Download 12. Probability

12. Probability
Colin McDiarmid
What are ‘typical’ properties of a graph? How can random methods yield
existence results for graphs? What will randomised graph algorithms do? What
have isoperimetric inequalities for graphs to do with probability? This chapter
gives an introductory taste of the fertile area of graphs and probability.
1
Introduction
Since Paul Erdős brought together the subjects of graph theory and probability
some forty years ago, the relationship has flowered marvellously. Here we attempt to
give something of the flavour of this still burgeoning area – no attempt is made at a
complete survey!
We discuss first some typical properties of random graphs and of algorithms on
random graphs; for example, what can we say about the chromatic number and about
methods to find good colourings? Then we see how to use random graphs in order to
obtain deterministic results, and continue along this line with the help of the Lovász
local lemma. The remaining two sections concern deterministic graphs. First we
consider randomised algorithms on graphs – that is, procedures that operate on a
deterministic graph and involve flipping coins. Finally we discuss the relationship
between results on concentration of measure and isoperimetric inequalities for graphs,
focussing on the n-cube Qn .
2
Random graphs: usual behaviour
There are several natural models of random graph: we
³ ´shall focus on two of these. The
random graph Gn,p has vertices v1 , . . . , vn , and the n2 possible edges occur independently, each with probability p. When p = 12 , this corresponds to sampling uniformly
from all labelled graphs on n vertices. The random graph Gn,m also has vertices
v1 , . . . , vn : each such graph with exactly³m´ edges is equally likely. Its behaviour is
very similar to that of Gn,p , with p = m/ n2 .
When we investigate random graphs, sometimes it is because we want to know
whether a property usually holds, or how well an algorithm usually works, and some-
1
times our aim is to prove a deterministic result. In this section we discuss the usual
behaviour.
We shall focus on four topics: colourings, Hamiltonian cycles, minimum spanning
trees and perfect graphs. For references not cited here, and for many further results,
see [2]. When considering colourings we shall use the model Gn,p with the edgeprobability p constant (not depending on n).
Vertex colouring
The natural place to start is with the independence (or stability) number α(Gn,p ).
We might guess that it should
usually be close to r, where r is such that the expected
³ ´
r
n
number of stable r-sets, r (1 − p)(2) , is about 1. Such an r = r(n) is asymptotically
equal to 2 log n/ log(1/(1 − p)), which equals 2 log2 n when p = 21 . The following result
shows that α is remarkably predictable.
Theorem 2.1 The independence number α satisfies α(Gn,p ) = r − 1 or r a.s.
Here ‘a.s.’ (almost surely) means that the statement holds with probability that tends
to 1 as n → ∞. The upper bound in the theorem is easy: it follows directly from
the ‘first moment method’ – that, if always X ≥ 0, then Pr(X ≥ 1) is at most
the expected value EX. The lower bound follows after some computations from the
‘second moment method’ – that if EX ≥ 0, then Pr(X > 0) ≥ (EX)2 /E(X 2 ), or a
similar inequality. Here the random variable X counts the number of stable k-sets in
Gn,p , where we take k as r + 1 or r − 1.
The chromatic number χ(G) is at least the number of vertices divided by the
stability number α(G). Hence, by Theorem 2.1, χ(Gn,p ) ≥ n/r a.s. It was conjectured
in 1975 that this bound is about the correct value, and in 1988 this was proved by
Bollobás [4]. In fact, we have the following rather precise result [21]:
Theorem 2.2
χ(Gn,p ) = n/(r + O(1)) a.s.
The key step in the proof is to establish ‘concentration of measure’. We find that a.s.
the random graph Gn,p has the property that every induced subgraph with at least
n/(log n)2 vertices contains a stable set of nearly r vertices; and so we can repeatedly
remove such a stable set and colour it. The last few vertices can each get a new colour.
The simple greedy or sequential colouring algorithmusually uses about twice as
many colours as necessary. It is not known whether any polynomial-time algorithm
usually achieves a ratio strictly less than 2.
The situation is quite different for another model of random graph. Consider a
fixed k, and suppose that the random graph Gn is obtained by sampling uniformly at
random from the graphs on vertices v1 , . . . , vn which have no subgraphKk+1 . Then Gn
is a.s. k-colourable, and there is a polynomial expected time algorithm which colours
Gn optimally – see Prömel and Steger [30] and the references therein.
