The max cut problem
Veronica Piccialli∗
ESI 2010 - Klagenfurt
∗ University of Rome Tor Vergata
1 / 16
Max Cut
Introduction
• Max Cut
An application in
statistical physics
The Goemans and
Williamson algorithm
Bibliography
find S ⊆ V such that the cut
S, V \ S is maximum.
graph G(V, E),
|V | = n with weights wij for
all (i, j) ∈ E
Given
a
Introducing the variables:
xi =
1
−1
if i ∈ S
if i ∈ V \ S
2 / 16
Max Cut
Introduction
• Max Cut
An application in
statistical physics
The Goemans and
Williamson algorithm
Bibliography
find S ⊆ V such that the cut
S, V \ S is maximum.
graph G(V, E),
|V | = n with weights wij for
all (i, j) ∈ E
Given
a
Introducing the variables:
xi =
1
−1
if i ∈ S
if i ∈ V \ S
The max cut problem can be expressed as
1X
max
wij (1 − xi xj )
2 i<j
(MC)
x2i = 1 i = 1, . . . , n
2 / 16
Max Cut
Introduction
• Max Cut
An application in
statistical physics
The Goemans and
Williamson algorithm
Bibliography
find S ⊆ V such that the cut
S, V \ S is maximum.
graph G(V, E),
|V | = n with weights wij for
all (i, j) ∈ E
Given
a
Introducing the variables:
xi =
1
−1
if i ∈ S
if i ∈ V \ S
The max cut problem can be expressed as
1X
max
wij (1 − xi xj )
2 i<j
(MC)
x2i = 1 i = 1, . . . , n
It can be rewritten as
max xT Cx
x2i = 1 i = 1, . . . , n
(MC)
where C = 41 (Diag(W e) − W ) = 14 L (L is the Laplacian of the graph)
2 / 16
Spin glasses
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
A very interesting problem in statistical physics (within theory of magnetism) is
the determination of ground states of spin glasses
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
3 / 16
Spin glasses
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
A very interesting problem in statistical physics (within theory of magnetism) is
the determination of ground states of spin glasses
A spin glass is an alloy of magnetic impurities diluited in a non magnetic metal
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
3 / 16
Spin glasses
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
A very interesting problem in statistical physics (within theory of magnetism) is
the determination of ground states of spin glasses
A spin glass is an alloy of magnetic impurities diluited in a non magnetic metal
A peak in magnetic susceptibility at a certain temperature means a phase
transition. It is not known what happens at low temperature.
Bibliography
3 / 16
Spin glasses
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
A very interesting problem in statistical physics (within theory of magnetism) is
the determination of ground states of spin glasses
A spin glass is an alloy of magnetic impurities diluited in a non magnetic metal
A peak in magnetic susceptibility at a certain temperature means a phase
transition. It is not known what happens at low temperature.
There are some theories on spin glasses behavior but it is impossible to realize
by practical experiments the situation predicted by the theory, so
mathematical models are used
3 / 16
Mathematical model
Introduction
Assume a spin glass contains n magnetic impurities (atoms).
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
4 / 16
Mathematical model
Introduction
An application in
statistical physics
Assume a spin glass contains n magnetic impurities (atoms). The magnetic
orientation of each atom is represented by Si ∈ R3 , kSi k = 1, i = 1, . . . , n
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
4 / 16
Mathematical model
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Assume a spin glass contains n magnetic impurities (atoms). The magnetic
orientation of each atom is represented by Si ∈ R3 , kSi k = 1, i = 1, . . . , n
For each pair i, j of atoms there is an interaction Jij depending on the material
and on the distance (decreases quickly with the distance), an example is
cos(Drij )
Jij = A
,
3
B 3 rij
A, B, D depending on the material
Bibliography
4 / 16
Mathematical model
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Assume a spin glass contains n magnetic impurities (atoms). The magnetic
orientation of each atom is represented by Si ∈ R3 , kSi k = 1, i = 1, . . . , n
For each pair i, j of atoms there is an interaction Jij depending on the material
and on the distance (decreases quickly with the distance), an example is
cos(Drij )
Jij = A
,
3
B 3 rij
A, B, D depending on the material
Bibliography
If atom i and j have spins Si and Sj , the energy interaction between the two
atoms is given by
Hij = Jij hSi , Sj i
Given a configuration ω , the energy of the system is given by the hamiltonian
H(ω) = −
n−1
X
n
X
i=1 j=i+1
Jij hSi , Sj i − h
n
X
Jij hSi , F i
i=1
where F ∈ R3 , kF k = 1 represents the orientation of an exterior magnetic field
of strength h.
