply that the -invariance ratio of is at most logarithmic. On the other hand, Chappell et al. also show that their logarithmic bound for the integrality gap is tight up to a constant factor. In Section 6 we directly show an lower bound for the -invariance ratio of for vertex graphs. This provides a different perspective for the integrality gap lower bound [8].

Related work.

Atserias and Dawar [2] have shown that the -invariance ratio for the vertex cover number is at most 2. This bound also follows from the -invariance of (which implies the -invariance of as by LP duality) combined with a standard rounding argument. They [2] use a different argument111Their approach [2] is based on constructing weighted graphs such that if . The vertex cover number

is estimated from below and from above in terms of weighted vertex covers of

, and appears as a technical tool in the argument. which alone does not yield -invariance of the fractional vertex cover .

The bound of 2 for the -invariance ratio of is optimal. Atserias and Dawar [2] also show that the -invariance ratio for is at least for each . This implies an unconditional inapproximability result for Vertex Cover in the model of encoding-invariant computations expressible in FPC. It remains open if similar lower bounds on the invariance ratios for Dominating Set and Triangle Packing can be shown. However, we mainly focus on applications of -invariance to proving lower bounds for the integrality gap (Section 6).

Notation and formal definitions.

For a tuple in , let be the matrix with if , if and otherwise. We also augment

by the vector of the colors of

if the graph is vertex-colored. encodes the ordered isomorphism type of in and serves as an initial coloring of for . In each refinement round, computes , where is the neighborhood of and denotes a multiset. If , refines the coloring by , where is the tuple . If has vertices, the color partition stabilizes in at most rounds. We define and . Now, if .

The color partition of according to is equitable: for any color classes and , each vertex in has the same number of neighbors in . Moreover, if is vertex-colored, then the original colors of all vertices in each are the same. If , then exactly when and have a common equitable partition [35, Theorem 6.5.1] (the coarsest such partition is actually the complete -invariant for .)

Let and be graphs with vertex set , and let and be the adjacency matrices of and , respectively. Then and are isomorphic if and only if for some permutation matrix . The linear programming relaxation allows

to be a doubly stochastic matrix. If such an

exists, and are said to be fractionally isomorphic. If and are colored graphs with the same partition of the vertex set into color classes, then it is additionally required that whenever and are of different colors. Building on [36], it is shown by [32] that two graphs are indistinguishable by color refinement if and only if they are fractionally isomorphic.

2 Reductions between linear programs

A linear program (LP) is an optimization problem of the form “maximize (or minimize) subject to ”, where , , is an matrix , and varies over all vectors in with nonnegative entries (which we denote by ). Any vector satisfying the constraints , is called a feasible solution and the function is called the objective function. We denote an LP with parameters by , where , if the goal is to minimize the value of the objective function, and , if this value has to be maximized. The optimum of the objective function over all feasible solutions is called the value of the program and denoted by .

Our goal now is to introduce an equivalence relation between LPs ensuring equality of their values. We begin with a motivating discussion.

Isomorphic and isometric LPs.

By duality, we can restrict our attention to maximization problems. For the current discussion, suppose that an is in an augmented form, that is, feasible solutions fulfill the equality

(which can always be assumed at the cost of introducing extra slack variables). With this LP we associate the linear transformation

defined by . Let be another LP in an augmented form with associated linear transformation , where .

We call and isomorphic under two conditions. First, there is a permutation of the variables changing the set of equations of to the set of equations of . The equality of the two sets of equations means that the corresponding lists are mapped by some permutation . More precisely:

  1. there are permutation matrices and such that the linear transformations and make the diagram

    commutative, that is, .

The second condition is

  1. and .

Suppose that and are isomorphic. If is a feasible solution of , i.e., , then is a feasible solution of . Indeed,

(3)

Moreover,

(4)

implying that . Since the isomorphism of LPs is clearly an equivalence relation, we conclude by symmetry that .

More generally, call and isometric if Conditions (A)–(B) are fulfilled for some orthogonal transformations and , i.e., and , where denotes the identity matrix. Equations (3) and (4) remain true as

We conclude that also for isometric and . Note that this conclusion is based on the symmetry of the isometry relation, which in its turn follows from the invertibility of the matrices and . We now suggest a more general equivalence concept ensuring the equality of LP values and yet not assuming non-singularity of the involved matrices.

Equivalence of LPs.