2
Edge colouring
Vizing’s theorem says that the edge-chromatic number χ0 of a graph with maximum
degree ∆ is equal to ∆ (Class 1) or ∆ + 1 (Class 2). It is of great interest (and
NP-hard) to determine which case occurs for a given graph or class of graphs.
How common are Class 2 graphs? Almost all graphs Gn,p have a unique vertex of
maximum degree, and so are in Class 1. Thus Pr(χ0 = ∆ + 1) = o(1). In fact, this
probability decreases to 0 very quickly [11].
Theorem 2.3 As n → ∞,
1
1
n−( 2 +o(1))n ≤ Pr(χ0 = ∆ + 1) ≤ n−( 8 +o(1))n .
The lower bound here arises from regular graphs of odd order. The upper bound
arises from a polynomial-time algorithm that attempts to edge-colour the graph with
∆ colours, and only rarely fails on Gn,p . It works by repeatedly removing near-perfect
matchings. Very recently, Perkovic and Reed [28] have proposed an algorithm that
will edge-colour Gn,p optimally in polynomial expected time.
Total colouring
For the total chromatic number χ00 , we can follow an approach similar to the above.
It is clear that χ00 ≥ ∆ + 1. The well-known total colouring conjecture asserts that
χ00 = ∆ + 1 or ∆ + 2.
Theorem 2.4 As n → ∞,
1
Pr(χ00 > ∆ + 1) ≤ n−( 8 +o(1))n ;
and, for some constant c with 0 < c < 1,
2
Pr(χ00 > ∆ + 2) = o(cn ).
In the second inequality above, the bound on the probability is very small, and so
we obtain some support for the total colouring conjecture, but we would prefer the
bound to be zero!
Connectedness and Hamiltonian cycles
How many edges do we need to make a random graph be connected or be Hamiltonian?
The threshold for connectivity in Gn,m was given by Erdős and Rényi [10] in 1960 in
a seminal paper on random graphs. The proof involves estimating expected numbers
of small components.
Theorem 2.5 If m = 12 n(log n + cn ), and cn → c as n → ∞, then
Pr(Gn,m is connected) → e−e
3
−c
as n → ∞.
It follows that if cn → −∞ then the probability → 0, and if cn → ∞ then the
probability → 1.
Determining the threshold for the existence of a Hamiltonian cycle was a major
open problem for many years. The key idea turned out to be rotation-extension.
Suppose that we have a path of length k, with vertices v0 , v1 , . . . , vk . If v0 or vk
has a neighbour not in the path, then we can extend the path by simply adding such
a neighbour. Failing this, suppose that vk has a neighbour vi , where 0 ≤ i ≤ k − 2.
If i = 0 and there is an edge vi w joining the cycle v0 , v1 , . . . , vk , v0 to the rest of the
graph, then the path w, vi , vi+1 , . . . , vk , v0 , . . . , vk−1 has length k + 1. If i 6= 0, then we
perform a rotation and construct the path v0 , v1 , . . . , vi , vk , vk−1 , . . . , vi+1 ; this path
still has length k, but has a different endpoint vi+1 from which we can again look
for an extension. Using rotations and extensions Komlós and Szemerédi eventually
solved the threshold problem in 1983 – see for example [2].
Theorem 2.6 If m = 12 n(log n + log log n + cn ), and cn → c as n → ∞, then
Pr(Gn,m is Hamiltonian) → e−e
−c
as n → ∞.
There is in fact a polynomial-time algorithm, based on the rotation-extension idea,
that finds a Hamiltonian cycle with probability as in the theorem above – see Bollobás,
Fenner and Frieze [5]. It has recently been shown that, for any fixed r ≥ 3, a random
r-regular graph on n vertices is a.s. Hamiltonian – see Robinson and Wormald [31, 32].
The proofs are quite different, and involve a delicate use of the second moment method.
The probabilities in the last two theorems are in fact the limiting probabilities
that the minimum degree δ satisfies δ ≥ 1 and δ ≥ 2 respectively. Indeed there is a
beautifully precise result, as follows.
Consider
the random graph process that starts with n isolated vertices and adds
³ ´
n
the 2 possible edges one at a time, uniformly at random. The hitting time τn (δ ≥ 1)
is the random number of edges that have to be added to make each degree at least
1, and similarly for other graph properties, such as being connected or Hamiltonian.
The next result shows that, in the random graph process, the edge whose addition
kills the last isolated vertex usually makes the graph connected, and similarly the
edge whose addition brings the last vertex up to degree 2 usually makes the graph
Hamiltonian.
