Abstract
The k-dimensional Weisfeiler-Leman procedure (
) 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 
-equivalent if dimention k does not suffice to distinguish them. 
-equivalence is known as fractional isomorphism of graphs, and the 
-equivalence relation becomes finer as k increases.
We investigate to what extent standard graph parameters are preserved by 
-equivalence, focusing on fractional graph packing numbers. The integral packing numbers are typically NP-hard to compute, and we discuss applicability of 
-invariance for estimating the integrality gap of the LP relaxation provided by their fractional counterparts.
Keywords: Computational complexity, The Weisfeiler-Leman algorithm, Fractional packing
Introduction
The 1-dimensional version of the Weisfeiler-Leman procedure is the classical color refinement applied to an input graph G. Each vertex of G is initially colored by its degree. The procedure refines the color of each vertex 
in rounds, using the multiset of colors of vertices u in the neighborhood N(v) of the vertex v. In the 2-dimensional version [25], all vertex pairs 
are classified by a similar procedure of coloring them in rounds. The extension of this procedure to a classification of all k-tuples of G is due to Babai (see historical overview in [4, 5]) and is known as the k-dimensional Weisfeiler-Leman procedure, abbreviated as 
. Graphs G and H 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 G and H are isomorphic (denoted 
). We say that 
is a 
-invariant graph parameter if the equality 
is implied even by the weaker condition 
. The smallest such k will be called the Weisfeiler-Leman (WL) dimension of 
.
If no such k 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 expressible in fixed-point logic with counting (FPC), a robust framework for study of encoding-invariant (or “choiceless”) computations; see the survey [7].
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, 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
. For example, the largest eigenvalue of the adjacency matrix has WL dimension 1 (see [24]), and the whole spectrum of a graph has WL dimension 2 (see [8, 13]), which implies that 
subsumes distinguishing non-isomorphic graphs by spectral methods.
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 [24]. The fractional counterpart of a graph parameter 
is denoted by 
. While 
is often hard to compute, 
provides, sometimes quite satisfactory, a polynomial-time computable 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 G. A striking result 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 
(see [21]), that each such 
is 
-invariant for some k. While this general theorem is applicable to many graph parameters of interest, it is not easy to extract an explicit value of k from this argument, and in any case such value would hardly be optimal.
We are interested in explicit and, possibly, exact bounds for the WL dimension. A first question here would be to pinpoint which fractional parameters 
are 
-invariant. This natural question, using the concept of fractional isomorphism [24], 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 [23].
We show that the fractional matching number 
is also a fractional parameter preserved by fractional isomorphism. Indeed, the matching number is an instance of the F-packing number
of a graph, corresponding to 
. Here and throughout, we use the standard notation 
for the complete graphs, 
for the path graphs, and 
for the cycle graph on n vertices. In general, 
is the maximum number of vertex-disjoint subgraphs 
of G that are isomorphic to the fixed pattern graph F. While the matching number is computable in polynomial time, computing 
is NP-hard whenever F has a connected component with at least 3 vertices [19], in particular, for 
. Note that 
-packing is the optimization version of the archetypal NP-complete problem Partition Into Triangles [14, 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 F-packing numbers 
. We begin in Sect. 2 with introducing a concept of equivalence between two linear programs 
and 
ensuring that equivalent 
and 
have equal optimum values. Next, in Sect. 3, we consider the standard optimization versions of Set Packing and Hitting Set [14, SP4 and SP8], two of Karp’s 21 NP-complete problems. These two generic problems generalize F-Packing and Dominating Set respectively. Their fractional versions have thoroughly been studied in hypergraph theory [12, 20]. 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 [23] on the 
-invariance of the fractional domination number and also shows that, if the pattern graph F has 
vertices, then the fractional F-packing number 
is 
-invariant for some 
. This bound for k comes from a logical definition of the instance of Set Packing corresponding to F-Packing in terms of an input graph G (see Corollary 6). 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 the fractional matching. We present a detailed treatment of the fractional matching in this exposition (Theorem 4), while the proof of our general result on the fractional F-packing numbers (Theorem 5), which includes the aforementioned cases of 
, is postponed to the full version of the paper [2].
