Spatial Mixing of Coloring Random Graphs
We study the strong spatial mixing (decay of correlation) property of proper -colorings of random graph with a fixed . The strong spatial mixing of coloring and related models have been extensively studied on graphs with bounded maximum degree. However, for typical classes of graphs with bounded average degree, such as , an easy counterexample shows that colorings do not exhibit strong spatial mixing with high probability. Nevertheless, we show that for with and sufficiently large , with high probability proper -colorings of random graph exhibit strong spatial mixing with respect to an arbitrarily fixed vertex. This is the first strong spatial mixing result for colorings of graphs with unbounded maximum degree. Our analysis of strong spatial mixing establishes a block-wise correlation decay instead of the standard point-wise decay, which may be of interest by itself, especially for graphs with unbounded degree.
A proper -coloring of a graph is an assignment of colors to the vertices so that adjacent vertices receive different colors. Each coloring corresponds to a configuration in the -state zero-temperature antiferromagnetic Potts model. The uniform probability space, known as the Gibbs measure, of proper -colorings of the graph, receives extensive studies from both Theoretical Computer Science and Statistical Physics.
An important question concerned with the Gibbs measure is about the mixing rate of Glauber dynamics, usually formulated as: on graphs with maximum degree , assuming , the lower bounds for and to guarantee rapidly mixing of the Glauber dynamics over proper -colorings. (See  for a survey.)
Recently, much attention has been focused on the spatial mixing (correlation decay) aspect of the Gibbs measure, which is concerned with the case where the site-to-boundary correlations in the Gibbs measure decay exponentially to zero with distance. In Statistical Physics, spatial mixing implies the uniqueness of infinite-volume Gibbs measure. In Theoretical Computer Science, a stronger notion is considered: the strong spatial mixing introduced in Weitz’s thesis . Here, the exponential decay of site-to-boundary correlations is required to hold even conditioning on an arbitrarily fixed boundary. Strong spatial mixing is interesting to Computer Science because it may imply efficient approximation algorithms for counting and sampling. This implication was fully understood for two-state spin systems. For multi-state spin systems such as coloring, this algorithmic implication of strong spatial mixing is only known for special classes of graphs, such as neighborhood-amenable (slow-growing) graphs . Strong spatial mixing of proper -coloring has been proved for classes of degree-bounded graphs, including regular trees , lattices graphs , and finally the general degree-bounded triangle-free graphs , all with the same bound where is the unique solution to .
All these temporal and spatial mixing results are established for graphs with bounded maximum degree. It is then natural to ask what happens for classes of graphs with bounded average degree. A natural model for the “typical” graphs with bounded average degree is the Erdös-Rényi random graph . In this model, the Gibbs measure of proper -colorings becomes more complicated because the maximum degree is unbounded and the decision of colorability is nontrivial. Nevertheless, it was discovered in  that for the rapid mixing of (block) Glauber dynamics over the proper -colorings can be guaranteed by a , much smaller than the maximum degree of . This upper bound on the number of colors was later reduced to a constant in  and independently in [15, 16], and very recently to a linear with in .
On the spatial mixing side, the strong spatial mixing of the models which are simpler than coloring has been studied on random graph , or other classes of graphs with bounded average degree. Recently in , such average-degree based strong spatial mixing is established for the independent sets of graphs with bounded connective constant. Since has connective constant with high probability, this result is naturally translated to .
It is then an important open question to ask about the conditions for the spatial mixing of colorings of graphs with bounded average degree. The following simple example shows that this can be very hard to achieve: Consider a long path of vertices, each adjacent to isolated vertices, where is the number of colors. When the path is sufficiently long, the connective constant of this graph can be arbitrarily close to 1. However, colors of those isolated vertices can be properly fixed to make the remaining path effectively a 2-coloring instance, which certainly has long-range correlation, refuting the existence of strong spatial mixing.
More devastatingly, it is easy to see that for any constant , with high probability the random graph contains a path of length in which every vertex has degree . As in the above example, even in a weaker sense of site-to-site correlation which was considered in , this forbids the strong spatial mixing up to a distance . Meanwhile, it is well known that the diameter of is with high probability. So the strong spatial mixing of colorings of random graph cannot hold except for a narrow range of distances in .