Theorem 2.7
τn (connected) = τn (δ ≥ 1) and τn (Hamiltonian) = τn (δ ≥ 2) a.s.
4
Minimum spanning trees
Let G be any connected graph. Suppose that, for each edge e of G, we have a random
length (or weight) Xe uniformly distributed on [0, 1], and these random variables are
independent. Let msp(G) denote the expected value of the minimum total length of
a spanning tree.
The asymptotic behaviour of this quantity for the complete graph Kn was determined by Frieze in 1985, using early results of Erdős and Rényi on the emergence of
a ‘giant component’ in sparse random graphs.
Theorem 2.8 As n → ∞, msp(Kn ) → ζ(3), where ζ(3) =
P∞
k=1
k −3 ∼ 1.2.
For corresponding results for msp(Kn,n ), see [12].
Almost all Berge graphs are perfect
A graph is perfect if for each induced subgraph the chromatic number equals the
maximum clique size. A Berge graph is a graph with no induced subgraph isomorphic
to an odd cycle or the complement of an odd cycle. All perfect graphs are Berge
graphs: the strong perfect graph conjecture asserts that all Berge graphs are perfect.
Let Pn be the number of perfect graphs, and let Bn be the number of Berge graphs,
on vertices v1 , . . . , vn . Then Pn ≤ Bn and the strong perfect graph conjecture asserts
that Pn = Bn . Prömel and Steger [29] have recently shown that Pn /Bn → 1 as
n → ∞.
3
Random graphs: deterministic results
We now consider some examples of the use of random graphs to obtain deterministic
results.
Ramsey numbers
The Ramsey number R(k, l) is the least n such that, in any red-blue colouring of the
edges of Kn , there is either a red Kk or a blue Kl . An easy induction shows that these
numbers are finite, and indeed that R(k, k) ≤ (4 + o(1))k .
What about lower bounds? Consider a random 2-colouring of the edges of Kn . The
probability that there ³is´a monochromatic Kk is at most the expected number of such
k
k
subgraphs, which is 2 nk 2−(2) . But this quantity is less than 1 if n ≤ 2 2 , for k ≥ 3.
It follows that in this case there must exist a 2-colouring with no monochromatic Kk ;
and so R(k, k) > 2k/2 . Indeed, the argument yields that R(k, k) = Ω(k2k/2 ) – see also
Section 4 below. This beautifully simple proof is from Erdős in 1947 (see for example
[1]). The lower bound has never been improved, apart from the constant factor (see
the next section).
5
Graphs with large chromatic number and large girth
The proof of the following theorem is another classic early example of the probabilistic
method, due to Erdős in 1959 (see for example [1]). Deterministic constructions had
to await Lovász in 1968.
Theorem 3.1 For each k and g, there is a graph H with χ(H) ≥ k and girth(H) ≥ g.
To prove the theorem it suffices to show that there is a graph G on n vertices such
that
√
(i) each set of s√= dn/ke vertices spans at least n edges, and
(ii) fewer than n cycles have length less than g.
For in this case we may obtain H by deleting an edge from each cycle in G of length
less than g: clearly then H has girth at least g, and α(H) < s, so that χ(H) ≥
n/α(H) ≥ k.
Is there a graph G as required? Consider Gn,p where p = 2k 2 /n, and n is large. The
probability √
that property (i) fails is at most the expected number of s-sets spanning
fewer than n edges, and this quantity is o(1). For property (ii), observe that, if En
denotes the expected number of cycles of length less than g, then
En ≤
g−1
X
ni pi ≤ (2k 2 )g .
i=3
√
Thus the probability that (ii) fails is at most En / n = o(1). Hence, for n sufficiently
large, there is a graph G as required.
The Hajós conjecture
A natural strengthening of Hadwiger’s conjecture was the conjecture of Hajós in 1961,
that if χ(G) = k then G has a subgraph homeomorphic to Kk . In 1979 Catlin
produced counter-examples, but the conjecture was blown out of the water shortly
afterwards by Erdős and Fajtlowicz (see [2]). Let p = 1/2. We have already seen
that χ(Gn,p ) ≥ n/(2 log2 n) a.s. Let k = dn2/3 e . The probability that Gn,p has a
subgraph homeomorphic to Kk is at most the probability that there is a k-set of
vertices ‘missing’ at most n − k edges: but this is at most the expected number of
such k-sets, which is o(1). Thus a.s. Gn,p has no subgraph homeomorphic to Kk .
Hence, for all large n, there are graphs with chromatic number at least n/(2 log2 n)
and no subgraph homeomorphic to Kk for any k ≥ n2/3 (which is much smaller).