4 / 16
Ising spins
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
A possible simplification is to replace Si ∈ R3 , i = 1, . . . , n and F ∈ R3 with
si ∈ {−1, 1}, i = 1, . . . , n and f ∈ {−1, 1}, meaning magnetic north pole up
and magnetic north pole down
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
5 / 16
Ising spins
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
A possible simplification is to replace Si ∈ R3 , i = 1, . . . , n and F ∈ R3 with
si ∈ {−1, 1}, i = 1, . . . , n and f ∈ {−1, 1}, meaning magnetic north pole up
and magnetic north pole down
The resulting model is called Ising spin glasses (correct model for some
substances).
The Goemans and
Williamson algorithm
Bibliography
5 / 16
Ising spins
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
A possible simplification is to replace Si ∈ R3 , i = 1, . . . , n and F ∈ R3 with
si ∈ {−1, 1}, i = 1, . . . , n and f ∈ {−1, 1}, meaning magnetic north pole up
and magnetic north pole down
The resulting model is called Ising spin glasses (correct model for some
substances).
Two possibilities:
Bibliography
5 / 16
Ising spins
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
A possible simplification is to replace Si ∈ R3 , i = 1, . . . , n and F ∈ R3 with
si ∈ {−1, 1}, i = 1, . . . , n and f ∈ {−1, 1}, meaning magnetic north pole up
and magnetic north pole down
The resulting model is called Ising spin glasses (correct model for some
substances).
Two possibilities:
1. Long range model: considering the interaction between all pairs of atoms
5 / 16
Ising spins
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
A possible simplification is to replace Si ∈ R3 , i = 1, . . . , n and F ∈ R3 with
si ∈ {−1, 1}, i = 1, . . . , n and f ∈ {−1, 1}, meaning magnetic north pole up
and magnetic north pole down
The resulting model is called Ising spin glasses (correct model for some
substances).
Two possibilities:
1. Long range model: considering the interaction between all pairs of atoms
2. Short range model: considering the interaction only between “close”
atoms, setting to zero the interaction between atoms being far apart
5 / 16
Ising spins
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
A possible simplification is to replace Si ∈ R3 , i = 1, . . . , n and F ∈ R3 with
si ∈ {−1, 1}, i = 1, . . . , n and f ∈ {−1, 1}, meaning magnetic north pole up
and magnetic north pole down
The resulting model is called Ising spin glasses (correct model for some
substances).
Two possibilities:
1. Long range model: considering the interaction between all pairs of atoms
2. Short range model: considering the interaction only between “close”
atoms, setting to zero the interaction between atoms being far apart
The short range Ising spin glass model allowed to predict a phase transition that
was experimentally observed (Nobel prize to Onsager)
5 / 16
Ising spins
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
A possible simplification is to replace Si ∈ R3 , i = 1, . . . , n and F ∈ R3 with
si ∈ {−1, 1}, i = 1, . . . , n and f ∈ {−1, 1}, meaning magnetic north pole up
and magnetic north pole down
The resulting model is called Ising spin glasses (correct model for some
substances).
