The MAX-CUT of sparse random graphs
Hervé Daudé∗
Conrado Martı́nez†
Vonjy Rasendrahasina‡
1
Abstract
Vlady Ravelomanana§
Introduction
1.1 Context and previous results Let G = (V, E)
be a graph. A k-cut of G is a partition of its vertex
set V into V1 , V2 , · · · , Vk . The size of a cut is the
number of edges connecting vertices in Vi and Vj for
i 6= j. The MAX-k-CUT problem asks for an optimal
cut that is a cut of maximum size. The particular
case k = 2 is denoted MAX-CUT and has received
the most attention. The last decades have seen a
growth of interest in MAX-CUT. Solving exactly MAXCUT is NP-hard and Hastad [14] showed that it is not
approximable to within a 16/17+ε factor of the optimal
cut unless P = NP.
When random graphs G(n, m) (see Janson, Luczak
and Ruciński [16]) are considered this maximisation
problem exhibits a transition as the density of graphs
2m/n increases above 1. More specifically, Coppersmith, Hajiaghayi, Gamarnik and Sorkin [4] studied the
expectation of MAX-CUT by means of method of moments, algorithmic analysis and martingale arguments,
showing that DistBip (G) jumps from Θ(1) to Θ(n).
Scott and Sorkin [20] proved that random instances of
µn2/3
MAX-CUT can be solved in expected linear time as long
2 ) we show that DistBip (G(n, m)) is a.a.s about
as G(n, m = n2 + O(n2/3 )). For dense cases i.e. large
(2m−n)3
1
1
+ 12 log n− 4 log µ for any 1 µ ≤ O(n1/3−ε ). densities, probabilistic
6n2
and algorithmic techniques are
preferred too, see for instance [11] and [3].
In the last eighties, seminal works of Flajolet, Janson, Knuth, Luczak, Pittel on random graphs ([9] and
[15]) opened a new road to reach very precise informations on various parameter of random graphs G(n, m).
Using powerful analytic combinatorics techniques (see
[15]), these authors gave deep and fruitful results on
the cyclic structure of random graphs in the so-called
scaling window, specifically when 2m − n n. More
recently these techniques has been successfully applied
[6], [18] to get new insights on the phase transition associated to random 2-XOR formula [5]. Recall that a
2-XOR formula is a conjunction of Boolean equations
∗ LATP - UMR CNRS 6622, Université de Provence. 13453
or clauses) of the form x ⊕ y = 0 or x ⊕ y = 1. Then, in
Marseille Cedex 13, France. Email: daude-at-cmi.univ-mrs.fr
† Dept.
Llenguatges i Sistemes Informàtics, Universitat the context of maximisation Rasendrahasina and RavPolitècnica de Catalunya, E-08034 Barcelona, Spain. Email: elomanana [19]. obtained first and precise information
conrado-at-lsi.upc.es
on the distribution of the maximum number of clauses
‡ LIPN - UMR CNRS 7030, Université de Paris Nord. 93430
which can be satisfied by any assignment of the variables
Villetaneuse, France. Email: vonjy-at-lipn.univ-paris13.fr
§ LIAFA - UMR CNRS 7089, Université Denis Diderot. 75205
in a 2-XOR formula, namely on he MAX-2-XORSAT
A k-cut of a graph G = (V, E) is a partition of
its vertex set into k parts; the size of the k-cut is
the number of edges with endpoints in distinct parts.
MAX-k-CUT is the optimization problem of finding
a k-cut of maximal size and the case where k =
2 (often called MAX-CUT) has attracted a lot of
attention from the research community. MAX-CUT
—more generally, MAX-k-CUT— is NP-hard and it
appears in many applications under various disguises.
In this paper, we consider the MAX-CUT problem
on random connected graphs C(n, m) and on ErdősRényi random graphs G(n, m). More specifically, we
consider the distance from bipartiteness of a graph
G = (V, E), the minimum number of edge deletions
needed to turn it into a bipartite graph. If we denote
this distance DistBip (G), the size of the MAX-CUT
of a graph G = (V, E) is clearly given by |E| −
DistBip (G). Fix ε > 0. For random connected
graphs, we prove that asymptotically almost surely
whenever m = n +
(a.a.s) DistBip (C(n, m)) ∼ m−n
4
O(n1−ε ). For sparse random graphs G(n, m = n2 +
Paris Cedex 13, France. Email: vlad-at-liafa.jussieu.fr
265
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problem. Observe that if we restrict the above clauses
to the form x ⊕ y = 1, the associated maximisation
problem is equivalent to MAX-CUT. In this work we
give new and precise information on the distribution of
the minimum number of edges that has to be cut in order to produce a bipartite graph from a random graph
G(n, m). Based on analytic tools mentionned above,
we thus consider the MAX-CUT problem in the scaling
window 2m − n n.