Expanders
Expanders are very important in theoretical computer science – for example, in sorting
networks and for economising in space or randomness. Roughly speaking, an expander
is a graph in which each set S of vertices that is not too big has many neighbours
outside S. There are many variations on the precise conditions. Generally it is easy
to show that a suitable random graph is an expander with high probability, but hard
to give an explicit construction (see for example [1, 2]).
6
4
The Lovász local lemma
This section continues in the direction of the last one. The Lovász local lemma [9] is
remarkably useful for proving existence results in graph theory and elsewhere. Let
A1 , . . . , An be events in a probability space, and let G be a graph on the vertex set
V = {1, . . . , n}. We call G a dependency graph for these events if, for each i ∈ V , the
event Ai is independent of the family of all the other events Aj for j not adjacent to i.
Theorem 4.1 (Lovász Local Lemma: symmetric version) Let A1 , . . . , An be events
in a probability space Ω, with a dependency graph G. Suppose that each event Ai has
probability at most p and that each vertex degree in G is at most d. If ep(d + 1) ≤ 1
T
then P ( ni=1 Ai ) > 0.
We discuss two applications of this result below: see [1] for extensions, references not
given here and further applications.
Hypergraph colouring
The classic application [9] of the above result is to hypergraph 2-colouring.
Theorem 4.2 Suppose that we have a family of subsets Ei of a set S, such that each
subset Ei has size at least k and meets at most d other subsets Ej . If e(d + 1) ≤ 2k−1
then there is a 2-colouring of S such that none of the subsets Ei is monochromatic.
This result gives a slight improvement in the lower bound for the Ramsey number
R(k, k) discussed earlier: we
take S as the edge set of Kn , and the subsets Ei of
³ ´
n
S are the edge sets of the k possible subgraphs Kk . We obtain only a factor 2
improvement, but it gives the best bound known.
To prove theorem 4.2, we colour the points of S blue or red independently and with
equal probability. For each subset Ei , let Ai be the event that Ei is monochromatic.
Then Pr(Ai ) ≤ 2−(k−1) . We may form a natural dependency graph G by letting
distinct vertices i and j be adjacent just when the corresponding subsets Ei and Ej
intersect. Then each vertex in G has degree at most d. But e(d + 1) 2−(k−1) ≤ 1, and
so by the local lemma there must be a colouring as desired.
The condition in theorem 4.2 can be improved slightly to e(d + 2) ≤ 2k (see [23]).
It follows that, given a 7-uniform hypergraph (where each subset Ei has size 7 and
each point is in 7 subsets Ei ), there must be a 2-colouring with no monochromatic
edges. Recently Beck caused excitement when he gave an efficient general method to
find a colouring as in theorem 4.2, although under more restrictive conditions on k
and d.
7
Linear arboricity
A linear forest is a forest in which each component is a path. The linear arboricity
la(G) of a graph G is the minimum number of linear forests needed to cover its
edges. The linear arboricity conjecture asserts that, if G has maximum degree d, then
la(G) ≤ d(d + 1)/2e . It is clear that la(G) ≥ d/2, and for regular graphs it is easy to
check that la(G) ≥ d(d + 1)/2e . The best asymptotic upper bound known for la(G)
is due to Alon (see [1]).
Theorem 4.3 There is a constant c such that, for every graph G with maximum
degree d,
2
1
la(G) ≤ d/2 + c d 3 (log d) 3 .
The proof uses the local lemma in different places. The following result supplies a
key step. Note that any graph with n vertices and maximum degree d has a stable
set of size at least n/(d + 1). At the cost of decreasing the size of the stable set by a
constant factor, we can insist that it has a certain structure.
Theorem 4.4 Let H be a graph with maximum degree d, and let V1 , V2 , . . . be pairwise
disjoint subsets of the vertex set V , each of size s ≥ 2ed. Then there is a stable set
W that contains a vertex from each set Vi .
To prove this theorem, suppose that from each Vi independently we pick exactly
one vertex uniformly at random, and let W be the random set of vertices picked. For
each edge f of G, let Af be the event that W contains both end-vertices of f . Then
each event Af has probability s−2 , and there is a natural dependency graph G in
which each degree is less than 2sd. (There is a vertex of G for each edge of H, and
the vertices of G corresponding to distinct edges f and f 0 of H are adjacent if some
Vi contains an end-vertex of both these edges.) But es−2 2sd = 2ed/s ≤ 1. So, by the
local lemma, there is some set W such that none of the events Af holds – that is, W
is stable.
