GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Complexity Results for the Gap Inequalities
Laura Galli
Konstantinos Kaparis
Adam N. Letchford
16th Combinatorial Optimization Workshop, Aussois, France
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Motivation
In 1996, Laurent & Poljak introduced a very general class of valid
inequalities for the cut polytope, called gap inequalities.
Gap inequalities were largely ignored thereafter and
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Motivation
In 1996, Laurent & Poljak introduced a very general class of valid
inequalities for the cut polytope, called gap inequalities.
Gap inequalities were largely ignored thereafter and
• the relevant complexity results are very limited,
• while there is not known separation algorithm for them.
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Outline
1
Gap Inequalities for the Cut Polytope
2
On Extreme Gap Inequalities
3
On the Complexity of Separation
4
Open Questions
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
The Cut Polytope
Given a graph G = (V , E ) the edge set
δ(S) = {(i, j) ∈ E : i ∈ S, j ∈ (V \ S), S ⊆ V } ,
defines a cut of G .
n
A vector x ∈ {0, 1}(2) is the incidence vector of a cut in Kn iff it
satisfies the following set of triangle inequalities:
xij + xik + xjk
≤ 2 (1 ≤ i < j < k ≤ n)
xij − xik − xjk
≤ 0 (1 ≤ i < j ≤ n; k 6= i, j)
The cut polytope (CUTn ), is the convex hull of such vectors.
CUTn has been studied intensively in the literature and many
classes of valid inequalities are known (see Deza & Laurent, 1997).
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Definition (Laurent & Poljak, 1997)
Gap inequalities for CUTn , take the following form:
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
(∀b ∈ Zn ).
1≤i<j≤n
Here, σ(b) denotes
P
bi , and
n
o
γ(b) := min |z T b| : z ∈ {±1}n
i∈V
is the so-called gap of b.
Every gap inequality defines a proper face of CUTn .
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Definition (Laurent & Poljak, 1997)
Gap inequalities for CUTn , take the following form:
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
(∀b ∈ Zn ).
1≤i<j≤n
Here, σ(b) denotes
P
bi , and
n
o
γ(b) := min |z T b| : z ∈ {±1}n
i∈V
is the so-called gap of b.
Every gap inequality defines a proper face of CUTn .
Conjecture
All facet defining gap inequalities have γ(b) = 1.
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Example
1
x
@
@
x
3
Complexity Results for the Gap Inequalities
2
x
@
@
@x
4
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Example
S
' $
V \S
' $
hhhh
1 t
((t3
hhh
((((
(
(
h
(
h
(
hhhh
t (((
ht4
2 (
& %
Complexity Results for the Gap Inequalities
& %
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Example
σ(b) =
Pi=4
i=1
V \S
S
' $
b1
b2
hhhh
1 t
bi
' $
((t3
hhh
((((
(
b1 b3
(
h
(
h
(
hhhh
t (((
ht4
2 (
b1 b4
& %
Complexity Results for the Gap Inequalities
b3
b4
& %
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Example
σ(b) =
P
i∈S
Pi=4
i=1
P
bi = σ(b)/2
' $
b2
hhhh
1 t
i∈V \S
bi = σ(b)/2
V \S
S
b1
bi
' $
((t3
hhh
((((
(
b1 b3
(
h
(
h
(
hhhh
t (((
ht4
2 (
b1 b4
& %
P
Complexity Results for the Gap Inequalities
1≤i<j≤n
b3
b4
& %
bi bj xij ≤ σ(b)2 /4
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Example
σ(b) =
P
i∈S
Pi=4
i=1
bi
P
bi = 2
=4
i∈V \S
bi = 2
V \S
S
' $
' $
hhhh
1 t
((t3
hhh
((((
(
(
h
(
h
(
hhhh
t (((
ht4
2 (
b1= 1
b2= 1
& %
P
Complexity Results for the Gap Inequalities
1≤i<j≤4 xij
b3 = 1
b4 = 1