Two possibilities:
1. Long range model: considering the interaction between all pairs of atoms
2. Short range model: considering the interaction only between “close”
atoms, setting to zero the interaction between atoms being far apart
The short range Ising spin glass model allowed to predict a phase transition that
was experimentally observed (Nobel prize to Onsager)
At 0 K the spin glasses configuration attains a minimum energy configuration, that
can be found by minimizing the Hamiltonian associated with the system. This
problem can be formulated as a max cut problem
5 / 16
Ising model and max cut I
Introduction
An application in
statistical physics
We define a graph G with n + 1 nodes (n atoms plus an external magnetic field
represented by the node 0)
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
6 / 16
Ising model and max cut I
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
We define a graph G with n + 1 nodes (n atoms plus an external magnetic field
represented by the node 0)
For a pair i, j G contains an edge ij if the interaction between i and j is not zero,
and the weight of the edge is equal to Jij .
The Goemans and
Williamson algorithm
Bibliography
6 / 16
Ising model and max cut I
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
We define a graph G with n + 1 nodes (n atoms plus an external magnetic field
represented by the node 0)
For a pair i, j G contains an edge ij if the interaction between i and j is not zero,
and the weight of the edge is equal to Jij .
There exist n edges 0i with weigth J0i = h
Bibliography
6 / 16
Ising model and max cut I
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
We define a graph G with n + 1 nodes (n atoms plus an external magnetic field
represented by the node 0)
For a pair i, j G contains an edge ij if the interaction between i and j is not zero,
and the weight of the edge is equal to Jij .
There exist n edges 0i with weigth J0i = h
Then the Hamiltonian can be represented as a quadratic function
H(ω) = −
X
ij∈E, i,j>0
Jij si sj − h
n
X
i=1
si = −
X
Jij si sj
ij∈E
6 / 16
Ising model and max cut I
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
We define a graph G with n + 1 nodes (n atoms plus an external magnetic field
represented by the node 0)
For a pair i, j G contains an edge ij if the interaction between i and j is not zero,
and the weight of the edge is equal to Jij .
There exist n edges 0i with weigth J0i = h
Then the Hamiltonian can be represented as a quadratic function
H(ω) = −
X
ij∈E, i,j>0
Jij si sj − h
n
X
i=1
si = −
X
Jij si sj
ij∈E
Each spin configuration corresponds to a partition of the nodes into
V + = {i : si = 1}, V − = {i : si = −1}
6 / 16
Ising model and max cut II
Introduction
An application in
statistical physics
For any W ⊆ V , define E(W ) = {ij ∈ E : i ∈ W, j ∈ W },
δ(W ) = {ij ∈ E : i ∈ W, j ∈ V \ W }
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
7 / 16
Ising model and max cut II
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
For any W ⊆ V , define E(W ) = {ij ∈ E : i ∈ W, j ∈ W },
δ(W ) = {ij ∈ E : i ∈ W, j ∈ V \ W }
X
X
H(ω) = −
Jij si sj = −
Jij −
ij∈E
ij∈E(V + )
X
ij∈E(V − )
Jij +
X
Jij
ij∈δ(V + )
The Goemans and
Williamson algorithm
Bibliography
7 / 16
Ising model and max cut II
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