The edge-disjoint version of F-Packing is another problem that has intensively been studied in combinatorics and optimization. Since it is known to be NP-hard for any pattern F containing a connected component with at least 3 edges [10], fractional relaxations have received much attention in the literature [17, 26]. We show that our techniques work well also in this case. In particular, the WL dimension of the fractional edge-disjoint triangle packing number 
is 2 (Theorem 7).
Integrality Gap via Invariance Ratio. Furthermore, we discuss the approximate invariance of integral graph parameters expressible by integer linear programs. For a first example, recall Lovász’s inequality [12, Theorem 5.21] 
. As 
is 
-invariant, it follows that 
for any pair of nonempty 
-equivalent graphs G and H. This bound is tight, as seen for the 
-equivalent graphs 
and 
. Consequently, the above relation between 
and 
is also tight. This simple example demonstrates that knowing, first, the exact value k of the WL dimension of a fractional parameter 
and, second, the discrepancy of the integral parameter 
over 
-invariant graphs implies a lower bound for the precision of approximating 
by 
.
Specifically, recall that the maximum 
, (respectively 
for minimization problems) is known as the integrality gap of 
. The integrality gap is important for a computationally hard graph parameter 
, as it bounds how well the polynomial-time computable parameter 
approximates 
.
On the other hand, we define the 
-invariance ratio for the parameter 
as 
, where the quotient is maximized over all 
-equivalent graph pairs (G, H). If 
is 
-invariant, then the 
-invariance ratio bounds the integrality gap from below. The following question suggests itself: How tight is this lower bound? In this context, we now consider the fractional domination number 
.
A general bound by Lovász [20] on the integrality gap of the fractional covering number for hypergraphs implies that the integrality gap for the domination number is at most logarithmic, specifically, 
for a non-empty graph G with n vertices. This results in an LP-based algorithm for approximation of 
within a logarithmic factor, which is essentially optimal as 
is hard to approximate within a sublogarithmic factor assuming 
[22]. As shown by Rubalcaba [23], 
is 
-invariant. Along with the Lovász bound, this implies that the 
-invariance ratio of 
is at most logarithmic. On the other hand, Chappell et al. [6] have shown that the logarithmic upper bound for the integrality gap of 
is tight up to a constant factor. In Sect. 6 we prove an 
lower bound even for the 
-invariance ratio of 
over n-vertex graphs. This implies the integrality gap lower bound [6], reproving it from a different perspective. In Sect. 6 we also discuss the additive integrality gap of the fractional edge-disjoint triangle packing.
Related Work. Atserias and Dawar [3] have shown that the 
-invariance ratio for the vertex cover number 
is at most 2. Alternatively, this bound also follows from the 
-invariance of 
(which implies the 
-invariance of 
as 
by LP duality) combined with a standard rounding argument. The approach of [3] uses a different 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 [3] also show that the 
-invariance ratio for 
is at least 7/6 for each k. This implies an unconditional inapproximability result for Vertex Cover in the model of encoding-invariant computations expressible in FPC.
Notation and Formal Definitions. For 
in 
, let 
be the 
matrix 
with 
if 
, 
if 
and 
otherwise. We also augment 
by the vector of the colors of 
if the graph G is vertex-colored. 
encodes the ordered isomorphism type of 
in G and serves as an initial coloring of 
for 
. In each refinement round, 
computes 
, where N(x) is the neighborhood of x and 
denotes a multiset. If 
, 
refines the coloring by 
, where 
is the tuple 
. If G has n vertices, the color partition stabilizes in at most 
rounds. We define 
and 
. Now, 
if 
.
The color partition of V(G) according to 
is equitable: for any color classes C and 
, each vertex in C has the same number of neighbors in 
. Moreover, if G is vertex-colored, then the original colors of all vertices in each C are the same. If 
, then 
exactly when G and H have a common equitable partition [24, Theorem 6.5.1].
Let G and H be graphs with vertex set 
, and let A and B be the adjacency matrices of G and H, respectively. Then G and H are isomorphic if and only if 
for some 
permutation matrix X. The linear programming relaxation allows X to be a doubly stochastic matrix. If such an X exists, G and H are said to be fractionally isomorphic. If G and H are colored graphs with the same partition of the vertex set into color classes, then it is additionally required that 
whenever u and v are of different colors. It turns out that two graphs are indistinguishable by color refinement if and only if they are fractionally isomorphic [24, Theorem 6.5.1].