3. For m =
(1.4)
Theorem 1.1. Let C(n, m) be a random connected
graph built with n vertices and m = n + ` edges. For
any fixed real number ε ∈ (0, 1) if m = n + O(n1−ε ) but
` = m − n → ∞ as n → ∞, we have
i
h
P 4` − O(`1−ε/4 ) ≤ DistBip (C) ≤ 4` + O log` `
DistBip (G) −
q
log n
12
D
−→ N (0, 1) .
4. For m = n2 (1 + µn−1/3 ), with 1 µ = O(n1/3−ε )
for any ε > 0 fixed,
(1.5)
6 DistBip (G1 ) D
−→ 1,
µ3
where G1 denotes the complex part1 of G. Moreover, if log n ≤ µ ≤ n1/12 then
(1.6)
DistBip (G2 ) −
q
log n
12 −
log n
12
+
log µ
4
D
−→ N (0, 1).
log µ
4
where G2 = G \ G1 is the non-complex part (trees
and unicycles) of the graph G.
MAX-CUT(G) = |E| − DistBip (G) .
Pittel and Yeum [18] have quantified the probabilities that random graphs are 2-colorable at their phase
transition; the current paper deals with the related hard
optimization problem MAX-CUT in the same range.
More precisely, we consider here the MAX-CUT
of random connected graphs C(n, m) and of ErdősRényi random graphs G(n, m) for values of m in the
same ranges as Pittel and Yeum. Our results rely on
enumerative and analytic combinatorics [10] and we
obtain the following two theorems, which give a very
precise characterization of the asymptotic behavior of
DistBip (C(n, m)) and DistBip (G(n, m)).
+ Θ(n2/3 ),
log n
12
1.2 Our contribution and results Rather than investigating the MAX-CUT proper, we have analyzed the
(edit) distance of a graph G = (V, E) from bipartiteness,
denoted DistBip (G). This distance is the minimum
number of edge deletions needed to turn it into a bipartite graph. It is immediate from the definition of
MAX-CUT and DistBip (G) that for any given graph
G
(1.1)
n
2
Remark. Observe that (1.2)–(1.6) show that the random variable DistBip (G) has a “continuous” behaviour during the evolution of random graphs in the
subcritical phase. These results are more accurate and
complete those given by Theorems 19 and 22 in [4, Section 8.2].
The rest of the paper focuses in the proof of the two
theorems above. This extended abstract omits many of
the technical details and thus it only provides a highlevel description of these proofs.
2
Proof of Theorem 1.1
We start with a few definitions that we will need in the
sequel.
Definition 1. An `-component is a connected graph
with
n vertices and n + ` edges (` ≥ −1). The excess
1−ε/2
)
≥ 1 − e−O(`
.
of a connected component is the number of its edges
minus the number of its vertices (thus an `-component
Theorem 1.2. Let G = G(n, m) be a random uniform
has excess `).
graph with n vertices and m edges. Then the following
holds
Definition 2. A cactus (also known as an Husimi
1. For any constant c, 0 < c < 12 if m = c.n then
tree [13]) is a connected graph in which any two cycles
(1.2)
have at most one vertex in common.
1
1 + 2c
D
DistBip (G) −→ Poisson
log
−c .
Definition 3. A smooth graph is a graph without
4
1 − 2c
vertices of degree one.
2. For m = n2 (1 − µn−1/3 ) where 1 µ = o(n1/3 ),
(1.3)
n
DistBip (G) − log
12 +
q
log n
log µ
12 − 4
log µ
4
D
−→ N (0, 1) .
1 A complex component of a graph G is a connected component
of G of excess ≥ 1 (see the definitions in next Section), that is, a
component which is not a tree nor an unicycle. The complex part
of a graph G is the set of all complex components of G.
266
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Our proof makes heavy use of exponential gener- obtain exactly s fundamental cycles of odd length in the
ating functions or EGFs (see for instance Flajolet and new graph G0 . These
map each bipartite
P`+1 constructions
`+1
Sedgewick [10] or Harary and Palmer [12]).
`-component to s=0 `+1
=
2
`-components.
s
In the rest of the paper we will use T (z) to denote
the EGF of rooted Cayley trees, W` (z) for the EGF of
4
4
7
7
`-components (see [21, 1, 17]), Bi,` (z) for the EGF of
`-components such that their edit distance to bipartite1
1
3
2
3
2
ness is i, Qi,` (z) for the EGF of `-components graphs
which are cacti with i simple cycles of odd length, and
Q` (z) for the EGF of connected cactus of excess `.
