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arXiv:2208.14229v1 [cs.DC] 30 Aug 2022
Lower bound for constant-size local certification⋆
Virgina Ard´evol Mart´ınez1, Marco Caoduro2, Laurent
Feuilloley3[0000−0002−3994−0898] , Jonathan Narboni4[0000−0002−3087−5073], Pegah
Pournajafi5, and Jean-Florent Raymond6[0000−0003−4646−7602]
1LAMSADE, Universit´e Paris Dauphine, France
2Laboratoire G-SCOP, Univ. Grenoble Alpes
3CNRS, LIRIS, Univ. Lyon, France
4LaBRI, Universit´e de Bordeaux, CNRS, Bordeaux, France
5ENS Lyon, LIP, Lyon, France
6CNRS, LIMOS, Universit´e Clermont Auvergne, France
Abstract. Given a network property or a data structure, a local cer-
tification is a labeling that allows to efficiently check that the property
is satisfied, or that the structure is correct. The quality of a certifica-
tion is measured by the size of its labels: the smaller, the better. This
notion plays a central role in self-stabilization, because the size of the
certification is a lower bound (and often an upper bound) on the memory
needed for silent self-stabilizing construction of distributed data struc-
tures. From the point of view of distributed computing in general, it is
also a measure of the locality of a property (e.g. properties of the network
itself, such as planarity).
When it comes to the size of the certification labels, one can identify three
important regimes: the properties for which the optimal size is polyno-
mial in the number of vertices of the graph, the ones that require only
polylogarithmic size, and the ones that can be certified with a constant
number of bits. The first two regimes are well studied, with several upper
and lower bounds, specific techniques, and active research questions. On
the other hand, the constant regime has never been really explored, at
least on the lower bound side.
The main contribution of this paper is the first non-trivial lower bound
for this low regime. More precisely, we show that by using certification on
just one bit (a binary certification), one cannot certify k-colorability for
k≥3. To do so, we develop a new technique, based on the notion of score,
and both local symmetry arguments and a global parity argument. We
hope that this technique will be useful for establishing stronger results.
We complement this result by a discussion of the implication of lower
bounds for this constant-size regime, and with an upper bound for a
related problem, illustrating that in some cases one can do better than
the natural upper bound.
⋆J.-F. Raymond was supported by the ANR project GRALMECO (ANR-21-CE48-
0004-01). L. Feuilloley was supported by ANR project GrR (ANR-18-CE40-0032).
2 Ard´evol Mart´ınez, Caoduro, Feuilloley, Narboni, Pournajafi, Raymond
1 Introduction
Local certification consists in assigning labels to the nodes of a network,
to allow them to check locally that some property holds [9]. Historically, the
concept appeared implicitly in the study of self-stabilization, where in addition to
computing the solution of the problem, the nodes would compute some additional
information that would allow fast checking of the solution (i.e. a certification of
the solution). The most classic example is maybe the problem of computing
a spanning tree, where in addition to computing the pointer to its parents,
every node stores its distance to the root [1]. Such additional information is an
overhead in the memory used, hence, it is a natural goal to minimize its size.
In [3], Blin, Fraigniaud, and Patt-Shamir proved that for a standard notion of
self-stabilization called silent stabilization and up to some hypothesis, the space
needed for certification is the same as the space needed for self-stabilization
More recently, the question of the certification of properties of the network
itself, and not the correctness of a data structure built on top of it, has attracted
a lot of attention. This paper follows this direction, and we will only consider
certification of graph properties.
Let us now give a more precise intuition of what a local certification is (proper
definitions will be given in Section 2). We denote the number of vertices of a
graph by n. For a graph property P, we will say that it has a local certification
of size sif:
–for any graph Gsuch that Pholds, there exists an assignment of labels of
size s(n) that can “convince” all the nodes that Pis satisfied,
–for any graph Gsuch that Pdoes not hold, for any assignment of labels of
size s(n), there is at least one node that detects that the property is not
satisfied.
At the level of the nodes, the behavior is the following ; every node runs the
same local decision algorithm that takes as input all the information available
in a neighborhood (i.e. a local view), and outputs a decision: accept or reject.
For positive instances, all the nodes are convinced, that is, they all accept. For
negative instances, at least one node rejects.
There are actually many possible models, depending on the notion of neigh-
borhood considered, the presence of identifiers and how nodes can use them,
etc. Two classic models are proof-labeling schemes [18] and locally checkable
proofs [16]. The precise model is not essential for the discussion that follows,
hence we delay their definitions to the model section.