For any W ⊆ V , define E(W ) = {ij ∈ E : i ∈ W, j ∈ W },
δ(W ) = {ij ∈ E : i ∈ W, j ∈ V \ W }
X
X
H(ω) = −
Jij si sj = −
Jij −
ij∈E
Setting C =
P
ij∈E
ij∈E(V + )
X
ij∈E(V − )
Jij +
X
Jij
ij∈δ(V + )
Jij , we get
H(ω) + C = 2
X
Jij
ij∈δ(V + )
7 / 16
Ising model and max cut II
Introduction
An application in
statistical physics
• Spin glasses
• Mathematical model
• Ising spins
• Ising model and max
cut I
• Ising model and max
cut II
The Goemans and
Williamson algorithm
Bibliography
For any W ⊆ V , define E(W ) = {ij ∈ E : i ∈ W, j ∈ W },
δ(W ) = {ij ∈ E : i ∈ W, j ∈ V \ W }
X
X
H(ω) = −
Jij si sj = −
Jij −
ij∈E
Setting C =
P
ij∈E
ij∈E(V + )
X
ij∈E(V − )
Jij +
X
Jij
ij∈δ(V + )
Jij , we get
H(ω) + C = 2
X
Jij
ij∈δ(V + )
Setting cij = −Jij for all ij ∈ E , then minimizing H(ω) is equivalent to solve
max
V + ⊆V
X
cij
ij∈δ(V + )
7 / 16
Relaxation
Introduction
An application in
statistical physics
Noting that
xT Cx = hC, xxT i = hC, Xi, for X = xxT
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
• Nesterov’s result
Bibliography
8 / 16
Relaxation
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
Noting that
xT Cx = hC, xxT i = hC, Xi, for X = xxT
The max cut problem is then equivalent to:
max
hC, Xi
diag(X) = e
X0
rank (X) = 1
• Nesterov’s result
Bibliography
8 / 16
Relaxation
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
• Nesterov’s result
Bibliography
Noting that
xT Cx = hC, xxT i = hC, Xi, for X = xxT
The max cut problem is then equivalent to:
max
hC, Xi
diag(X) = e
X0
rank (X) = 1
Dropping the rank constraint, the standard SDP relaxation of max cut can be
obtained:
max
hC, Xi
diag(X) = e
X0
8 / 16
Properties of the SDP relaxation
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
1. Slater is satisfied for both primal and dual so strong duality holds
2. The constraints diag(X) = e, X 0 imply that −1 ≤ Xij ≤ 1 for all
i, j .
3. If X is feasible, and |Xij | = 1, then xik = sgn(xij )xjk
4. The only feasible matrix of rank 1 satisfying Xij ∈ {−1, 1} are of the form
xxT , with x ∈ {−1, 1}n
• Nesterov’s result
Bibliography
9 / 16
Alternative way of deriving the SDP relaxation
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
The max cut problem can be written as
X
1
max
wij (1 − xi xj )
2
i<j
(MC)
x2i = 1 i = 1, . . . , n
• Nesterov’s result
Bibliography
10 / 16
Alternative way of deriving the SDP relaxation
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
The max cut problem can be written as
X
1
max
wij (1 − xi xj )
2
i<j
(MC)
x2i = 1 i = 1, . . . , n
The feasible region can be enlarged:
• Nesterov’s result
Bibliography
10 / 16
Alternative way of deriving the SDP relaxation
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
The max cut problem can be written as
X
1
max
wij (1 − xi xj )
2
i<j
(MC)
x2i = 1 i = 1, . . . , n
The feasible region can be enlarged:
xi
⇒ vi ∈ Rk , k ≤ n
x2i = 1 ⇒ kvi k = 1
• Nesterov’s result
Bibliography
10 / 16
Alternative way of deriving the SDP relaxation
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
• Nesterov’s result
The max cut problem can be written as
X
1
max
wij (1 − xi xj )
2
i<j
(MC)
x2i = 1 i = 1, . . . , n
The feasible region can be enlarged:
xi
⇒ vi ∈ Rk , k ≤ n
x2i = 1 ⇒ kvi k = 1
vi is the relaxation of xi ∈ {−1, 1} to the n-dimensional unit sphere.