8
8
5
5
6
6
B/
A/
We will also use a tilde on top of an EGF to
4
4
indicate the EGF for the smooth counterparts: namely,
7
7
we denote by W̃` (resp. B̃i,` and Q̃i,` ) the EGF for
1
1
e2
e2
e1
2
2
3
3
the families of smooth graphs obtained by pruning, i.e.,
e1
c2
reducing recursively all vertices of degree 1, the families
c1
c2
c01
of graphs counted by W` (resp. Bi,`P
and Qi,` ).
8
8
zn
5
5
n
6
6
D/
C/
For any EGF A with A(z) =
n an n! , [z ]A(z)
th
denotes the n coefficient of the series A(z), that is,
4
7
[z n ]A(z) = an /n!. If A and B are two EGFs, we write
1
e1
2
3
A(z) B(z) (or simply A B) if there exists some n0
e2
such that for all n ≥ n0 we have [z n ]A(z) ≤ [z n ]B(z).
0
c1
Lemma 2.1. For all ` ≥ −1,
6
W` (z) 2`+1 B0,` (z).
Proof. To prove this lemma, we show that each bipartite component of excess ` can be associated to at least
2`+1 `-components. Let G be a bipartite `-component,
i.e., DistBip (G) = 0. Let T be a spanning tree of G.
Denote by ei , 1 ≤ i ≤ `+1, the sequence of (`+1) edges
in G \ T . The graph T ∪ ei has a unique fundamental
cycle ci and in graph terminology (see for instance [7])
the (` + 1) cycles ci form a basis of the cycle space associated to G. From the bipartite original graph G, we
build a graph G0 of the same size but with exactly s
fundamental cycles of odd length (for any s ∈ [0, ` + 1])
as follows. For each fundamental cycle ci corresponding to the edge ei = {i1 , i2 } ∈ G \ T , without loss of
generality, we can assume that i1 < i2 and give the
same orientation (i.e. i1 ← i2 ) to the cycle ci . For instance in the figure below in the cycle c1 (graph C/),
i2 = 3 and i1 = 2. Once the orientation of ci is fixed,
ci can be rewritten as [i1 → i2 → x1 → x2 · · · xj → i1 ]
where the x1 x2 · · · xj are the vertices following i2 according to the order of the orientation (for example
c1 = [i1 = 2 → i2 = 3 → x1 = 6 → x2 = 8 → i1 = 2] in
graph C/). Next, we modify the length of ci by removing the edge between i1 and i2 and by adding an edge
between i1 , x1 . Then, we have a new fundamental cycle
of odd length c0i = (i1 → x1 → x2 · · · xj → i1 ) in the
transformed graph G0 . More generally, we can transform as described above s cycles (with 0 ≤ s ≤ ` + 1)
among all the (`+1) fundamental cycles of G, in order to
c2
8
E/
5
Figure 1: Creating a connected graph with exactly 1
fundamental cycle of odd length (graphs D/ and E/)
from a bipartite component (graph A/). The graph B/
depicts an arbitrary spanning tree of the graph A/.
Similar results hold for cactus graphs.
Lemma 2.2. For all ` ≥ 0,
Q` (z) =
`+1
X
Qi,` (z) 2`+1 Q0,` (z) .
i=0
We also have bounds for the EGFs B̃s,` counting smooth
`-components at distance s from bipartiteness:
Lemma 2.3. For all ` ≥ 0 and all s ∈ [0, ` + 1]
2s X
`+1
(2.7) B̃s,` (z) z −s + · · · + z s B̃0,` (z) .
i
i=s
Proof. Denote by B̃s,` (s ∈ [0, ` + 1]) the family of
connected smooth graphs of excess ` and with edit
distance from bipartiteness equal to s. We want to
(s)
(s)
build a family B̃` such that B̃s,` ⊂ B̃` , obtained as
below from each element of B̃0,` by changing the parity
of the length of s Wright paths2 . Let C ∈ B̃0,` and recall
2A
Wright path in a smooth connected component is a simple
path such that all vertices in the path have degree 2, except the
endpoints which have degree ≥ 3.
267
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that it can be associated to its ` + 1 fundamental cycles
(cf. Lemma 2.1). Note that for any Wright path: 1)
either it belongs to exactly one fundamental cycle, or
2)it is shared by at least 2 fundamental cycles.