1.1 Three typical regimes for the certificate sizes
As said earlier, a natural goal in the study of local certification is to minimize
the size of the certificates. It is well-known that the optimal size is always in
O(n2), since one can always use the adjacency matrix as a certificate and make
the node check the consistency of this matrix with their neighborhoods, as well as
Lower bound for constant-size local certification 3
check that the property holds in the graph described by the matrix [18, Theorem
3.2].
Also, for any subquadratic function f, it is possible to engineer a property
for which the optimal size is f(n) [18, Corollary 2.4]. In other words, if we
consider the certificate size as the complexity of a property, there is no gap in the
complexity of certification. Nevertheless, for all the natural properties that have
been studied, the optimal certificate size only belongs to one of the following
three regimes (already identified in [16]): polynomial, (poly)logarithmic, and
constant size. For example, there is no known natural properties with certificate
size Θ(log log n), or Θ(log∗n), or Θ(2√log n).
In this paper, we are interested in lower bounds for the constant size regime.
But let us provide a quick overview of the three regimes, in order to give the full
picture and later discuss the novelty of our techniques.
Polynomial regime. It is known that the Θ(n2) size is needed for some specific
properties, such as having a non-trivial automorphism [16, Theorem 6.1] or hav-
ing chromatic number at least 4 (up to polylog factors) [16, Theorem 6.4]. Even
innocent-looking properties such as having diameter 3 or being triangle-free re-
quire certificates of size Ω(n) [6, Theorem 1] and n/eO(√n)[7, Proposition 5],
respectively.
Polylogarithmic regime. The regime of (poly)logarithmic certificate size has at-
tracted a lot of attention recently, and is often referred to as compact certifica-
tion (or LogLCP in [16]). The best-known local certification is the certification
of acyclicity (that is, the vertification of the class of trees) for which the optimal
size is Θ(log n) (by a straightforward adaptation of [18, Lemma 2.2]). It has
been proved recently that planarity and bounded-genus [12,11,8] have logarith-
mic certification, and that MSO properties on graphs of bounded treedepth [5]
and bounded treewidth [15] have respectively Θ(log n) and O(log2n) local cer-
tifications. An important open question in the area is to establish whether any
graph class defined by a set of forbidden minors has a compact certification.
Partial results are known for small minors [4] or minors with specific shapes
(namely paths [5, Corollary 2.7] and planar graphs [15, Corollary 3]). Finally,
let us mention one key result of the area, even if it is concerned with a data
structure instead of a graph property: the optimal certificate size for a minimum
spanning tree is O(log nlog W), where Wis the maximum weight [17].
Constant size regime. First, let us note that for some properties, no certificate is
needed. For example, checking that the graph is a cycle can be done by simply
having every node check that it has exactly two neighbors (we will always assume
that the graph is connected, thus there must be only one cycle).
Now, let us make a connection with a class of construction problems. A key
class of problems in distributed computing is the construction of locally checkable
labelings [20], or LCLs for short, that are the constant-size labelings that can
be checked by inspecting a local neighborhood. Examples of LCLs are maximal
independent sets, maximal matchings, and minimal dominating sets at some
4 Ard´evol Mart´ınez, Caoduro, Feuilloley, Narboni, Pournajafi, Raymond
distance d. There is now a very large literature on computing such labelings.
The problem that has attracted the most interest is the one of vertex coloring:
given an integer k(that typically depends on the maximum degree), how fast
can we compute an assignment of colors to the vertices, such that for every edge
the endpoints have different colors. We refer to the monograph [2] for distributed
graph coloring.
For any LCL, we can design a certification question:
Question 1. How many bits are needed to certify that the graph has a solution?
Most classic LCL problems are designed so that any graph has a solution;
for example, any graph has a maximal matching. But it is not true for any LCL;
for example, given a positive integer k∈N, not all graphs are k-colorable.
When a solution exists, and can be checked by inspecting each node and
its direct neighbors, it is trivial to design a certification for the question above:
get a solution, give every node its label as a certificate, and for the verification,
let the vertices run the local checking. Specifically, to certify that a graph is
k-colorable, one can find a proper k-coloring, and then give to every node its
color. It is then easy for the vertices to check this certification: every node checks
that no neighbor has been given the same color as itself. This certification uses
certificates of ⌈log k⌉bits, and the key question that we would like to answer is:
Question 2. Can we do better than ⌈log k⌉bits to certify that a graph is k-
colorable?
This question was already listed in [9] as one of the key open questions in
the field, and in the following section, we will review a few reasons why it is a
question worth studying.