Reductions Between Linear Programs
A linear program (LP) is an optimization problem of the form “maximize (or minimize) 
subject to 
”, where 
, 
, M is an 
matrix 
, and x varies over all vectors in 
with nonnegative entries (which we denote by 
). Any vector x satisfying the constraints 
, 
is called a feasible solution and the function 
is called the objective function. We denote an LP with parameters a, M, b by LP(a, M, b, opt), 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 val(L). Our goal now is to introduce an equivalence relation between LPs ensuring equality of their values.
Equivalence of LPs. Let 
and 
be linear programs (in general form), where 
, 
, 
and 
. We say that 
reduces to 
(
for short), if there are matrices 
and 
such that
, where 
and 
are said to be equivalent (
for short) if 
and 
.
Theorem 1
If 
, then 
.
Proof
Let 
and 
and assume 
via (Y, Z). We show that for any feasible solution x of 
we get a feasible solution 
of 
with 
, where 
is as in the definition:
 |
Thus, 
implies 
and the theorem follows. 
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 M and N
fractionally isomorphic if there are doubly stochastic matrices 
and 
such that
 |
1 |
Grohe et al. [16, Eqs. (5.1)–(5.2) in arXiv version] discuss similar definitions. They use fractional isomorphism fractional isomorphism to reduce the dimension of linear equations and LPs. The meaning of (1) will be clear from the proof of Theorem 3 below.
Lemma 2
If M and N are fractionally isomorphic 
matrices, then
 |
where 
denotes the n-dimensional all-ones vector.
Proof
Since the matrices Y and Z in (1) are doubly stochastic, 
and 
. Along with the first equality in (1), these equalities imply that 
. The reduction 
follows similarly from the second equality in (1) as 
and 
are doubly stochastic. 
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 maximum is called in combinatorics the matching number of hypergraph
and denoted by 
. The fractional version is given by 
where M is the 
incidence matrix of 
, namely
 |
The optimum value 
is called the fractional matching number of 
.
Let 
denote the incidence graph of 
. Specifically, this is the vertex-colored bipartite graph with biadjacency matrix M on two classes of vertices; m vertices are colored red, n vertices are colored blue, and a red vertex j is adjacent to a blue vertex i if 
.
Theorem 3
Let 
and 
be two families each consisting of n subsets of the set 
. If 
, then 
.
Proof
Denote the incidence matrices of 
and 
by M and N respectively. Let
 |
be the adjacency matrices of 
and 
respectively. Since 
and 
are indistinguishable by color refinement, by [24, Theorem 6.5.1] we conclude that these graphs are fractionally isomorphic, that is, there is a doubly stochastic matrix X such that
 |
2 |
and 
whenever u and v are from different vertex color classes. The latter condition means that X is the direct sum of an 
doubly stochastic matrix Y and an 
doubly stochastic matrix Z, that is, Equality (2) reads
 |
yielding 
. Thus, M and N are fractionally isomorphic. Lemma 2 implies that 
. Therefore, these LPs have equal values by Theorem 1. 
-invariance of the Fractional Matching Number
Recall that a set of edges 
is a matching in a graph G if every vertex of G is incident to at most one edge from M. The matching number
is the maximum size of a matching in G. Note that this terminology and notation agrees with Sect. 3 when graphs are considered hypergraphs with hyperedges of size 2. Fractional Matching is defined by the LP
 |
whose value is the fractional matching number
. The above LP is exactly the linear program 
for the instance 
of Fractional Set Packing formed by the edges of G as 2-element subsets of V(G), that is, 
.
Theorem 4
The fractional matching number is 
-invariant.
Proof
Given 
, we have to prove that 
or, equivalently, 
where 
is as defined above. By Theorem 3, it suffices to show that 
. To this end, we construct a common equitable partition of 
and 
, appropriately identifying their vertex sets. Recall that 
and a red vertex 
is adjacent to a blue vertex 
if 
.
For 
, let 
and define 
on V(H) similarly. First, we identify V(G) and V(H) (i.e., the red parts of the two incidence graphs) so that 
for every x 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 G and H.