Bibliography
10 / 16
Alternative way of deriving the SDP relaxation
Introduction
The max cut problem can be written as
An application in
statistical physics
X
1
max
wij (1 − xi xj )
2
i<j
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
• Nesterov’s result
Bibliography
(MC)
x2i = 1 i = 1, . . . , n
The feasible region can be enlarged:
xi
⇒ vi ∈ Rk , k ≤ n
x2i = 1 ⇒ kvi k = 1
vi is the relaxation of xi ∈ {−1, 1} to the n-dimensional unit sphere. We get the
problem
max
XX
wij (1 − viT vj )
i,j i<j
kvi k2 = 1 i = 1, . . . , n, vi ∈ Rn
(the same as X = V V T )
10 / 16
Goemans and Williamson algorithm
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
S1 First, solve
max
XX
wij (1 − viT vj )
i,j i<j
2
kvi k = 1 i = 1, . . . , n, vi ∈ Rn
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
• Nesterov’s result
Bibliography
11 / 16
Goemans and Williamson algorithm
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
S1 First, solve
max
XX
wij (1 − viT vj )
i,j i<j
2
kvi k = 1 i = 1, . . . , n, vi ∈ Rn
S2 Choose a random vector h on the unit sphere
• Nesterov’s result
Bibliography
11 / 16
Goemans and Williamson algorithm
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
S1 First, solve
max
XX
wij (1 − viT vj )
i,j i<j
2
kvi k = 1 i = 1, . . . , n, vi ∈ Rn
S2 Choose a random vector h on the unit sphere
S3 Set V = {i : hT vi ≥ 0}
• Nesterov’s result
Bibliography
11 / 16
Goemans and Williamson algorithm
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
S1 First, solve
max
XX
wij (1 − viT vj )
i,j i<j
2
kvi k = 1 i = 1, . . . , n, vi ∈ Rn
S2 Choose a random vector h on the unit sphere
S3 Set V = {i : hT vi ≥ 0}
Let W be the corresponding cut
• Nesterov’s result
Bibliography
11 / 16
Performance of the algorithm
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
1. The expected value of the produced cut is:
arcos(viT vj )
E[W ] =
wij
π
i<j
X
• Nesterov’s result
Bibliography
12 / 16
Performance of the algorithm
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
1. The expected value of the produced cut is:
arcos(viT vj )
E[W ] =
wij
π
i<j
X
2. Assume wij ≥ 0. Then E(W ) ≥
α = min
α 21
0≤θ≤π
P
i<j
wij (1 − viT vj ), with
θ
2
π 1 − cos(θ)
• Nesterov’s result
Bibliography
12 / 16
Performance of the algorithm
Introduction
An application in
statistical physics
1. The expected value of the produced cut is:
arcos(viT vj )
E[W ] =
wij
π
i<j
The Goemans and
Williamson algorithm
X
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
2. Assume wij ≥ 0. Then E(W ) ≥
α = min
α 21
0≤θ≤π
P
i<j
wij (1 − viT vj ), with
θ
2
π 1 − cos(θ)
• Nesterov’s result
Bibliography
3. α > 0.87856
12 / 16
Performance of the algorithm
Introduction
An application in
statistical physics
1. The expected value of the produced cut is:
arcos(viT vj )
E[W ] =
wij
π
i<j
The Goemans and
Williamson algorithm
X
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
2. Assume wij ≥ 0. Then E(W ) ≥
α = min
α 21
0≤θ≤π
P
i<j
wij (1 − viT vj ), with
θ
2
π 1 − cos(θ)
• Nesterov’s result
Bibliography
3. α > 0.87856
4. Assume wij ≥ 0. Then
∗
zMC
∗ −z ∗
zMC
SDP
> 0.87856
12 / 16
Performance of the algorithm
Introduction
An application in
statistical physics
1. The expected value of the produced cut is:
arcos(viT vj )
E[W ] =
wij
π
i<j
The Goemans and
Williamson algorithm
X
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
2. Assume wij ≥ 0. Then E(W ) ≥
α = min
α 21
0≤θ≤π
P
i<j
wij (1 − viT vj ), with
θ
2
π 1 − cos(θ)
• Nesterov’s result
Bibliography
3. α > 0.87856
4. Assume wij ≥ 0. Then
∗
zMC
∗ −z ∗
zMC
SDP
> 0.87856
5. Note that there is no polynomial approximation algorithm with constant
< 0.9412 unless P=NP [Hästad 1997] .