By using Alon’s methods and the local lemma again, we can obtain a result on
random graphs that lends some support to the linear arboricity conjecture [25, 27].
Theorem 4.5 (i) For fixed degree r, a random r-regular graph on n vertices has linear
arboricity d(r + 1)/2e a.s.
(ii) For fixed edge probability p, the random graph Gn,p has linear arboricity d∆/2e
a.s.
5
Deterministic graphs: random methods
Now we suppose that we are given some (deterministic) graph, and consider some
random things we can do with it!
8
Finding a minimum-cut
Let G be a connected graph with n (≥ 3) vertices. There are well-known efficient
deterministic methods to find a minimum cut based on network flow ideas, but here
is a far simpler randomised method [15, 17].
while at least three vertices remain
choose an edge uniformly at random and contract it
(keeping any multiple edges formed, but not loops).
output the set of edges remaining.
We claim that the probability that we obtain a minimum cut is at least 2/n2 , and so
the probability that a minimum cut is not found in tn2 independent attempts is at
2
most (1 − 2/n2 )tn ≤ e−2t .
To see why this claim is true, consider a fixed minimum cut C, of size k. The
algorithm returns C if none of the edges in C is chosen. Now G has least nk/2 edges,
since each vertex degree is at least k, and so the probability that the first edge chosen
is not in C is at least 1 − (2/n). Conditional on this happening, the probability that
the second edge chosen is not in C is at least 1 − 2/(n − 1) and so on. Hence the
probability that C is returned is at least
n−2
Y
i=1
(1 −
2
2
)=
.
n−i+1
n(n − 1)
Finding a maximum cut
Suppose that we have a graph G = (V, E) with non-negative weights wij = wji on
its edges. The maximum (weighted) cut problem is to find a set S of vertices that
maximises the weight of the edges between S and V \ S. This problem is NP-hard.
The following elegant random rounding idea has recently been shown to give good
approximations [13].
The maximum cut problem may be modelled by the quadratic integer program
max 12
X
wij (1 − yi yj ) subject to yi ∈ {−1, 1} for all i ∈ V,
i<j
where we take G as Kn without loss of generality. Here yi = 1 corresponds to vertex
i being in S say. We relax this program by replacing the variables yi ∈ {−1, 1} by
variables vi ∈ Sn , where Sn is the unit Euclidean ball in <n . This gives the semidefinite program
max 21
X
wij (1 − vi .vj ) subject to vi ∈ Sn for all i ∈ V,
i<j
which can be solved in polynomial time.
9
The randomised method takes a (near) optimal solution v1 , . . . , vn to the relaxed
program, picks a random vector r ∈ Sn , and outputs S = {i : vi .r ≥ 0}. It turns out
that the expected weight of the corresponding cut is at least 0.878 times the optimal
value. Similar ideas lead to an algorithm that will properly colour a 3-colourable
graph using O(n1/4 ) colours [16]. Both of these algorithms are naturally randomised,
but may be de-randomised to run in polynomial time and beat the best previously
known such approximation algorithms.
Random walks on graphs
A random walk on a graph G starts at some vertex v0 , picks a neighbour v1 uniformly
at random, from v1 picks a neighbour v2 uniformly at random, and so on. There are
intimate connections between random walks and electrical networks. Given vertices u
and v in G, let the commute time Cuv be the expected number of steps starting from
u to reach v plus the expected number of steps starting from v to reach u; and let the
effective resistance Ruv be the overall electrical resistance between u and v when each
edge has resistance 1.
Seemingly unrelated quantities like commute times and effective resistances satisfy
the same harmonic relations, and we find that, if G is connected and has n vertices
and m edges, then Cu,v = 2mRu,v : it follows that Cu,v < n3 [6].
Now suppose that we wish to test whether u and v are in the same component
of G (when n is very large). Starting at vertex u, perform a random walk for 2n3
steps. If we ever reach v we say ‘yes’ (correctly) and if not we say ‘no’. By Markov’s
inequality, the probability that we say ‘no’ incorrectly is less than 21 . This leads to a
randomised algorithm that works in logarithmic space, since we need remember only
where we are and how many steps we have taken.
Random walks on (large) expander graphs G are particularly important. The random walk is ‘rapidly mixing’, in that it converges exponentially fast to the stationary
distribution. If G is also regular then the stationary distribution is uniform. But
then we can sample almost uniformly from the vertices of G, and from this we can
approximately count the number of vertices. This approach allows us to estimate, for
example, the number of perfect matchings in a dense graph [14], and to estimate the
volume of a convex body[8].