1.2 Motivation for studying the constant-size regime
In local certification, and more generally in theoretical computer science, the
focus is usually not on the precise constants in the complexities, thus one might
consider the questions above to be non-essential. Let us list a few reasons why
Questions 1and 2are actually important.
An arena for new lower bound techniques. The lower bound techniques that we
have for local certification are mainly of two types (see the survey [9] for precise
citations and more detailed sketches).
First, there are the techniques based on counting arguments, also called cut-
and-plug techniques, that can be rather sophisticated but boils down to the
following fact: if we use o(log n)-bit labels in yes-instances (that is, correct in-
stances) then some (set of) labels will appear in different places of an instance
(or in different instances), because there are nvertices and o(n) different labels.
Using this, one can build a no-instance and derive a contradiction. This is the
technique used to show almost all lower bounds in the logarithmic regime.
Lower bound for constant-size local certification 5
The second classic technique is a reduction from communication complexity,
which we will not sketch here, but simply mention that it works better for the
polynomial regime.
At this point, two things are clear: (1) we have only two main techniques, and
they are now very well understood, and (2) they do not solve all our problems. In
particular, they do not seem to apply to the o(log n) regime. One can hope that
by trying to give a negative answer to Question 2, we will create new techniques,
and that these techniques could be useful to establish new lower bounds.
A point of view on the encoding of LCLs. As mentioned earlier, the study of LCLs
now plays a key role in our understanding of locality in distributed computing. By
asking Question 1, we are basically asking about how we express such problems.
If it is possible to use fewer bits than the obvious encoding to certify that a
solution exists, what does this tell us? If we cannot “compress” the encoding,
can we say that it is the core of the problem? Such consideration could have
an impact on techniques that heavily rely on precise encodings, such as round
elimination [22] and local conflict colorings [19,14].
Beyond the constant regime. Up to now, we have considered the question of
k-coloring with constant k, thus Question 2was about the constant regime per
se. But actually, one could let kdepend on n, and the question is meaningful for
labels of size up to Θ(log n) (which corresponds to coloring with Θ(n) colors).
Hence, we are not only playing with constants when studying Question 2in the
general case. Note that if we could show that k-coloring requires labels of size
Θ(log k) for all the range of k, then we would also have fairly natural problems
strictly between constant and logarithmic, which would be new and interesting
in itself.
A candidate for disproving the trade-off conjecture. One of the remaining im-
portant open questions in local certification is the following, which we will call
the trade-off conjecture.
Question 3. Suppose that there exists a local certification with labels of size
f(n) for some property, where every node would check its radius at distance 1.
Is it true that there always exists a certification with labels of size O(f(n)/t) if
we allow the nodes to see at distance tin the graph?
This question and variants of it were raised in [21,10,13]. In these papers,
the authors basically prove (among other results) that the answer is positive
if the certification uses only spanning trees and a uniform certification (giving
the same information to every node). Since these are the main tools used for
certification in the logarithmic and polynomial regimes, the constant regime
(and its extension beyond constant, discussed above) seems to be the place to
find potential counterexamples. Note that for the constant regime, one could
try to disprove the conjecture with αf (n)/t, for some given constant α. And it
seems like a reasonable approach, since it is difficult to imagine how the trade-
off conjecture could be true for coloring-like problems: even if the nodes can see
6 Ard´evol Mart´ınez, Caoduro, Feuilloley, Narboni, Pournajafi, Raymond
further, how can we save bits to certify colorability? In order to prove such a
counterexample, we first need to have proper lower bounds for distance 1, that
is, to answer Question 2.
1.3 Our results and techniques
In this paper, we give the first non-trivial lower bound for the certification
of k-colorability. This is the first step of a research direction that we hope to be
successful, and in terms of result it is a small step, in the sense that our answer
to Question 2is restricted in several ways. The first restriction we make is that
instead of proving a log klower bound, we prove that it is not possible to certify
k-colorability for any k≥3, if we use only one bit (that is, only two different
labels). We will call this a binary certification. The second restriction is that
we take a model that is not the most powerful one. In our result, a vertex has
access to the following information: its identifier, its label, and the multiset of a
of its neighbors. That is, a node cannot see further than its direct neighbors and
cannot access the identifiers of its neighbors (which corresponds to the original
proof-labeling scheme model [18], but not to the generalizations, such as locally
checkable proofs [16]), and there is no port number.
To prove this result, we introduce a new technique. Similarly to the cut-and-
plug technique mentioned earlier, we reason about one or several yes-instances
and prove that we can craft a no-instance where the vertices would accept. But
the reasoning is different, since counting arguments based on the pigeon-hole
principle applied to the certificates can only lead to Ω(log n) lower bounds. First,
we define the notion of score for a neighborhood and prove that if two vertices
have been given different labels but have the same score, then we can build a
no-instance that is accepted. Then we prove that this necessarily happens in
some well-chosen graphs, thanks to a series of local symmetry arguments, and a
global parity argument.