In this case, inspired by the studies of spatial mixing in rooted trees, where only the decay of correlation to the root is considered, we propose to study the strong spatial mixing with respect to a fixed vertex, instead of all vertices.
We make following assumptions:
is fixed, and for and sufficiently large ( is fine);
is arbitrarily fixed and is a random graph drawn from , where is sufficiently large.
Note that vertex is fixed independently of the sampling of random graph. With these assumptions we prove the following theorem.
Let and satisfy Assumption 1.1, and an arbitrary super-constant function. With high probability, is -colorable and the following holds: for any region containing , whose vertex boundary is , for any feasible colorings partially specified on which differ only at vertices that are at least distance away from in , for some constants depending only on and , it holds that
for a uniform random proper -coloring of and any , where is the vertex set on which and differ, and denotes the shortest distance in between and any vertex in .
This is the first strong spatial mixing result for colorings of graphs with unbounded maximum degree. Our technique is developed upon the error function method introduced in , which uses a cleverly designed error function to measure the discrepancy of marginal distributions, and the strong spatial mixing is implied by an exponential decay of errors measured by this function.
In all existing techniques for strong spatial mixing of colorings, when the degree of a vertex is unbounded, a multiplicative factor of is contributed to the decay of correlation, which unavoidably ruins the decay. However, in the real case for colorings of graphs with unbounded degree, a large-degree vertex may at most locally “freeze” the coloring, rather than nullify the existing decay of correlation. This limitation on the effect of large-degree vertex has not been addressed by any existing techniques for spatial mixing.
We address this issue by considering a block-wise correlation decay, so that within a block the coloring might be “frozen”, but between blocks, the decay of correlation is as in that between vertices in the degree-bounded case. This analysis of block-wise correlation decay can be seen as a spatial analog to the block dynamics over colorings of random graphs, and is the first time that such an idea is used in the analysis of spatial mixing.
As one of the most important random CSP, the decision problem of coloring sparse random graphs has been extensively studied, e.g. in [1, 3]. Monte Carlo algorithms for sampling random coloring in sparse random graphs were studied in [4, 8, 15, 16, 7], and in , a non-Monte-Carlo algorithm was given for the same problem which uses less colors but has worse error dependency than the Monte-Carlo algorithms. In [10, 14] the correlation decay on computation tree for coloring was studied which implies FPTAS for counting coloring.
Let be an undirected graph. For each vertex , let denote the degree of . For any , let denote the distance between and in ; and for any vertex sets , let and . The subscripts can be omitted if graph is assumed in context. For any vertex set , we use to denote the vertex boundary of , and use to denote the edge boundary of .
We consider the list-coloring problem, which is a generalization of -coloring problem. Let be a finite integer, a pair is called a list-coloring instance if is an undirected graph, and is a sequence of lists where for each vertex , is a list of colors from associated with vertex . A is a proper coloring of if for every vertex and no two adjacent vertices in are assigned with the same color by . A list-coloring instance is said to be feasible or colorable if there exists a proper coloring of . A coloring can also be partially specified on a subset of vertices in . For , let denote the set of all possible colorings (not necessarily proper) of the vertices in . A coloring partially specified on a subset of vertices is said to be feasible if there is a proper coloring of such that and are consistent over set . A coloring partially specified on a subset of vertices is said to be proper or locally feasible if is a proper coloring of where is the subgraph of induced by and denotes the sequence of lists restricted on set of vertices. For any , we use to denote the set of proper colorings of .
When for all vertices , a list-coloring instance becomes an instance for -coloring, which we denote as .
Self-avoiding walk (SAW) tree.
Given a graph and a vertex , a tree rooted by can be naturally constructed from all self-avoiding walks starting from so that each walk corresponds to a vertex in , and each walk is the parent of walks where is a vertex. We use to denote this tree constructed as above, and call it a self-avoiding walk tree (SAW) of graph .
Gibbs measure and strong spatial mixing.