12 / 16
Nonlinear reformulation of max cut I
Introduction
The max cut problem is equivalent to
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
maxhCσ(V u), σ(V u)i
kvi k = 1 i = 1, . . . , n
kuk = 1
 T
v1
 . 
where for any a ∈ Rn , σ(a) = sgn(a), and V =  .. 
vnT
• Nesterov’s result
Bibliography
13 / 16
Nonlinear reformulation of max cut I
Introduction
The max cut problem is equivalent to
An application in
statistical physics
maxhCσ(V u), σ(V u)i
kvi k = 1 i = 1, . . . , n
kuk = 1
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
• Nesterov’s result
Bibliography
 T
v1
 . 
where for any a ∈ Rn , σ(a) = sgn(a), and V =  .. 
vnT
It is also equivalent to
max Eu (hCσ(V u), σ(V u)i)
kvi k = 1 i = 1, . . . , n
13 / 16
Nonlinear reformulation of max cut II
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
The max cut problem can be rewritten as
2
hC, arcsin(X)i
π
diag(X) = e
X0
max
• Nesterov’s result
Bibliography
14 / 16
Nonlinear reformulation of max cut II
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
The max cut problem can be rewritten as
2
hC, arcsin(X)i
π
diag(X) = e
X0
max
1
Sketch of the proof: Choose V = X 2 . Then we need to prove
2
Eu (hCσ(V u), σ(V u)i) = hC, arcsin(X)i
π
• Nesterov’s result
Bibliography
14 / 16
Nesterov’s result
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
If C 0, then
∗
2
zMC
> = 0.6366
∗
∗
− zSDP
zMC
π
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
• Nesterov’s result
Bibliography
15 / 16
Nesterov’s result
Introduction
An application in
statistical physics
If C 0, then
The Goemans and
Williamson algorithm
• Relaxation
• Properties of the SDP
relaxation
• Alternative way of
deriving the SDP
relaxation
• Goemans and
Williamson algorithm
• Performance of the
algorithm
• Nonlinear
reformulation of max cut
I
• Nonlinear
reformulation of max cut
II
Define X ◦k = X ◦ X ◦(k−1) and consider that
1 X ◦3
1 · 3 X ◦5
arcsin(X) = X +
+
+ ...
2 3
2·4 5
and −1 ≤ Xij ≤ 1 then arcsin(X) − X 0. Since C 0, we get
2
hC, arcsin(X) − Xi ≥ 0
π
• Nesterov’s result
Bibliography
∗
2
zMC
> = 0.6366
∗
∗
− zSDP
zMC
π
and hence
∗
zM
C
2
2 ∗
2 ∗
2
≥ hC, arcsin(X)i ≥ hC, Xi = zSDP ≥ zM C .
π
π
π
π
15 / 16
Some references
Introduction
An application in
statistical physics
The Goemans and
Williamson algorithm
Bibliography
• Some references
1 F Barahona, M Grötschel, M Jünger, G Reinelt. An application of combinatorial optimization to
statistical physics and circuit layout design. Operations Research 36(3): 493–513 (1988).
2 C. Simone, M. Diehl, M. Jünger, P. Mutzel, G. Reinelt and G. Rinaldi. Exact ground states of Ising
spin glasses: New experimental results with a branch-and-cut algorithm. Journal of statistical
physics 80(1-2): 487–496 (1995)
3 F. Liers, M. Jünger, G. Reinelt and G. Rinaldi, Computing Exact Ground States of Hard Ising Spin
Glass Problems by Branch-and-Cut, in New Optimization Algorithms in Physics, edited by A.
Hartmann and H. Rieger, pages: 47-68, (2004).
4 M. Laurent and S. Poljak On a positive semidefinite relaxation of the cut polytope. Linear Algebra
and its Application 223/224:439-461 (1995).
5 M. Laurent, S, Poljak and F. Rendl. Connections between semidefinite relaxations of the max-cut
and stable set problems. Math. Programming 77(2):225-246 (1997).
6
M. X. Goemans and D.P. Williamson. Improved approximation algorithms for maximum cut and
satisfiability problems using semidefinite programming J. ACM 42:1115-1145 (1995)
7 Y. Nesterov Semidefinite relaxation and nonconvex quadratic optimization Optimitization Methods
and Software, 9:141-160 (1998) .
16 / 16