We complement this main result by proving that in some cases (namely
distance-2 3-colorability) one can actually go below the size of the natural encod-
ing. As we will see, this happens because graphs that are distance-2 3-colorable
have a very specific shape. This illustrates why establishing lower bounds for
such problems is not so easy: the fact that the graph can (or cannot) be colored
with a given number of colors implies that it has a given structure, and this
structure could in theory be used in the certification to compress the natural
log k-bit encoding.
2 Models and definitions
We denote by Nthe set of non-negative integers, and by |A|the cardinality
of a set A. All graphs in this paper are simple and connected. The vertex-set
and the edge-set of a graph Gare denoted by V(G) and E(G), respectively.
The closed neighborhood of a vertex v∈V(G), denoted by N[v], is defined by
N[v] = N(v)∪ {v}where N(v) = {u∈V(G) : uv ∈E(G)}is the neighborhood
Lower bound for constant-size local certification 7
of v. We denote the complete graph on nvertices by Kn. A proper k-coloring of
the vertex-set of a graph Gis a function φ:V(G)→ {1,2,...,k}such that if
xy ∈E(G), then φ(x)6=φ(y). In other words, it is an assignment of colors to the
vertices of Gusing at most kcolors, such that the endpoint of every edge receive
different colors. We say that Gis k-colorable if it admits a proper k-coloring.
Let f:N→N∪ {∞} be a function. We say that a graph Gon nvertices
is equipped with an identifier assignment of range f(n) if every vertex is given
an integer in [1, f (n)] (its identifier, or ID for short) such that no two vertices
of the graph are given the same number. Typically, f(n) is some polynomial of
n, and in this paper it has to be at least n3(but we did not try to optimize this
parameter).
Acertificate assignment of size sof a graph Gis a labeling of the vertices
of Gwith strings of length s, that is, a function ℓ:V(G)→ {0,1}s. A binary
certificate assignment is a certificate assignment with s= 1.
As hinted earlier, there are many variants for the definition of local certifi-
cation. An important aspect is the type of algorithm that the node run. This is
a local algorithm, in the sense that the nodes can see only a neighborhood in
the graph, but this neighborhood can be at distance 1, constant, non-constant
etc. Another important aspect is the symmetry-breaking hypothesis: whether
there are identifiers, whether the nodes can see the identifiers of their neighbors,
whether they can distinguish these neighbors, etc. In this paper, we use the
following notion.
Definition 1. Alocal decision algorithm is an algorithm that runs on every
vertex of a graph. It takes as input the identifier of the node, the certificate of
the node, and the multiset of certificates of its neighbors, and outputs a decision,
accept or reject.
Definition 2. Fix a function f:N→N∪ {∞}. A proof-labeling scheme of size
sfor a property Pis a local decision algorithm Asuch that the following holds:
for every graph Gand every identifier assignment of range f(|V(G)|)of Gthere
exists a certificate assignment of size sof Gsuch that Aaccepts on every vertex
in V(G), if and only if, the graph Ghas property P.
Notice that the proof-labeling scheme depends on the chosen function f.
In the proofs, as a first step, we will prove the result in a weaker anonymous
model.
Definition 3. An anonymous proof-labeling scheme is the same as a proof-
labeling scheme, but the graphs are not equipped with identifiers (or equivalently,
the outcome of the local decision algorithm is invariant by a change of the iden-
tifiers).
For a graph Gwith identifier id and labeling ℓ, the view of a vertex vin
(G, id, ℓ) in the proof-labeling scheme is a tuple (Mv,id(v)) where Mvis the
multiset
{(ℓ(u),id(u)) : u∈N(v)}.
In the anonymous case, the view of a vertex vis only the multiset defined above.
8 Ard´evol Mart´ınez, Caoduro, Feuilloley, Narboni, Pournajafi, Raymond
3k-colorability does not have a binary certification
This section contains our main contribution. We prove that k-colorability
does not have a binary certification when k≥3. Recall that for k= 2, k-
colorabily indeed has a binary certification (take the colors as certificates).
3.1 Indistinguishability setting
Let us first clarify the proof strategy with Lemma 1. It is a classic strategy,
that we detail for completeness.