A feasible list-coloring instance gives rise to a natural probability distribution , which is the uniform distribution over all proper list-colorings. This distribution is also called the Gibbs measure of list-colorings. We also a notation of to evaluate probability of an event defined on a uniform random proper coloring of . Let and . For any feasible coloring partially specified on vertex set , we use to denote the marginal distribution over colorings of vertices in conditioning on that the coloring of vertices in is as specified by . And when , we write . The list-coloring instance in the subscripts can be omitted if it is assumed in context. Formally, for a uniformly random proper coloring of , we have
Definition 2.1 (Strong Spatial Mixing).
The Gibbs measure on proper -colorings of a family of finite graphs exhibits strong spatial mixing (SSM) if there exist constants such that for any graph , any , and any two feasible -colorings , we have
where is the subset on which and differ, and is the total variation distance.
When the exponential bound relies on instead of , the definition becomes weak spatial mixing (WSM). The difference is SSM requires the exponential correlation decay continues to hold even conditioning on the coloring of a subset of vertices being arbitrarily (but feasibly) specified.
Random graph model
The Erdös-Rényi random graph is the graph with vertices and random edges where for each pair , the edge is chosen independently with probability . We consider with fixed .
We say an event occurs with high probability (w.h.p.) if the probability of the event is .
3 Correlation decay along self-avoiding walks
In this section, we analyze the propagation of errors between marginal distributions measured by a special norm introduced in  in general degree-unbounded graphs. Throughout this section, we assume to be a list-coloring instance with and where each .
The following error function is introduced in .
Definition 3.1 (error function).
Let and be two probability measures over the same sample space . We define
with the convention that and .
We assume to be feasible so that for vertex set and feasible colorings of vertex set , the marginal probabilities and are well-defined. The strong spatial mixing is proved by establishing a propagation of errors . Note that unlike in bounded-degree graphs, in general the value of can be infinite, which occurs when the possibility of a particular coloring of is changed by conditioning on and . This is avoided when a vertex cut with certain “permissive” property separating from the boundary.
If there is a such that for every and removing disconnects and , then is finite for any feasible colorings .
It is sufficient to show that whenever . Suppose that removing separates the graph into subgraphs and where contains and contains . For any proper coloring of and any proper coloring of , we must have , because for every , and hence it is always possible to coloring in a greedy fashion to complete a proper coloring of along with a proper coloring of to a proper coloring of the entire graph . Note that this implies the lemma because now a coloring of is possible if and only if it can be completed to a proper coloring of , a property independent of and . ∎
This motivates the following definition of permissive vertex and vertex set.
Given a list-coloring instance , a vertex is said to be permissive in if for all neighbors of and , it holds that . A set of vertices is said to be permissive if all vertices in are permissive.
Let be the self-avoiding walk tree of graph expanded from vertex . Recall that every vertex in can be naturally identified (many-to-one) with the vertex in at which the corresponding self-avoiding walk ends (which we also denote by the same letter ).
Given a list-coloring instance , let , , and a set of vertices in . Suppose that the root has children in and for , let denote the subtree rooted by . The quantity is recursively defined as follows
where is a piecewise function defined as that for any vertex in ,
where is the degree in the original graph instead of the degree in SAW-tree .
In particular, when is a -coloring instance , we denote this quantity as .
To state the main theorem of this section, we need one more definition.
Let , , , and . A set of vertices in is a cutset in for and if: (1) no vertex in is identified to or any vertex with by ; and (2) any self-avoiding walk from to a vertex in must intersect in . A cutset in for and is said to be permissive in if every vertex in is identified with a permissive vertex in by .
The following theorem is the main theorem of this section, which bounds the error function by the defined in Definition 3.3 when there is a good cutset in the SAW tree.
Let be a feasible list-coloring instance where and . Let , and be arbitrary, and . If there is a permissive cutset in for and , then for any feasible colorings which differ only on , it holds that
This theorem is implied by the following weak spatial mixing version of the theorem.