The case s = 1 is described below followed by
the general case. For s = 1 in the case 1), a Wright
path from C is chosen and the parity of its length is
modified by either deleting or inserting a vertex on this
path. In terms of the EGF if C(z) is the monomial
corresponding
−1
to C, such transformation is given by
`+1
z
+
z
C(z). Similarly in case 2), the length of
1
a Wright path shared by at least 2 cycles is modified by
the deletion or theinsertion of a vertex. In this last case,
the operator `+1
z −1 + z C(z) counts more graphs
2
than those we want to enumerate since the choices `+1
2
include all pairs of fundamental cycles (whether sharing
or not a path) and shared paths can be counted more
z2
z3
than twice. Let E(z) = 1−z
2 and ω(z) = 1−z 2 be the
EGFs of paths of respectively even and odd length, the
multiplication of z ±1 means that an even path becomes
an odd path and vice-versa by extending or reducing its
length
E(z) =
(1)
B̃`
z3
z2
deletion or insertion
←−−−−−−−−−−−→ ω(z) =
2
1−z
1 − z2
graphs using the enumerative and analytic approaches.
The lemma below follows their results (see [15, eq. (3.9),
(19.13) and (19.14)]) and it will be necessary for our
purposes. The details of the proof are based on saddlepoint methods (see Flajolet and Sedgewick [10]).
Lemma 2.5. For any functions a ≡ a(n) and b ≡ b(n),
let
τn (a, b) = n![z n ]
(2.8)
T (z)an (1
1
.
− T (z))bn
As n is large, if 0 ≤ a < b 1 but bn 1, then τn (a, b)
satisfies
n!
(1 − u0 )2
p
τn (a, b) = √
2πn (1 + a)(1 − u0 )2 + bu20
exp(nu0 )
1
× (a+1)n
1+O
,
bn
bn
u0
(1 − u0 )
where
(2.9)
u0 ≡ u0 (a, b) =
1
1
1p 2
a+ b+1−
a + 2 ab + b2 + 4 b.
2
2
2
The corollary below follows from well-known results
due to Wright [21, 22] and Lemma 2.5.
By the constructions above,
B̃1,` ⊂ Corollary 2.1. If s,
it followsthat
` ∈ N are such that 0 ≤ s ≤ `
1−ε
`+1
`+1
−1
and
1
`
=
O
n
then
+ 2
and B̃1,` (z) (z + z )
B̃0,` (z).
1
s
X
3/2 −1/2
[z n ]T (z)j W` (z)
)
≤ eO(` n
.
n ]W (z)
[z
`
j=−s
The general case is very similar to the case s = 1.
Starting from a bipartite graph C, the operator
2s X
`+1 i=s
|
i
z −s + z −(s−1) + · · · + z s−1 + z s
{z
}
Lemmas 2.1 and 2.3, together with the previous
corollary, are the key ingredients to get our next corollory.
Corollary 2.2. Let C be an `-component built with
applied to C(z) counts all the transformations of C, n vertices. For any fixed real number ε ∈ (0, 1), ` =
including the insertions and/or the suppressions of s O(n1−ε ) and for all s ≤ `+1 − O `1−ε/4 we have for
4
vertices changing the parity of s Wright paths in C.
large `
The details for the basic case ` = 1 are omitted in
1−ε/2
this extended abstract.
P (DistBip (C) = s) ≤ e−α `
,
Similar results hold for cactus graphs.
for some constant α > 0.
Lemma 2.4. For all ` ≥ 0 and all s ∈ [0, ` + 1],
`+1
Q̃s,` (z) z −s + · · · + z s Q̃0,` (z).
s
The following result tells us that with high probability
an `-component is at least ∼ 4` far from being bipartite.
Corollary 2.3. Let C an `-component of size n. For
any
fixed real number ε ∈ (0, 1), as 1 ` = O(n1−ε ) ,
Proof. The proof is similar to the one of the previous
lemma, but in this case s paths belonging to exactly s we have
1−ε/2
distinct cycles are chosen among the ` + 1 cycles.
`
1−ε/4
P DistBip (C) ≥ − O `
≥ 1 − e−α`
.
4
As shown by Janson, Knuth, Luczak and Pittel in [15]
tree polynomials are very useful when studying random for some constant α > 0.