Lemma 1. Let sbe a positive integer, f:N→N∪ {∞} be a function, and
Λ⊆Nbe a set of indices. If for every i∈Λthere exist a connected graph Gi
with identifier idi:V(Gi)→[1, f (|V(Gi)|)], and there exists a connected graph
Hsuch that
1. Giis k-colorable for every i∈Λ,
2. His not k-colorable,
3. for every set of labelings {ℓi}i∈Λ, where ℓi:V(Gi)→ {0,1}sis a labeling of
size sof Gi, there exists a labeling ℓ:V(H)→ {0,1}sof size sof Hand an
identifier id : V(H)→[1, f (|V(H)|)] such that for every view in (H, id, ℓ)
there exists i∈Λsuch that the view is the same as some view in (Gi,idi, ℓi),
then k-colorability cannot be certified by certificates of size s.
In case Gand Hdo not have identifiers, the same holds with removing the
identifier functions and ffrom the statement of the Lemma.
Proof. Suppose there exists a local certification of size sfor k-colorability. Then,
in particular, for every i∈Λ, there exists a labeling ℓifor the graph Gisuch that
the verifier algorithm accepts on every vertex. For this set of labelings {ℓi}i∈Λ,
consider the labeling ℓand the identifier assignment id of Hdescribed in item (3)
of the Lemma. The verifier algorithm accepts on every vertex of (H, id, ℓ) because
its view is the same as a view in (Gi,idi, ℓi) for some i∈Λ. This contradicts the
fact that His not k-colorable. ⊓⊔
3.2 Notion of score
Let ℓbe a binary labeling of a graph G, and let v∈V(G). The score of vin ℓ,
denoted by scoreℓ(v) or score(v) if there is no confusion, is defined as follows:
scoreℓ(v) = |{u∈N[v] : ℓ(u) = 1}|.
Given a k-regular graph Gand a binary labeling ℓof G, the score matrix
of (G, ℓ) is a 2 ×(k+ 2) matrix Swith rows labeled with 0 and 1 and columns
labeled from 0 to k+ 1. Let Si,j denote the (i, j ) element of S. We set
S1,0=S0,k+1 = 1,
and for i= 0,1, j= 0,1,...,k+ 1, and (i, j)6= (1,0),(0, k + 1) we set
Si,j =|{v∈V(G) : ℓ(v) = i, score(v) = j}|.
Lower bound for constant-size local certification 9
3.3 Our graph construction and its properties
Fix an integer k≥3. We build a graph as follows: take the disjoint union of
three copies of Kk+1 . For i= 1,2,3, let aiand bibe two distinct vertices in the
i-th copy. Then, remove the edges a1b1,a2b2, and a3b3from the graph, and add
the edges b1a2,b2a3, and b3a1to it. We denote the resulting graph by Nk. See
Figure 1. In the figure, each set Ct,t∈ {1,2,3}induces a Kk−1in the graph
and atand btare complete to Ct, i.e. every vertex of Ctis connected by an edge
to atand to bt.
a2
b1
a1
b3a3
b2
C2
C1
C3
Fig. 1. The graph Nk.
Lemma 2. For every k≥3, the graph Nkis k-colorable.
Proof. For every t∈ {1,2,3}, color atand btwith color tand color the k−1
vertices in Ctwith the k−1 colors {1,2,...,k}\{t}. It is easy to check that
this is a proper coloring of the vertex-set of Nkand as k≥3 = t, we have used
exactly the kcolors {1,2,...,k}.⊓⊔
Lemma 3. Assume k≥3is an integer and set G=Nk. If ℓ:V(G)→ {0,1}
is a binary labeling of G, and Sis the score matrix of (G, ℓ), then there exists
an integer j∈ {0,1,...,k+ 1}such that S0,j S1,j 6= 0.