Next, extend the coloring 
to E(G) (the blue part of 
) by 
, and similarly extend 
to E(H). Denote the color class of 
containing 
by 
, the color class containing x by 
etc. Note that 
is equal to the number of edges in G between 
and 
(or the number of edges within 
if 
). Since 
is a common equitable partition of G and H, we have 
whenever 
(note that 
does not need to be an edge in H, nor 
needs to be an edge in G). This allows us to identify E(G) and E(H) so that 
for every e in 
.
Now, consider the partition of 
into the color classes of 
(or the same in terms of H) 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 y of G. Note that, if considered on 
, the classes C and D also must contain 
and 
adjacent in 
(take 
and any 
adjacent to x in H such that 
). Denote 
(it is not excluded that 
). The vertex x has exactly as many D-neighbors in 
as it has 
-neighbors in G. This number depends only on C and 
or, equivalently, only on C and D. The same number is obtained also while counting the D-neighbors of 
in 
.
On the other hand, e has exactly one neighbor x in C if 
and exactly two C-neighbors x and y if 
. What is the case depends only on D and C, and is the same in 
and 
. Thus, we do have a common equitable partition of 
and 
. 
As was discussed in Sect. 1, we are able to generalize Theorem 4 to any fractional F-packing number 
. For a graph G, let 
be the family of subsets of V(G) consisting of the vertex sets 
of all subgraphs 
of G isomorphic to the pattern graph F. Now, 
. Dell et al. [9] establish a close connection between homomorphism counts and 
equivalence, which motivates the following definition. The homomorphism-hereditary treewidth of a graph F, denoted by 
, is the maximum treewidth 
over all homomorphic images 
of F. The proof of the following result can be found in the full version of the paper [2].
Theorem 5
If 
, then 
is 
-invariant.
First-Order Interpretability. Our approach to proving Theorem 4 was, given an instance graph G of Fractional Matching Problem, to define an instance 
of Fractional Set Packing Problem having the same LP value. The following definition concerns many similar situations. Given a correspondence 
, we say that an istance 
of Fractional Set Packing is definable over a graph G with excess
e if 
implies 
.
This definition includes a particular situation when 
is first-order interpretable in G in the sense of [11, 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 k in terms of the adjacency relation of G. The number k is called width of the interpretation. In this case, if there is a first-order sentence over s variables that is true on 
but false on 
, then there is a first-order sentence over sk variables that is true on G but false on H. Cai, Fürer, and Immerman [5] showed that two structures are 
-equivalent iff they are equivalent in the 
-variable counting logic 
. Therefore, Theorem 3 has the following consequence.
Corollary 6
Let 
be a fractional graph parameter such that 
, where 
admits a first-order interpretation of width k in G (even possibly with counting quantifiers). Under these conditions, 
is definable over G with excess 
and, hence, 
is 
-invariant.
To obtain 
-invariance via Corollary 6, we would need an interpretation of width 1. This is hardly possible in the case of the fractional matching number, and an interpretation of width 2 could only give us 
-invariance of 
. Thus, our purely combinatorial argument for Theorem 4 is preferable here.
Fractional Edge-Disjoint Triangle Packing
We now show that the approach we used in the proof of Theorem 4 works as well for edge-disjoint packing. Given a graph G, let T(G) denote the family of all sets 
consisting of the edges of a triangle subgraph in G. We regard T(G) as a family 
of subsets of the edge set E(G). The optimum value of Set Packing Problem on 
, which we denote by 
, is equal to the maximum number of edge-disjoint triangles in G. Let 
be the corresponding fractional parameter.
Theorem 7
The fractional packing number 
is 
-invariant.
Proof
Given a graph G, we consider the coloring 
of 
defined by 
on E(G) and 
on T(G). Like in the proof of Theorem 4, the upper case notation 
will be used to denote the color class of 
.
Suppose that 
. This condition means that we can identify the sets E(G) and E(H) so that 
for every e in 
. Moreover, the 
-equivalence of G and H implies that 
for any 
and 
with 
. This allows us to identify T(G) and T(H) so that 
for every t in 
. As in the proof of Theorem 4, it suffices to argue that 
is a common equitable partition of the incidence graphs 
and 
. The equality 
will then follow by Theorem 3.
