Lovász number explained
In graph theory, the Lovász number of a graph is a real number that is an upper bound on the Shannon capacity of the graph. It is also known as Lovász theta function and is commonly denoted by
, using a script form of the Greek letter
theta to contrast with the upright theta used for Shannon capacity. This quantity was first introduced by
László Lovász in his 1979 paper
On the Shannon Capacity of a Graph.
Accurate numerical approximations to this number can be computed in polynomial time by semidefinite programming and the ellipsoid method.The Lovász number of the complement of any graph is sandwiched between the chromatic number and clique number of the graph, and can be used to compute these numbers on graphs for which they are equal, including perfect graphs.
Definition
Let
be a
graph on
vertices. An ordered set of
unit vectors
is called an
orthonormal representation of
in
, if
and
are orthogonal whenever vertices
and
are not adjacent in
:
Clearly, every graph admits an orthonormal representation with
: just represent vertices by distinct vectors from the
standard basis of
.
[1] Depending on the graph it might be possible to take
considerably smaller than the number of vertices
.
The Lovász number
of graph
is defined as follows:
where
is a unit vector in
and
is an orthonormal representation of
in
. Here minimization implicitly is performed also over the dimension
, however without loss of generality it suffices to consider
.
[2] Intuitively, this corresponds to minimizing the half-angle of a rotational
cone containing all representing vectors of an orthonormal representation of
. If the optimal angle is
, then
and
corresponds to the symmetry axis of the cone.
Equivalent expressions
Let
be a graph on
vertices. Let
range over all
symmetric matrices such that
whenever
or vertices
and
are not adjacent, and let
denote the largest
eigenvalue of
. Then an alternative way of computing the Lovász number of
is as follows:
The following method is dual to the previous one. Let
range over all
symmetric
positive semidefinite matrices such that
whenever vertices
and
are adjacent, and such that the
trace (sum of diagonal entries) of
is
. Let
be the
matrix of ones. Then
Here,
is just the sum of all entries of
.
. Let
be a unit vector and
be an orthonormal representation of
. Then
Value for well-known graphs
The Lovász number has been computed for the following graphs:
Properties
If
denotes the strong graph product of graphs
and
, then
If
is the complement of
, then
with equality if
is
vertex-transitive.
Lovász "sandwich theorem"
The Lovász "sandwich theorem" states that the Lovász number always lies between two other numbers that are NP-complete to compute. More precisely,where
is the clique number of
(the size of the largest
clique) and
is the chromatic number of
(the smallest number of colors needed to color the vertices of
so that no two adjacent vertices receive the same color).
The value of
can be formulated as a
semidefinite program and numerically approximated by the
ellipsoid method in time bounded by a polynomial in the number of vertices of
G.
[3] For
perfect graphs, the chromatic number and clique number are equal, and therefore are both equal to
. By computing an approximation of
and then rounding to the nearest integer value, the chromatic number and clique number of these graphs can be computed in polynomial time.
Relation to Shannon capacity
The Shannon capacity of graph
is defined as follows:
where
is the
independence number of
graph
(the size of a largest
independent set of
) and
is the strong graph product of
with itself
times. Clearly,
. However, the Lovász number provides an upper bound on the Shannon capacity of graph, hence
For example, let the confusability graph of the channel be
, a
pentagon. Since the original paper of it was an open problem to determine the value of
. It was first established by that
.
Clearly,
\Theta(C5)\ge\alpha(C5)=2
. However,
, since "11", "23", "35", "54", "42" are five mutually non-confusable messages (forming a five-vertex independent set in the strong square of
), thus
.
To show that this bound is tight, let
be the following orthonormal representation of the pentagon:
and let
. By using this choice in the initial definition of Lovász number, we get
. Hence,
.
However, there exist graphs for which the Lovász number and Shannon capacity differ, so the Lovász number cannot in general be used to compute exact Shannon capacities.
Quantum physics
The Lovász number has been generalized for "non-commutative graphs" in the context of quantum communication. The Lovasz number also arises in quantum contextuality in an attempt to explain the power of quantum computers.
See also
References
Notes and References
- A representation of vertices by standard basis vectors will not be faithful, meaning that adjacent vertices are represented by non-orthogonal vectors, unless the graph is edgeless. A faithful representation in
is also possible by associating each vertex to a basis vector as before, but mapping each vertex to the sum of basis vectors associated with its closed neighbourhood.
- If
then one can always achieve a smaller objective value by restricting
to the subspace spanned by vectors
; this subspace is at most
-dimensional.
- .