Proof. If there exists a vertex with label 0 whose neighbors all have label 0 as
well, then S0,0= 1, and since by definition, S1,0= 1, choosing j= 0 gives us the
required result. Similarly, if there exists a vertex with label 1 whose neighbors
all have label 1 as well, then S1,k+1 = 1, and as S1,k+1 = 1, choosing j=k+ 1
gives us the required result. Hence, from now on, we may assume that
there exists no vertex in Gthat has the same label as all its neighbors. (1)
10 Ard´evol Mart´ınez, Caoduro, Feuilloley, Narboni, Pournajafi, Raymond
For t∈ {1,2,3}, set Ct=N(at)∩N(bt). Notice that Ctinduces a clique of
size k−1 in G. For all t∈ {1,2,3}, for all u, v ∈Ct, we have N[u] = N[v], thus
score(u) = score(v). If two distinct vertices uand vof Cthave different labels,
then by choosing j= score(v), we have S0,jS1,j 6= 0. So, we may assume that
for every t∈ {1,2,3},
all the vertices of Cthave the same label. (2)
Thanks to (2), for the rest of this proof and by abuse of notion, we use the term
the label of Ctfor referring to the common label of the vertices of Ct, and we
denote it by ℓ(Ct). Also, notice that by (1), if ℓ(Ct) = i, where i∈ {0,1}, then
at least one of the vertices atand btmust receive the label 1 −i. Thus, for every
t∈ {1,2,3},
at least one of atand bthas a label different from the one of Ct. (3)
Since there are three indices {1,2,3}, but only two labels {0,1}, there exist
t, t′∈ {1,2,3}such that t6=t′and ℓ(at) = ℓ(at′). By symmetry, we may
assume ℓ(a1) = ℓ(a2) = ˆ
ifor some ˆ
i∈ {0,1}. Notice that for every u∈C1,
N[u]\ {a1}=N[b1]\ {a2}, so score(u) = score(b1). Thus if the label of C1
is different from the label of b1, then choosing j= score(b1) completes the
proof. So, we may assume that ℓ(C1) = ℓ(b1). Therefore, by (3), we must have
ℓ(C1)6=ℓ(a1). Consequently, ℓ(b1) = ℓ(C1) = 1 −ˆ
i.
Now, if ℓ(b3) = 1 −ˆ
i, then for every u∈C1we have score(a1) = score(u).
And because ℓ(C1)6=ℓ(a1), choosing j= score(a1) completes the proof. Hence
we assume ℓ(b3) = ˆ
i. Now, because of (1), there must be a vertex uin the
neighborhood of b3with label 1 −ˆ
i. As we already have ℓ(a3) = ˆ
i, we must have
u∈C3, and therefore by (2), ℓ(C3) = 1 −ˆ
i.
Moreover, if ℓ(C2) = 1 −ˆ
i, then we have score(a2) = score(b1), and as a2
and b1have different labels, choosing j= score(a2) completes the proof. So, we
also assume that ℓ(C2) = ˆ
i. Thus ℓ(C2) = ℓ(a2) = ˆ
i. Therefore, by (3), we must
have ℓ(b2) = 1 −ˆ
i. Now, notice that the neighbors of a3all have label 1 −ˆ
i, thus
by (1), we must have ℓ(a3) = ˆ
i.
The labels of vertices of G, with all the assumptions so far, are as shown in
Figure 2.
To conclude, consider a1and a vertex u∈C3. In their closed neighbor-
hoods, they both have twice the label ˆ
iand k−1 times the label 1 −ˆ
i, so
score(a3) = score(u). Moreover, they have different labels. So, by choosing
j= score(a3) we have the required result. ⊓⊔
3.4 Anonymous case
Theorem 1. For every k≥3,k-colorability is not certifiable by binary certifi-
cates in the anonymous model.
Proof. Let Gbe the graph Nkand Hbe a complete graph on k+ 1 vertices.
Let ℓbe a binary labeling of G, and let Sbe the score matrix of (G, ℓ). Since G
Lower bound for constant-size local certification 11
ℓ(a2) = ˆ
i
ℓ(b1) = 1 −ˆ
i
ℓ(a1) = ˆ
i
ℓ(b3) = ˆ
i ℓ(a3) = ˆ
i
ℓ(b2) = 1 −ˆ
i
C2
C1
C3
ℓ(C2) = ˆ
iℓ(C1) = 1 −ˆ
i
ℓ(C3) = 1 −ˆ
i
Fig. 2. The labels at the end of the proof of Lemma 3.
is k-colorable and His not, by Lemma 1, to prove the theorem, it is enough to
find a binary labeling ℓ′:V(H)→ {0,1}of Hsuch that every view in (H, ℓ′) is
a view in (G, ℓ).
By Lemma 3, there exists j∈ {0,1,...,k + 1}such that S0,j and S1,j are
non-zero, meaning that:
–if j= 0, then there exists a vertex u∈V(G) such that ℓ(u) = 0 and all its
neighbors have label 0,
–if j=k+ 1, then there exists a vertex v∈V(G) such that ℓ(v) = 1 and all
its neighbors have label 1, or,
–if 0 < j < k + 1, then there exist two distinct vertices u, v ∈V(G) such that
ℓ(u) = 0, ℓ(v) = 1, and there are jvertices of label 1 in N(u), and j−1
vertices of label 1 in N(v).