& %
≤4
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Example
˘
¯
γ(b) := min |z T b| : z ∈ {±1}n
P
σ(b) = i=3
i=1 bi = 3
P
i∈S
P
bi = (σ(b) + γ(b))/2
' $
' $
b1= 1
1 t
b2= 1
((((
(((
(
t
(
2
P
Complexity Results for the Gap Inequalities
1≤i<j≤n
bi = (σ(b) − γ(b))/2
V \S
S
& %
i∈V \S
((t3
(((
b3 = 1
& %
´
`
bi bj xij ≤ σ(b)2 − γ(b)2 /4
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Example
˘
¯
γ(b) := min |z T b| : z ∈ {±1}n
P
σ(b) = i=3
i=1 bi = 3
P
i∈S
P
bi = (σ(b) + γ(b))/2
' $
' $
b1= 1
1 t
b2= 1
((((
(((
(
t
(
2
& %
Complexity Results for the Gap Inequalities
1≤i<j≤3 xij
bi = (σ(b) − γ(b))/2
V \S
S
P
i∈V \S
((t3
(((
b3 = 1
& %
≤2
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Dominated or Implied Inequalities
Gap inequalities are remarkably general...
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Dominated or Implied Inequalities
Gap inequalities are remarkably general...
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
∀b ∈ Zn
1≤i<j≤n
gap
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Dominated or Implied Inequalities
Gap inequalities are remarkably general...
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
∀b ∈ Zn
1≤i<j≤n
X
n
bi bj xij ≤ bσ(b)2 /4c and σ(b) odd ∀ b ∈ Z
1≤i<j≤n
gap
- rounded psd
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Dominated or Implied Inequalities
Gap inequalities are remarkably general...
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
∀b ∈ Zn
1≤i<j≤n
X
bi bj xij ≤ σ(b)2 /4
∀b ∈ R
n
1≤i<j≤n
gap
- rounded psd
Complexity Results for the Gap Inequalities
- psd
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Dominated or Implied Inequalities
Gap inequalities are remarkably general...
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
∀b ∈ Zn
1≤i<j≤n
X
bi bj xij ≤ σ(b)2 /4 and γ(b) = 0
∀
n
b∈Z
1≤i<j≤n
gap
- rounded psd
Complexity Results for the Gap Inequalities
- psd
- gap-0
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Dominated or Implied Inequalities
Gap inequalities are remarkably general...
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
∀b ∈ Zn
1≤i<j≤n
X
bi bj xij ≤ 0 and σ(b) = 0
∀
n
b∈Z
1≤i<j≤n
gap
- rounded psd
Complexity Results for the Gap Inequalities
- psd
- gap-0
-negative-type
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Dominated or Implied Inequalities
Gap inequalities are remarkably general...
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
∀b ∈ Zn
1≤i<j≤n
X
`
´
bi bj xij ≤ σ(b)2 − γ(b)2 /4 and γ(b) = 1
∀
n
b∈Z
1≤i<j≤n
gap
- rounded psd
A
A
A
A
A
U
A
gap-1
Complexity Results for the Gap Inequalities
- psd
- gap-0
-negative-type
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Dominated or Implied Inequalities
Gap inequalities are remarkably general...
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
∀b ∈ Zn
1≤i<j≤n
X
`
´
bi bj xij ≤ σ(b)2 − γ(b)2 /4 and γ(b) = 1
∀
n
b∈Z
1≤i<j≤n
gap
- rounded psd
- psd
- gap-0
A
3
A
A
A
A
U
A
gap-1 Complexity Results for the Gap Inequalities
-negative-type
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Dominated or Implied Inequalities
Gap inequalities are remarkably general...
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
∀b ∈ Zn
1≤i<j≤n
X
bi bj xij ≤ 0 and σ(b) = 1
∀
n
b∈Z
1≤i<j≤n
gap
- rounded psd
- psd
- gap-0
-negative-type
A
3
A
A
A
A
U
A
- hypermetric
gap-1 Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Dominated or Implied Inequalities
Gap inequalities are remarkably general...