Let and be linear programs (in general form), where , , and . We say that linearly reduces to ( for short), if there are matrices and such that

  • , where

We call a linear reduction from to . and are said to be equivalent ( for short) if and .

Theorem 2.1

Equivalent linear programs have equal values .

Proof

Let and and assume via . We show that for any feasible solution of we get a feasible solution of with , where the relation symbol is as in the definition:

Thus, implies and the theorem follows.∎

Note that isometric LPs are equivalent. We now describe a different kind of equivalent LPs.

LPs with fractionally isomorphic matrices.

Recall that a square matrix is doubly stochastic if its entries in each row and column sum up to 1. We call two matrices and fractionally isomorphic if there are doubly stochastic matrices and such that

(5)

Grohe at al [22, Eq. (5.1)-(5.2)] discuss similar definitions. Their purpose is to use fractional isomorphism and color refinment to reduce the dimension of linear equations and LPs.

Lemma 1

If and are fractionally isomorphic matrices, then

where denotes the -dimensional all-ones vector.

Proof

Since the matrices and in (5) are doubly stochastic, and . Along with the first equality in (5), these equalities imply that . The reduction follows similarly from the second equality in (5) as and are doubly stochastic. ∎

3 Getting started

3.1 Fractional Set Packing

The Set Packing problem is, given a family of sets , where , to maximize the number of pairwise disjoint sets in this family. The fractional version is given by where is the incidence matrix of , namely

under

Let denote the incidence graph of . Specifically, this is the vertex-colored bipartite graph with biadjacency matrix on two classes of vertices; vertices are colored red, vertices are colored blue, and a red vertex is adjacent to a blue vertex if .

Theorem 3.1

Let and be two families each consisting of subsets of the set . Let and be their incidence matrices. If , then .

Proof

Let

be the adjacency matrices of and respectively. Since and are indistinguishable by color refinement, by Ramana et al. [32] we conclude that these graphs are fractionally isomorphic, that is, there is a doubly stochastic matrix such that

(6)

and whenever and are from different vertex color classes. The latter condition means that is the direct sum of an doubly stochastic matrix and an doubly stochastic matrix , that is,

Therefore, Equality (6) reads

yielding

that is, and are fractionally isomorphic. Lemma 1 implies that . Therefore, these LPs have equal values by Theorem 2.1.∎

The dual version of is the following minimization problem:

under

This is an LP relaxation of the Hitting Set problem: Find a smallest set having a non-empty intersection with each .

3.2 -invariance of the fractional domination number

The closed neighborhood of a vertex is defined as . A set is dominating in if . The domination number is the minimum cardinality of a dominating set in .

As a warm-up example, consider the fractional Dominating Set problem, whose -invariance was established in [34]:

under

The value of this LP is the fractional domination number . We can see this as the fractional Hitting Set problem for consisting of the closed neighborhoods of all vertices in . The incidence matrix of and the adjacency matrix of the graph are related by the equality . If , then we conclude by Ramana et al. [32] that and are fractionally isomorphic, that is, for a doubly stochastic , where is the adjacency matrix of . It follows that , where is the incidence matrix of . Similarly, . Therefore, by Lemma 1 and Theorem 2.1. This follows also from Theorem 3.1 and LP duality as .

3.3 Definability excess

As we have just seen, given an instance graph of the fractional Dominating Set problem, we can define an instance of the fractional Hitting Set problem having the same LP value. The following definition concerns many similar situations and applies to any logical formalism.

Definition 1

We say that an istance of Fractional Set Packing or its dual version is definable over a graph with excess if

This definition is very general. It includes a particular situation when is first-order interpretable in in the sense of [17, Chapter 12.3], which means that for the color predicates (to be red or blue respectively) as well as for the adjacency relation of we have first order formulas defining them on for some in terms of the adjacency relation of . The number is called width of the interpretation. In this case, if there is a first-order sentence over variables that is true on but false on , then there is a first-order sentence over variables that is true on but false on . Cai, Fürer, and Immerman [7] showed that two structures are -equivalent iff they are equivalent in the -variable counting logic . Therefore, Theorem 3.1 has the following consequence.

Corollary 1

Let be a fractional graph parameter such that , where admits a first-order interpretation of width in (even possibly with counting quantifiers). Under these conditions, is definable over with excess and, hence, is -invariant.