Let V(H) = U∪Vsuch that |U|=k+ 1 −jand |V|=j. If j∈ {0, k + 1},
then Uor Vis an empty set. Notice that the condition on the cardinalities of U
and Vimplies that U∩V=∅. Define:
ℓ′(w) = (0w∈U
1w∈V
Notice that if j= 0 or j=k+ 1, then ℓ′(·) = 0 or ℓ′(·) = 1 respectively.
The view of each vertex of Uin His the same as the view of uin Gand the
view of each vertex of Vin His the same as the view of vin G. So, every view
in His the same as some view in G.⊓⊔
12 Ard´evol Mart´ınez, Caoduro, Feuilloley, Narboni, Pournajafi, Raymond
3.5 Extension to identifiers
Theorem 2. For every k≥3,k-colorability is not certifiable by binary certifi-
cates in the proof-labeling scheme model when the range of the identifiers for a
graph on nvertices is f(n) = n3+ 3n.
Proof. Set K= (k+ 1)2+ 1 and Λ={1,2,...,K}.
Let G1, G2, . . . , GKbe Kconnected graphs each isomorphic to Nk. Notice
that |V(Gi)|= 3k+3. For i∈Λ, consider an identifier idi:V(Gi)→[1, f(|V(Gi)|)]
of Gisuch that the vertices of Gireceive ID’s in the range [(i−1)(3k+3)+1, i(3k+3)].
Notice that
(i−1)(3k+ 3) + 1 ≥(1 −1)(3k+ 3) + 1 = 1,
and
i(3k+ 3) ≤K(3k+ 3)
=(k+ 1)2+ 1(3k+ 3)
=(3k+ 3)3
27 + (3k+ 3)
≤(3k+ 3)3+ 3(3k+ 3)
=f(3k+ 3) = f(|V(Gi)|).
So, [(i−1)(3k+ 3) + 1, i(3k+ 3)] ⊆[1, f (|V(Gi)|)]. Thus the identifiers idiexist.
Let Hbe a complete graph on k+ 1 vertices. Notice that each Giis k-
colorable and His not. Hence, by Lemma 1, to prove the theorem, it is enough to
define an identifier id : V(H)→[1, f (|V(H)|)] and a binary labeling assignment
ℓ:V(H)→ {0,1}of Hsuch that every view in (H, id, ℓ) is the same as a view
in (Gi,idi, ℓi) for some i∈Λ.
Let S(i)be the score matrix of (Gi, ℓi). By Lemma 3, for every i∈Λ, there
exists j∈ {0,1, . . . , k + 1}such that S(i)
0,j S(i)
1,j 6= 0. Therefore, by the pigeonhole
principle, there exists an integer j∈ {0,1,...,k+ 1}and a subset Λ0of Λsuch
that |Λ0| ≥ k+ 1 and for all i∈Λ0, we have S(i)
0,j S(i)
1,j 6= 0.
Notice that j∈ {0,1,...,k+ 1}, hence
|Λ0| ≥ k+ 1 ≥k+ 1 −jand |Λ0| ≥ k+ 1 ≥j.
Thus, we can find jdistinct vertices v1, v2,...,vj∈Si∈Λ0V(Gi) with label 1
and score j, and k+ 1 −jdistinct vertices vj+1 , vj+2 ,...,vk+1 ∈Si∈Λ0V(Gi)
with label 0 and score j. Notice that in case that j= 0 (resp. j=k+ 1), then
the first (resp. the second) set of vertices in empty.
Now, assume V(H) = {u1, u2,...,uk+1}.
First, define an identifier id : V(H)→[1, f (|V(H)|)] as follows:
id(ut) = id(vt) for every t∈ {1,2,...,k+ 1}.
Notice that by this definition, for all t, we have:
id(ut)∈[
i∈Λ
[(i−1)(3k+ 3) + 1, i(3k+ 3)] = [1, K (3k+ 3)].
Lower bound for constant-size local certification 13
On the other hand:
K(3k+ 3) = (3k+ 3)3
27 + (3k+ 3)
= (k+ 1)3+ 3(k+ 1)
=f(k+ 1) = f(|V(H)|),
and therefore the image of id is a subset of [1, f (|V(H)|)].
Second, define a binary labeling ℓ:V(H)→ {0,1}as follows:
ℓ(ut) = (1t≤j
0j+ 1 ≤t.
Notice that if j= 0 or j=k+ 1, then ℓ′(·) = 0 or ℓ′(·) = 1 respectively.