The triangle problem
It is NP-hard to determine whether there is a 2-colouring of the vertices of a graph
G such that no triangle is monochromatic. But suppose that G has a proper vertex
3-colouring. Then clearly there must exist a 2-colouring of the vertices as above –just
amalgamate two of the colours in a proper 3-colouring: the triangle problem for G is
to find such a 2-colouring. Consider the following randomised ‘bit-flipping’ method
[22].
10
start with an arbitrary 2-colouring.
while we find a monochromatic triangle
pick a random vertex in the triangle, and flip its colour.
What could be simpler? It is also effective: if G has n vertices, then we find a 2colouring with no monochromatic triangles in expected number of iterations at most
3 2
n . To see this, fix a proper vertex colouring of G with the colours -1,0 and 1, and
8
use the colours -1 and 1 for the 2-colouring: then half the dot product of the two
colourings traces a symmetric random walk bounded within an interval of length n.
There are good deterministic ways to solve the triangle problem using linear algebra,
but there is no known ‘graph’ method [24].
Total colouring
Let us return briefly to total colourings, and prove an upper bound on χ00 (G) by a
simple randomised method [20, 26].
Theorem 5.1 Let G be graph with n vertices, and let k be an integer with k! ≥ n.
Then χ00 (G) ≤ ∆ + k + 2.
The algorithm starts with a vertex colouring with ∆ + 1 colours and an edge
colouring with ∆ + 1 different colours. We pick a random bijection π between the
two sets of colours, identify the corresponding colours, and let the ‘rejection graph’ R
contain those edges that would give a colour clash in G. Now χ00 (G) ≤ ∆ + 1 + χ0 (R).
But simple manipulations show that for each vertex, the probability that its degree
in R is at least k + 1 is less than 1/k!. Hence Pr(∆(R) ≥ k + 1) < n/k! ≤ 1, and so
for some bijection π we have χ0 (R) ≤ ∆(R) + 1 ≤ k + 1. The bound can in fact be
tightened to χ00 (G) ≤ χ0 (G) + k [7].
To block or to direct?
Two town planners, Blocker and Director, are bored and play a game on a graph
G. Blocker flips a coin independently for each edge e and blocks (deletes) it with
probability 12 . Let PB (hc) and PB (s → t) denote the corresponding probabilities
that there remains an (unblocked) Hamiltonian cycle or path from s to t. Director
similarly flips a coin for each edge e, and orients it so that each orientation is equally
likely. Let PD (hc) and PD (s → t) denote the corresponding probabilities that there
is a (directed) Hamiltonian cycle or path from s to t.
How do these probabilities compare? It turns out [18] that
PB (hc) ≤ PD (hc) and PB (s → t) = PD (s → t).
We may prove these results by an induction involving the edges, using the following
idea. Consider a particular edge e = {u, v} of G, and fix any pattern of blocking
or orienting on all the other edges. If there is a Hamiltonian cycle not using e (and
11
respecting any orientations), then what happens with e is irrelevant, so we assume
that this does not happen. Similarly we assume that there is a Hamiltonian path from
u to v that does not use e, or one from v to u (or both). But then the probability
that Blocker permits a Hamiltonian cycle is 12 , and the probability that Director does
is either 12 or 1.
6
Concentration of measure and isoperimetric inequalities for graphs
In 1979 Maurey used a ‘concentration of measure’ inequality to prove an isoperimetric
inequality for ‘permutation graphs’ (see below). That started a ball rolling which has
had exciting effects in the study of normed spaces, random graphs, the mathematics
of operational research and theoretical computer science (see [19], which contains
references not given here). This ball is still on the move – see [33].
Let us start with a special case of the ‘independent bounded differences inequality’.
Theorem 6.1 Let X1 , . . . , Xn be independent random variables, with Xk taking valQ
ues in a set Ak for each k. Suppose that the measurable function f : k Ak → <
satisfies |f (x) − f (x0 )| ≤ 1, whenever the vectors x and x0 differ only in one coordinate. Let Y be the random variable f (X1 , . . . , Xn ). Then, for any t > 0, both
Pr(Y − EY ≥ t) and Pr(Y − EY ≤ −t) are at most exp (−2t2 /n) .