268
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Proof. The proof is derived from Corollary 2.2. In fact, 3
Proof of Theorem 1.2
The proofs of (1.2), (1.3) and (1.4) are similar to
those of [19, equations 1,2 and 3 of Theorem 1] by
P (DistBip (C) = s) ≤
quantifying the r.v. counting the number of uncyclic
s=0
components where the cycle is of odd length (instead
`+1
− O `1−ε . of the cycle being of odd weight as in [19]) on random
`·P DistBip (C) =
4
Erdős-Rényi graphs G(n, m). In the ranges of (1.2),
Lemma 2.6. If DistBip (C) = s for an `-component (1.3) and (1.4), DistBip (all complex components) can
C, the component has a cactus with at most s cycles of be neglected. In [6, Theorem 3.2], it is shown that
the probability that a random graph built with n
odd length as a subgraph.
vertices and m edges has no complex components is
n2
Proof. Recall that an `−component C contains no cycle 1 − O
as n and m tend to ∞ but n −
(n−2m)3
of odd length iff it has no fundamental cycle of odd
2m n2/3 . Consequently, the total excess of the
length. If DistBip (C) = s, it has obviously at least s
graph is4 of order Op (n2 /(n − 2m)3 ). In the critical
fundamental cycles of odd length. Only s of these cycles
region m = n2 ± O(1)n2/3 , the excess of the complex
are disjoint3 . If at least s + 1 of these cycles are disjoint,
components is Op (1) (see e.g. Bollobás [2] and Janson,
they cannot be eliminated by removing at most s edges.
Luczak and Ruciński [16]). To prove (1.5), we use the
The following lemma tells us that in a cactus of excess fact that as the number of edges of a random graph G
reaches m = n2 + µ2 n2/3 with µ ≡ µ(n) 1, almost
`, roughly `/2 cycles are of odd length.
surely there is a unique giant component in G. Hence,
Lemma 2.7. For any fixed real number ε ∈ (0, 1), let ` we need to control the number of edge deletions in
such that ` = O(n1−ε ). For an `-component cactus C, order to obtain a bipartite component by the giant
we have
component. We know from the results of random graph
theory by Pittel and Wormald [17] that the excess of
`
`
− O `1−ε/4 ≤ DistBip (C) ≤ − O `1−ε/4
P
the a.s. unique giant component of G is Gaussian
2
2
(throughout the whole supercritical phase, viz. from
≥ 1 − exp −α`1−ε/2 ,
µ = o(n1/3 ) 1 to µ = εn1/3 ). Particularly, for any
fixed real number ε ∈ (0, 1) and µ = O(n1/3−ε ), the
for some constant α > 0.
expected excess of the giant component is ∼ 23 µ3 (see
also [15, Theorem 6]). Then, we use Theorem 1.1 to
Our last lemma in this section states that an `
quantify P DistBip (Giant component) ∼ µ3 /6 .
component is spanned by a cactus with 2` + O log` `
The proof of (1.6) is complete by characterizing
cycles with a probability at least 1 − e−O(`) .
the limiting distribution of the number of unicyclic
components with cycle of odd lentgh in a random graph
Lemma 2.8. Let c0 = 38 log 3 + 12 log 2 and let smax ≡
G ∈ G(n, m), for m in the corresponding range.
smax (n, `) the excess of a spanning cactus of an `component of size n. For any fixed real number ε ∈
4 Conclusion
(0, 1), any x > 0 and 1 ` = O(n1−ε ), we have
In this paper, we have studied the most probable
`
`
value of MAX-CUT in sparse instances of random
−2x`
P smax ≥ + (c0 + x)
≤O e
.
(connected) graphs. By means of enumerative and
2
log `
analytic combinatorics [10], we have shown how one
We are now almost ready to prove Theorem 1.1.
can quantify the size of the MAX-CUT as the graph
Proof. [Proof of Theorem 1.1] Lemma 2.8 states that is large. Our results apply to the sparse cases of both
a.a.s. an `-component is spanned by a cactus with at random connected graphs (Theorem 1.1) and random
most `/2 + (c0 + x) log` ` cycles for any x > 0. Then, by graphs (Theorem 1.2). They complete those given in
Coppersmith, Hajiaghayi, Gamarnik and Sorkin [4] and
Lemma 2.7, for such cactii DistBip (·) is smaller than
in Coja-Oghlan, Moore and Sanwalani [3] and give
`/4 + O (`/ log `), with high probability. This proves
that DistBip (C(n, m)) is smaller than `/4+O (`/ log `),
thus completing the result given by Corollary 2.3 and
4 Following Janson, Luczak and Rucinskı́ [16, p. 10], if X are
n
proving Theorem 1.1.
r. v. and a are real numbers we write X = O (a ) as n → ∞
(`+1)/4−O (`1−ε/2 )
X
n
3 We
say two cycles are disjoint iff they have no common edge.
n
p
n
if for every δ > 0 there exists constants Cδ and n0 such that
P(|Xn | ≤ Cδ an ) > 1 − δ for every n ≥ n0 .
269
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insight on the growth of the MAX-CUT around the References
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