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
∀b ∈ Zn
1≤i<j≤n
X
bi bj xij ≤ 0 and σ(b) = 1
∀
n
b∈Z
1≤i<j≤n
gap
- rounded psd
- psd
- gap-0
-negative-type
3
A
3
A
A
A
A
U
A
- hypermetric
gap-1 Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Dominated or Implied Inequalities
Gap inequalities are remarkably general...
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
∀b ∈ Zn
1≤i<j≤n
X
bi bj xij ≤ bσ(b)2 /4c and σ(b) odd ∀
b ∈ {0, |1|}n
1≤i<j≤n
gap
- rounded psd
- psd
- gap-0
-negative-type
3
A
3
A
A
A
A
U
A
- hypermetric
gap-1 J
^
Jodd clique
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Gap Inequalities: Dominated or Implied Inequalities
Gap inequalities are remarkably general...
X
bi bj xij ≤ σ(b)2 − γ(b)2 /4
∀b ∈ Zn
1≤i<j≤n
gap
xij + xik + xjk
≤2
(1 ≤ i < j < k ≤ n)
xij − xik − xjk
≤0
(1 ≤ i < j ≤ n; k 6= i, j)
- rounded psd
- psd
- gap-0
-negative-type
3
A
3
A
A
A
A
U
A
- hypermetric
gap-1 J
- triangle
^
Jodd clique
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On Extreme Gap Inequalities
Lemma 1: The encoding length of the coefficients of an
extreme rounded psd inequality is polynomially bounded in n.
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On Extreme Gap Inequalities
Lemma 1: The encoding length of the coefficients of an
extreme rounded psd inequality is polynomially bounded in n.
1
Rounded psd’s separation can be reduced to hypermetric
separation (Letchford & Sørensen, 2010)
2
The coefficients of extreme hypermetric inequalities are
bounded by a polynomial in n (Avis & Grishukin, 1993)
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On Extreme Gap Inequalities
Lemma 1: The encoding length of the coefficients of an
extreme rounded psd inequality is polynomially bounded in n.
Lemma 2: The following decision problem is N P-complete:
Given positive integers n and k and a vector b ∈ Zn , is γ(b) < k?’
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On Extreme Gap Inequalities
Lemma 1: The encoding length of the coefficients of an
extreme rounded psd inequality is polynomially bounded in n.
Lemma 2: The following decision problem is N P-complete:
Given positive integers n and k and a vector b ∈ Zn , is γ(b) < k?’
P
bi −
P
1
A set S such that
certificate.
2
Reduction from the ‘partition’ problem.
Complexity Results for the Gap Inequalities
i∈S
i∈V \S
bi < k is a short
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On Extreme Gap Inequalities
Lemma 1: The encoding length of the coefficients of an
extreme rounded psd inequality is polynomially bounded in n.
Lemma 2: The following decision problem is N P-complete:
Given positive integers n and k and a vector b ∈ Zn , is γ(b) < k?’
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On Extreme Gap Inequalities
Lemma 1: The encoding length of the coefficients of an
extreme rounded psd inequality is polynomially bounded in n.
Lemma 2: The following decision problem is N P-complete:
Given positive integers n and k and a vector b ∈ Zn , is γ(b) < k?’
Lemma 3: The following decision problem is Co N P-complete:
2
2
P
Given n, k ∈ N, b ∈ Zn , is 1≤i<j≤n bi bj xij ≤ σ(b)4 −k valid for
CUTn ?
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On Extreme Gap Inequalities
Lemma 1: The encoding length of the coefficients of an
extreme rounded psd inequality is polynomially bounded in n.
Lemma 2: The following decision problem is N P-complete:
Given positive integers n and k and a vector b ∈ Zn , is γ(b) < k?’