This result can be proved by elementary manipulations with the moment generating
function MY (s) = E(esY ) of Y . It extends the standard Chernoff bound for binomial
P
random variables: to see this take each set Ak = {0, 1} and let f (x1 , . . . , xn ) = k xk .
In the random graph Gn,p , let Xk give the edges present in the set of possible edges
{vi , vk } with i < k. Then we obtain the following result.
Theorem 6.2 Suppose that the graph function f satisfies |f (G) − f (G0 )| ≤ 1, whenever G0 can be obtained from G by changing edges incident with a single vertex. Then
the random variable Y = f (Gn,p ) satisfies
Pr(|Y − EY | ≥ t) ≤ 2 exp(−2t2 /n)
for any t > 0.
If we take f as the chromatic number χ, we see that χ(Gn,p ) is concentrated around
its mean.
Consider a finite metric space (V, d). We shall be interested in the case when
V is the vertex set of a graph and d measures distance in the graph. For A ⊆ V
and t ≥ 0, the t-neighbourhood At of A is the set {v ∈ V : d(v, A) ≤ t}, where
d(v, A) = min{d(v, x) : x ∈ A}. A ‘discrete isoperimetric inequality’ gives a lower
bound on |At | depending on |A| and t.
12
We shall focus on the n-cube Qn , in which the vertices are the 2n points in {0, 1}n ,
and two vertices are adjacent if they differ in exactly one coordinate. We shall also
discuss the ‘permutation graph’ associated with the symmetric group Sn , in which the
vertices are the n! permutations of {1, . . . , n}, and two vertices g and h are adjacent
if g −1 h is a transposition.
Theorem
6.3 Let A be any set of half of the 2n vertices of the n-cube Qn , and let
q
t0 = n/2. Then, for any t ≥ t0 ,
³
´
|At |/2n ≥ 1 − exp −2(t − t0 )2 /n .
We can deduce this isoperimetric inequality from Theorem 6.1 by taking each Xk
uniformly distributed on Ak = {0, 1} and letting f (x) = d(x, A) (see [19]). From a
similar bounded differences inequality, we can obtain what is essentially the result
of Maurey with which we started this section: we simply replace the phrase ‘the 2n
vertices of the n-cube Qn ’ in Theorem 6.3 by ‘the n! vertices of the permutation graph
of Sn ’.
In the cube Qn , a Hamming ball centred at a vertex v and of radius d consists of
all vertices w such that the distance from v to w is less than d, together with some
vertices at distance d. A seminal theorem of Harper in 1966 (see [3]) concerns two
sets A and C of vertices in Qn . The theorem asserts that there exist Hamming balls
B0 centred at the all-zero vector and B1 centred at the all-one vector, such that we
have |B0 | = |A|, |B1 | = |C| and d(B0 , B1 ) ≥ d(A, C). This yields the precise solution
to the isoperimetric problem for the n-cube. From Harper’s theorem and a Chernoff
bound, we can obtain the following cleaner version of Theorem 6.3.
Theorem 6.4 Let A be any set of 2n−1 vertices of the n-cube Qn . Then, for any
t ≥ 0,
|At |/2n ≥ 1 − exp(−2t2 /n).
The isoperimetric inequality Theorem 6.3 was obtained from Theorem 6.1 on concentration of measure. It is interesting that we can also reverse this process. Recall
that a number L is a median of a random variable Y if both Pr(Y ≤ L) ≥ 21 and
Pr(Y ≥ L) ≥ 21 .
Theorem 6.5 Let X be uniformly distributed over V = {0, 1}n , let the function f
on V satisfy |f (x) − f (x0 )| ≤ 1 whenever x and x0 differ only in one co-ordinate,
and let L be a median of Y = f (X). Then, for any t ≥ 0, both Pr(Y −L > t) and
Pr(Y −L < −t) are at most exp(−2t2 /n).
This result is similar to Theorem 6.1 with ‘mean’ replaced by ‘median’. To see how
it follows from Theorem 6.4, let A = {x ∈ V : f (x) ≤ L}. Then
Pr(Y − L ≤ t) ≥ Pr(X ∈ At ) ≥ 1 − exp(−2t2 /n).
13
References
[1] N. Alon and J.H. Spencer, The Probabilistic Method, Wiley, 1992.
[2] B. Bollobás, Random Graphs, Academic Press, 1985.
[3] B. Bollobás, Combinatorics, Cambridge University Press, 1986.
[4] B. Bollobás, The chromatic number of random graphs, Combinatorica 8 (1988), 49-55.