Thus, for every t, 1 ≤t≤k+ 1, the view of vertex utin (H, id, ℓ) is exactly
the view of vertex vtsome of the Gi’s, i∈Λ0.⊓⊔
4 Going below ⌈log k⌉
In this section, we illustrate that in some cases one can go below the natural
upper bound. More precisely, we exhibit an LCL whose natural encoding uses k
different labels, and for which one can find a certification that a solution exists
with strictly less than klabels. The problem we use is distance-2 3-coloring,
which is the same as 3-coloring except that even having two nodes at distance 2
colored the same is forbidden. The natural encoding consists in giving the colors
(that are between 1 and 3) to the vertices, but we show that one can actually
certify distance-2 3-colorability with just two different certificates.
Lemma 4. A connected graph Gis distance-2 3-colorable if and only if it is a
a cycle of length 0 mod 3 or a path.
Proof. First, let Gbe a connected distance-2 3-colorable graph. Each vertex and
its neighbors form a set of vertices that are pairwise at distance at most 2, so
they should all have different colors. Since there are only three available colors,
Ghas maximum degree at most 2. Thus, Gis a path or a cycle.
If Gis a path, then we are done, so assume Gis a cycle. Assume that the
cycle is v1, v2, . . . , vk, v1. Consider a proper distance-2 3-coloring of Gwith colors
{0,1,2}. Notice that if a vertex vof Ghas color imod 3, then necessarily, its
two neighbors have colors (i−1) mod 3 and (i+ 1) mod 3. Without loss of
generality, assume that the color of v1is 1 and the color of v2is 2. Let [t] denote
the remainder of tdivided by 3. So, [t]∈ {0,1,2}. We prove by induction on
tthat vthas color [t] for every t∈ {1,2,...,k}. This holds by assumption for
t= 1,2. Now let t≥3 and assume that vt′has color [t′] for every t′< t. By
the induction hypothesis, the color of vt−1is [t−1], hence the colors of its two
neighbors, namely vt−2and vt, are [t] and [t+1]. Again, by induction hypothesis,
the color of vt−2is [t−2] = [t+ 1]. Thus the color of vtmust be [t], proving the
14 Ard´evol Mart´ınez, Caoduro, Feuilloley, Narboni, Pournajafi, Raymond
statement. Therefore, the color of vkis [k]. Finally, notice that because v1has
color 1, and v2has color 2, the color of vkmust be 0. Therefore [k] = 0, meaning
that the length of the cycle, kis equal to 0 mod 3.
Second, in both cycles of length 0 mod 3 and in paths, it is possible to find
a proper distance-2 3-coloring, by simply choosing the color of one vertex and
propagating the constraints. ⊓⊔
Theorem 3. One can certify distance-2 3-colorable graphs with a binary certi-
fication.
Proof. We prove that paths and cycles of length 0 mod 3 can be recognized with
a binary certification, by Lemma 4, this will give the desired result.
The idea of the certification is to give certificates of the form:
...1,0,0,1,0,0,1,0,0...
Let us describe first the verifier algorithm of a vertex v:
1. If the degree is strictly more than 2, reject,
2. If the degree is 2, accept if and only if, score(v) = 1, that is, among vand
its neighbors, exactly one has a 1 as certificate.
3. If the degree is 1, accept.
Because of Item 1, only paths and cycles can be accepted. For paths, the following
labeling makes every vertex accept: choose one endpoint u, and give label 1 to a
vertex v, if and only if, the distance from uto vis 0 mod 3. For cycles of length 0
mod 3, the following labeling makes every vertex accept: choose an orientation
and an arbitrary vertex u, and again give label 1 if and only the distance from
uto vin the sense of the orientation is 0 mod 3. It is easy to see that in cycles
of length different from 0 mod 3, at least one vertex will reject with Item 2.
Hence, we have a proper certification that a graph is a path or a cycle of
length 0 mod 3, with only two different labels (that is, just one bit). ⊓⊔
5 Challenges and open questions
We have proved the one bit is not enough for certifying k-colorability for
k≥3. We conjecture that the answer to Question 2is negative, that is, that
⌈log k⌉is the optimal certification size for k-colorability, and this is the very first
step.
There are several challenges to overcome before one can hope to prove the
conjecture. First, it would be nice to have a more general model, where the
nodes can see their neighbors’ identifiers, or at least distinguish them, and even
better, where the nodes can see at a larger distance. At least the first step in
this direction might work by using some Ramsey argument, but then losing the
upper bound on the identifier range. Second, it might be necessary to use graphs
of large chromatic number that do not have large cliques as subgraph, and these
have complicated structures.
Lower bound for constant-size local certification 15
A different direction is to understand other LCLs, and to try to see which
ones have a certification that is more efficient than the natural encoding (such
as distance-2 3-coloring) and which do not. Maybe in this direction, one could
characterize exactly which properties can be certified with one bit.
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