Lemma 3: The following decision problem is Co N P-complete:
2
2
P
Given n, k ∈ N, b ∈ Zn , is 1≤i<j≤n bi bj xij ≤ σ(b)4 −k valid for
CUTn ?
Theorem 1 : Suppose that every gap inequality is either a
rounded psd, or implied by rounded psd’s. Then N P = Co N P.
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On Extreme Gap Inequalities
Lemma 1: The encoding length of the coefficients of an
extreme rounded psd inequality is polynomially bounded in n.
Lemma 2: The following decision problem is N P-complete:
Given positive integers n and k and a vector b ∈ Zn , is γ(b) < k?’
Lemma 3: The following decision problem is Co N P-complete:
2
2
P
Given n, k ∈ N, b ∈ Zn , is 1≤i<j≤n bi bj xij ≤ σ(b)4 −k valid for
CUTn ?
Theorem 1 : Suppose that every gap inequality is either a
rounded psd, or implied by rounded psd’s. Then N P = Co N P.
Corollary 1 : Suppose that every gap inequality is either a gap-1
inequality, or implied by gap-1 inequalities. Then N P = Co N P.
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On Extreme Gap Inequalities
Lemma 2: The following decision problem is N P-complete:
Given positive integers n and k and a vector b ∈ Zn , is γ(b) < k?’
Lemma 3: The following decision problem is Co N P-complete:
2
2
P
Given n, k ∈ N, b ∈ Zn , is 1≤i<j≤n bi bj xij ≤ σ(b)4 −k valid for
CUTn ?
Lemma 4 : γ(b) can be computed in O(n||b||1 ) time.
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On Extreme Gap Inequalities
Lemma 2: The following decision problem is N P-complete:
Given positive integers n and k and a vector b ∈ Zn , is γ(b) < k?’
Lemma 3: The following decision problem is Co N P-complete:
2
2
P
Given n, k ∈ N, b ∈ Zn , is 1≤i<j≤n bi bj xij ≤ σ(b)4 −k valid for
CUTn ?
Lemma 4 : γ(b) can be computed in O(n||b||1 ) time.
(
)!
X
X
n
γ(b) = ||b||1 −2 max
|bi |yi :
|bi |yi ≤ ||b||1 /2, y ∈ {0, 1}
i∈V
Complexity Results for the Gap Inequalities
i∈V
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On Extreme Gap Inequalities
Lemma 2: The following decision problem is N P-complete:
Given positive integers n and k and a vector b ∈ Zn , is γ(b) < k?’
Lemma 3: The following decision problem is Co N P-complete:
2
2
P
Given n, k ∈ N, b ∈ Zn , is 1≤i<j≤n bi bj xij ≤ σ(b)4 −k valid for
CUTn ?
Lemma 4 : γ(b) can be computed in O(n||b||1 ) time.
Theorem 2: Suppose that there exists a polynomial p(n) such
that every gap inequality is implied by gap inequalities with
||b||1 ≤ p(n). Then N P = Co N P.
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On the Separation Problem: Gap-1 Inequalities
Lemma 5: If a gap-1 inequality is extreme then the encoding
length of its coefficients is polynomially bounded in n.
Lemma 6 : The following problem is in N P: ‘Given an integer
n
n ≥ 2 and x ∗ ∈ [0, 1](2) , does x ∗ violate a gap-1 inequality?’.
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On the Separation Problem: Gap-1 Inequalities
Lemma 5: If a gap-1 inequality is extreme then the encoding
length of its coefficients is polynomially bounded in n.
Lemma 6 : The following problem is in N P: ‘Given an integer
n
n ≥ 2 and x ∗ ∈ [0, 1](2) , does x ∗ violate a gap-1 inequality?’.
Theorem 3: The separation problem for gap-1 inequalities can
be formulated as an IQP with O(n) variables and constraints.
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On the Separation Problem: Gap-1 Inequalities
Theorem 3: The separation problem for gap-1 inequalities can
be formulated as an IQP with O(n) variables and constraints.
min