In order to obtain -invariance via Theorem 3.1, we need definability with zero excess. If we use Corollary 1 for this purpose, this would require an interpretation of width 1. This is not always possible but, luckily, this is not the only way to get zero excess. As an example (in a slightly general setting), consider where is the adjacency matrix of . As easily seen, if , then there is a doubly stochastic such that and, as well, . Therefore, the value of this LP is -invariant (this was observed by Rubalcaba [34] for any polynomial in ). This contrasts with the fact that each entry of counts the number of 2-walks between two corresponding vertices, which cannot be captured by the logic . Another example where a combinatorial argument yields more than Corollary 1 is presented below.

3.4 -invariance of the fractional matching number

Recall that a set of edges is a matching in a graph if every vertex of is incident to at most one edge from . The matching number is the maximum size of a matching in . The fractional Matching Problem is defined by the LP

under

whose value is the fractional matching number . The above LP is just the linear program for the instance of Fractional Set Packing formed by the edges of as 2-element subsets of . If we want to interpret in the input graph , this will not be possible with width 1. Also, an interpretation of width 2 can only give -invariance. Nevertheless, we are able to show that is definable over with zero excess.

Theorem 3.2

The fractional matching number is -invariant.

Proof

Given , we have to prove that or, equivalently, where is as defined above, that is, and a red vertex is adjacent to a blue vertex if . By Theorem 3.1, it suffices to show that . To this end, we construct a common equitable partition of and , appropriately identifying their vertex sets.

For , let and define on similalrly. First, we identify and (i.e., the red parts of the two incidence graphs) so that for every in , which is possible because -equivalent graphs have the same color palette after color refinement. The color classes of now form a common equitable partition of and .

Next, extend the coloring to (the blue part of ) by , and similarly extend to . Denote the color class of containing by , the color class containing by etc. Note that is equal to the number of edges in between and (or the number of edges within if ). Since is a common equitable partition of and , we have whenever (note that does not need to be an edge in , nor needs to be an edge in ). This allows us to identify and so that for every in .

Now, consider the partition of into the color classes of (or the same in terms of ) and verify that this is an equitable partition for both and . Indeed, let and be color classes of such that there are and adjacent in , that is, for some vertex of . Note that, if considered on , the classes and also must contain and adjacent in (take and any adjacent to in such that ). Denote (it is not excluded that ). The vertex has exactly as many -neighbors in as it has -neighbors in . This number depends only on and or, equivalently, only on and . The same number is obtained also while counting the -neighbors of in .

On the other hand, has exactly one neighbor in if and exactly two -neighbors and if . What is the case depends only on and , and is the same in and . Thus, we do have a common equitable partition of and . ∎

The fractional matching number is precisely the fractional -packing number, and we generalize Theorem 3.2 to fractional -packing numbers in Section 4. In particular, there we will establish -invariance of Fractional Triangle Packing. The edge-disjoint version of -Packing is another problem that has intensively been studied in combinatorics and optimization. Since it is known to be NP-hard for any pattern containing a connected component with at least 3 edges [15], fractional relaxations have received much attention in the literature [16, 23, 38, 39]. The approach we used in the proof of Theorem 3.2 works as well for edge-disjoint packing, which we demonstrate in the next subsection.

3.5 -invariance of Fractional Edge-Disjoint Triangle Packing

Given a graph , let denote the family of all sets consisting of the edges of a triangle subgraph in . We regard as a family of subsets of the edge set . The optimum value of Set Packing Problem on , which we denote by , is equal to the maximum number of edge-disjoint triangles in . Let be the corresponding fractional parameter.

Theorem 3.3

The fractional edge-disjoint triangle packing number is -invariant.

Proof

Given a graph , define a coloring of by on and on . Like in the proof of Theorem 3.2, the upper case notation will be used to denote the color class containing .

Suppose that . This condition means that we can identify the sets and so that for every in . Moreover, the -equivalence of and implies that for any and with . This allows us to identify and so that for every in . As in the proof of Theorem 3.2, it suffices to argue that is a common equitable partition of the incidence graphs and . The equality will then follow by Theorem 3.1.

Let and be color classes of such that there is an edge between them in , that is, there are and such that . If considered on , the classes and also must contain and adjacent in (take, for example, the edge of and extend it to a triangle with other two edges and such that and , which must exists in because and are -equivalent). Denote and (it is not excluded that some of the classes , , and coincide).

Let , , and be the vertices of the triangle in , and suppose that . The number of -neighbors that has in is equal to the number of vertices such that is one of the up to pairs in