[5] B. Bollobás, T.I. Fenner and A.M. Frieze, An algorithm for finding Hamilton paths and
cycles in random graphs, Combinatorica 7 (1987), 327-341.
[6] A.K. Chandra, P. Raghavan, W.L. Ruzzo, R. Smolensky and P. Tiwari, The electrical
resistance of a graph captures its commute and cover times, Proc. 21st Annual ACM
Symposium on the Theory of Computing, 1989, pp. 574-586.
[7] A.G. Chetwynd and R. Häggkvist, Some upper bounds on the total and list chromatic
numbers of multigraphs, J. Graph Theory 16 (1992), 503-516.
[8] M.E. Dyer, A. Frieze and R. Kannan, A random polynomial algorithm for approximating the volume of convex bodies, Proc. 21st Annual ACM Symposium on the Theory
of Computing, 1989, pp. 375-381.
[9] P. Erdős and L. Lovász, Problems and results on 3-chromatic hypergraphs and some
related questions, in Infinite and Finite Sets, (ed. A. Hajnal et al.), North Holland,
Amsterdam, 1975, pp. 609-628.
[10] P. Erdős and A. Rényi (1960) On the evolution of random graphs, Publ. Math. Inst.
Hungar. Acad. Sci. 5 17-61.
[11] A. Frieze, W. Jackson, C. McDiarmid and B. Reed, Edge-colouring random graphs, J.
Combinatorial Theory B 45 (1988), 135-149.
[12] A. Frieze and C. McDiarmid, On random minimum length spanning trees, Combinatorica 9 (1989), 363-374.
[13] M.X. Goemans and D.R. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, to appear, 1993.
[14] M.R. Jerrum and A. Sinclair, Approximating the permanent, SIAM Journal on Computing 18 (1989), 1149-1178.
[15] D.R. Karger, Global min-cuts in RNC, Proc. 4th Annual ACM-SIAM Symposium on
Discrete Algorithms, 1993.
[16] D.R. Karger, R. Motwani and M. Sudan, Approximate graph coloring by semidefinite
programming, to appear, 1994.
14
[17] D.R. Karger and C. Stein, An O(n2 ) algorithm for minimum cuts, Proc. 26th Annual
ACM Symposium on the Theory of Computing, 1991, pp. 757-765.
[18] C. McDiarmid, General percolation and random graphs, Advances in Applied Probability 13 (1981), 40-60.
[19] C. McDiarmid, On the method of bounded differences, Surveys in Combinatorics, 1989
(ed. J. Siemons), London Mathematical Society Lecture Note Series 141, Cambridge
University Press, 1989, pp. 148-188.
[20] C. McDiarmid, Colourings of random graphs, Graph Colouring (eds. R.Nelson and
R.J.Wilson), Pitman Research Notes in Mathematics 218, Longman, 1990, pp. 79-86.
[21] C. McDiarmid, On the chromatic number of random graphs, Random Structures and
Algorithms 1 (1990), 435-442.
[22] C. McDiarmid, A random recolouring method for graphs and hypergraphs, Combinatorics, Probability and Computing 2 (1993), 363-365.
[23] C. McDiarmid, Hypergraph colouring and the Lovász Local Lemma, Combinatorics,
Probability and Computing, to appear.
[24] C. McDiarmid, A random bit-flipping method for seeking agreement, 1994.
[25] C. McDiarmid and B. Reed, Linear arboricity of random regular graphs, Random Structures and Algorithms 1 (1990), 443-445.
[26] C. McDiarmid and B. Reed, On total colourings of graphs, J.Combinatorial Theory B
57 (1993), 122-130.
[27] C. McDiarmid and B. Reed, Linear arboricity of random graphs, Combinatorics, Probability and Computing, to appear.
[28] L. Perkovic and B.A. Reed, Edge coloring in polynomial expected time, to appear.
[29] H.J. Prömel and A. Steger, Almost all Berge graphs are perfect, Combinatorics Probability and Computing 1 (1992), 53-79.
[30] H.J. Prömel and A. Steger, Coloring clique-free graphs in linear expected time, Random
Structures and Algorithms 3 (1992), 375-402.
[31] R.W. Robinson and N.C. Wormald, Almost all cubic graphs are Hamiltonian, Random
Structures and Algorithms 3 (1992), 117-126.
[32] R.W. Robinson and N.C. Wormald, Almost all regular graphs are Hamiltonian, Random
Structures and Algorithms, to appear.
[33] M. Talagrand, Concentration of measure and isoperimetric inequalities in product
spaces, manuscript, 1994.
15