n
X

i=1
bi2 +
X
1≤i<j≤n
Complexity Results for the Gap Inequalities
(2 − 4xij∗ )bi bj : γ(b) = 1, b ∈ [−U, U]n ∩ Zn



Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On the Separation Problem: Gap-1 Inequalities
Theorem 3: The separation problem for gap-1 inequalities can
be formulated as an IQP with O(n) variables and constraints.
min

n
X

i=1
bi2 +
X
(2 − 4xij∗ )bi bj : γ(b) = 1, b ∈ [−U, U]n ∩ Zn

1≤i<j≤n
X
i∈S
Complexity Results for the Gap Inequalities


bi −
X
bi = 1
i∈V \S
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On the Separation Problem: Gap-1 Inequalities
Theorem 3: The separation problem for gap-1 inequalities can
be formulated as an IQP with O(n) variables and constraints.
min

n
X

bi2 +
X
(2 − 4xij∗ )bi bj : γ(b) = 1, b ∈ [−U, U]n ∩ Zn

1≤i<j≤n
i=1
n
X
bi si −
n
X
i=1


bi (1 − si ) =
i=1
n
X
bi (2si − 1) = 1,
i=1
for bi ∈ {0, 1}n and bi = 1 iff i ∈ S.
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On the Separation Problem: Gap-1 Inequalities
Theorem 3: The separation problem for gap-1 inequalities can
be formulated as an IQP with O(n) variables and constraints.
min

n
X

X
bi2 +
i=1
(2 − 4xij∗ )bi bj : γ(b) = 1, b ∈ [−U, U]n ∩ Zn

1≤i<j≤n
Pn
1 (2pi
pi ∈
Zn ,
− bi ) = 1
pi ≤ Usi
i ∈ 1, . . . , n
pi ≥ −Usi
i ∈ 1, . . . , n
bi − pi + Usi ≤ U
i ∈ 1, . . . , n
−bi + pi + Usi ≤ U
i ∈ 1, . . . , n
bi ∈
Complexity Results for the Gap Inequalities


{0, 1}n ,
bi = 1 iff i ∈ S
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On the Separation Problem: General Gap Inequalities
A violated gap inequality exists iff :
min

n
X

i=1
bi2 +
X
1≤i<j≤n
Complexity Results for the Gap Inequalities
(2 − 4xij∗ )bi bj : γ(b) = 1, b ∈ Qn


<1

Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On the Separation Problem: General Gap Inequalities
A violated gap inequality exists iff :
min

n
X

bi2 +
i=1
X
(2 − 4xij∗ )bi bj : γ(b) ≥ 1, b ∈ Qn
i∈V \S
Complexity Results for the Gap Inequalities
<1

1≤i<j≤n

 

X
X
X
X

bi −
bi ≥ 1∨
bi −
bi ≤ −1
i∈S


i∈S
(∀S ⊂ V )
i∈V \S
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
On the Separation Problem: General Gap Inequalities
A violated gap inequality exists iff :
min

n
X

i=1
bi2 +
X
(2 − 4xij∗ )bi bj : γ(b) ≥ 1, b ∈ Qn
1≤i<j≤n


<1

Solve the following Convex Quadratic Program for all subsets
F 0 ⊆ F, where F = 2V
Pn
P
2
∗
min
i=1 bi +
1≤i<j≤n (2 − 4xij )bi bj
P
P
s.t.
(∀S ∈ F 0 )
i∈S bi −
i∈V \S bi ≥ 1
P
P
(∀S ∈ F \ F 0 )
i∈S bi −
i∈V \S bi ≤ −1
b ∈ Qn .
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Conclusions
• We prove that unless if N P = Co N P:
1 The gap inequalities with γ(b) > 1 do not imply all other gap
inequalities.
2 There must exist gap inequalities with exponentially large
coefficients that they are not implied by other gap inequalities.
• Gap-1 separation can be formulated as an IQP with O(n)
variables and constraints.
• There is a finite (doubly exponential) separation algorithm for
general gap inequalities.
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Open Questions
• Does there exist a singly-exponential separation algorithm for
gap inequalities?
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Open Questions
• Does there exist a singly-exponential separation algorithm for
gap inequalities?
• Do gap inequalities define a polyhedron?
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Open Questions
• Does there exist a singly-exponential separation algorithm for
gap inequalities?
• Do gap inequalities define a polyhedron?
• Is there a gap inequality with γ(b) > 1 that induces a facet of
CUTn ?
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUTn
Extreme GAPs
Separation
Open Questions
Thank you for your attention!
Questions?
Complexity Results for the Gap Inequalities
Laura Galli, Konstantinos Kaparis, Adam N. Letchford