Let be a feasible list-coloring instance where and . Let and be arbitrary, and . If there is a permissive cutset in for and , then for any feasible colorings , it holds that
The two feasible colorings can be expressed as and such that and are two feasible colorings of vertices in and is a feasible coloring of vertices in . Let be such a list-coloring instance where is obtained from by deleting all vertices in and incident edges, and is a color list for vertices in obtained from by deleting color from the lists for all neighbors of any . Clearly, is the instance obtained from by conditioning on that is colored as , thus must be feasible since is feasible. Let . Obviously is a subtree of . Let be obtained from permissive cutset in for and by excluding those vertices which are identified with a vertex in by . It is easy to see that is a cutset in for and , and is also permissive in because the operation applied by on the original instance never decreases the gap . Thus, by Lemma 3.3, we have
where and are the marginal distributions at in the new instance . It is easy to see that
thus and where and are marginal probabilities defined in the original instance . Therefore, we have . It remains to show that where , which is quite easy to see, because every self-avoiding walk in ended in must be a self-avoiding walk in ended in and also the operation applied by on the original instance never decreases the gap thus never increases the value of for any vertex in the SAW-tree. ∎
3.1 The block-wise correlation decay
Now our task is to prove Lemma 3.3. This is done by establishing the decay of along walks among blocks with the following good property.
Given a list-coloring instance , a vertex set is a permissive block around in if and for every vertex in the vertex boundary .
For permissive blocks , a coloring of is globally feasible if and only if it is locally feasible (i.e. proper on ).
Let and a permissive block such that . Then for any feasible coloring , for any coloring , it holds that if and only if is proper on .
Let . Note that with and must be a vertex cut separating and . Then the lemma can be proved by the same argument as in the proof of Proposition 3.1. ∎
Notations. We now define some notations which are used throughout this section. Let be a permissive block in a feasible list-coloring instance . Let be the edge boundary of . We enumerate these boundary edges as . For , we assume where and . Note that in this notation more than one or may refer to the same vertex in . Let be the subgraph of induced by vertex set . For a coloring and , we denote . For and , let be obtained from by removing the color from the list for all and removing the color from the list for all (if any of these lists do not contain the respective color then no change is made to them).
With this notation, the following lemma generalizes a recursion introduced in  for bounded-degree graphs to general graphs by using permissive blocks.
Let be a feasible list-coloring instance, a permissive block with edge boundary where for each , and any two proper colorings of . For every ,
if a vertex is permissive in , then it is permissive in the new instance ;
the new instance is feasible.
For any feasible coloring of a vertex set with , we have
When modifying to , for any vertex , every time a color is removed from , at least one of the neighbors of is also deleted, so never decreases, which means that any vertex is permissive in if it is permissive in .
We next show that is feasible. Let and a proper coloring of subgraph induced by (such a coloring must exist or otherwise is not feasible). Recall that every vertex must remain to have in since is a permissive block in and never reduces the gap , which means no matter what is, we can always properly color in a greedy fashion without conflicting with , giving us a proper coloring of .
We then prove the recursion. Due to Lemma 3.4, both and are positive since and are proper on . For any , observe that
As argued above we have in because is a boundary vertex of a permissive block in and never reduces the gap . This implies that the probability because conditioning on any particular coloring of neighbors of there are at least two colors in its list not used by its neighbors. Therefore, we have , and the following telescopic product is safe to apply:
The following bounds for marginal probabilities are quite standard.
Given a feasible list-coloring instance , if vertex has and , then for any feasible coloring and any , we have
If vertex is permissive in and , then for any feasible coloring and any , we have
For the first inequality, conditioning on any coloring of neighbors of , there are at least colors in not used by its neighbors, thus .
For the second inequality, note that for a permissive with , is a permissive block with . Applying the recursion in Lemma 3.5, we have
for any , where are the neighbors of , and each remains to have in each new list-coloring instance . Also by Lemma 3.5, all new instances are feasible. Thus by the first inequality, we have . Therefore, it holds that
Summing this over all , we have
which implies . ∎
Let be a feasible list-coloring instance, and a permissive block around with edge boundary where for each . Let be a vertex set with , and any two feasible colorings of . Assume to be two proper colorings of achieving the maximum in the error function:
It holds that
where and are the respective marginal distributions of coloring of vertex conditioning on and in the new list-coloring instance .
Let denote the two colors of achieving the maximum in
We then have
Due to Lemma 3.4, since is a permissive block and , we have that if and only if is proper on (and the same also holds for condition ). Recall that we use to denote the set of proper colorings of . Therefore, we have
where are the colorings of achieving the maximum in