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 C and D also must contain 
and 
adjacent in 
(take, for example, the edge 
of H and extend it to a triangle with other two edges 
and 
such that 
and 
, which must exist in H because H and G are 
-equivalent). Denote 
and 
(it is not excluded that some of the classes C, 
, and 
coincide).
Let x, y, and z be the vertices of the triangle t in G, and suppose that 
. The number of D-neighbors that e has in 
is equal to the number of vertices 
such that 
is one of the 8 pairs in 
, like 
(some of these pairs can coincide). Since the partition of 
by the coloring 
is not further refined by 
, this number depends only on C and D. We obtain the same number also while counting the D-neighbors of 
in 
.
On the other hand, t has exactly one neighbor e in C if C differs from both 
and 
, exactly two C-neighbors if C coincides with exactly one of 
and 
, and exactly three C-neighbors e, 
, and 
if 
. Which of the three possibilities occurs depends only on D and C, and is the same in 
and 
. This completes our verification that we really have a common equitable partition. 
Invariance Ratio and Integrality Gap
Recall the discussion in the introduction about the domination number 
.
Theorem 8
For infinitely many n, there are n-vertex 
-equivalent graphs G and H such that 
.
Proof
It suffices to show that the variation of the domination number among n-vertex d-regular graphs is logarithmic for an appropriate choice of the degree function 
.
Assuming that dn is even, let 
denote a random d-regular graph on n vertices. Given 
, let 
denote the Erdős–Rényi random graph with edge probability p. Kim and Vu [18] proved for certain degree functions 
that the distribution 
can be approximated from below and above, with respect to the subgraph relation, by distributions 
and 
with 
and 
. We need the part of this sandwiching result about the approximation from above.
For our purposes, we consider pairs n, d such that 
and, thus, 
. Applied to this case, the Kim-Vu theorem says that there is a joint distribution of 
and 
with 
such that 
with probability 
as n increases. It follows that
 |
with probability 
. Glebov et al. [15] proved that 
with probability 
whenever 
and 
. Hence 
with probability 
. As a consequence, there is an n-vertex d-regular graph G with 
.
On the other hand, consider 
, where 
stands for the complete bipartite graph with vertex classes of size s and t, and note that 
. Therefore, 
, which readily gives us the desired bound. 
We conclude with a discussion of Edge-Disjoint Triangle Packing. Haxell and Rödl [17] proved that 
is well approximated by 
on dense graphs as 
for n-vertex G. On the other hand, Yuster [26] showed that 
for infinitely many G, and it is open whether this lower bound is tight. Define the invariance discrepancy of 
as the function 
where the maximum is taken over 
-equivalent n-vertex graphs G and H. As follows from Theorem 7, this function provides a lower bound for the maximum integrality gap 
over n-vertex graphs. In this respect, it is reasonable to ask what the asymptotics of 
is. The following fact is a step towards this goal.
Proposition 9
.
Proof
Consider 
and 
, where S and R are the Shrikhande and 
rook’s graphs respectively. Both have vertex set 
, and (i, j) and 
are adjacent in S if (
and 
) or (
and 
) or (
and 
), where equality is in 
, while they are adjacent in R if 
(row 4-clique) or 
(column 4-clique). S is completely decomposable into edge-triangles 
and, hence, 
. On the other hand, in R the edges of each 
all belong to the same row or column 4-clique. Since a packing can take at most one 
from each row/column 
, we have 
. This yields 
as desired. 
Footnotes
O. Verbitsky was supported by DFG grant KO 1053/8–1. He is on leave from the IAPMM, Lviv, Ukraine.
Contributor Information
Alberto Leporati, Email: alberto.leporati@unimib.it.
Carlos Martín-Vide, Email: carlos.martin@urv.cat.
Dana Shapira, Email: shapird@g.ariel.ac.il.
Claudio Zandron, Email: zandron@disco.unimib.it.
Vikraman Arvind, Email: arvind@imsc.res.in.
Frank Fuhlbrück, Email: fuhlbfra@informatik.hu-berlin.de.
Johannes Köbler, Email: koebler@informatik.hu-berlin.de.
Oleg Verbitsky, Email: verbitsky@informatik.hu-berlin.de.
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