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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. A vector x ∈ {0, 1}(n

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 = 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:

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 (∀b ∈ Zn). Here, σ(b) denotes

i∈V bi, and

γ(b) := min

• |zTb| : z ∈ {±1}n

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:

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 (∀b ∈ Zn). Here, σ(b) denotes

i∈V bi, and

γ(b) := min

• |zTb| : z ∈ {±1}n

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 ① ① ①

• ❅

❅ ❅ ❅ ❅

1 2 3 4

Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford

GAPs for the CUTn Extreme GAPs Separation Open Questions

Example

t t t ✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

S V \ S

2 1 3

❤❤❤❤❤❤❤❤❤❤❤❤❤ ❤ 4 t

Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford

GAPs for the CUTn Extreme GAPs Separation Open Questions

Example

t t t ✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

S V \ S

2 1 3

❤❤❤❤❤❤❤❤❤❤❤❤❤ ❤ 4 t

b2 b1 b3 b4 σ(b) = Pi=4

i=1 bi

b1b3 b1b4

Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford

GAPs for the CUTn Extreme GAPs Separation Open Questions

Example

t t t ✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

S V \ S

2 1 3

❤❤❤❤❤❤❤❤❤❤❤❤❤ ❤ 4 t

b2 b1 b3 b4 σ(b) = Pi=4

i=1 bi

b1b3 b1b4 P

i∈S bi = σ(b)/2

P

i∈V \S bi = σ(b)/2

P

1≤i<j≤n bibjxij ≤ σ(b)2/4 Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford

GAPs for the CUTn Extreme GAPs Separation Open Questions

Example

t t t ✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

S V \ S

2 1 3

❤❤❤❤❤❤❤❤❤❤❤❤❤ ❤ 4 t

b2 b1 b3 b4 σ(b) = Pi=4

i=1 bi

= 1 = 1 = 1

P

i∈S bi = 2

P

i∈V \S bi = 2

P

1≤i<j≤4 xij ≤ 4

= 4 = 1

Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford

GAPs for the CUTn Extreme GAPs Separation Open Questions

Example

t t t ✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

S V \ S

2 1 3 b2 b1 b3 = 1 = 1 = 1

P

i∈S bi = (σ(b) + γ(b))/2

P

i∈V \S bi = (σ(b) − γ(b))/2

γ(b) := min ˘ |zT b| : z ∈ {±1}n¯ σ(b) = Pi=3

i=1 bi = 3

P

1≤i<j≤n bibjxij ≤

` σ(b)2 − γ(b)2´ /4

Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford

GAPs for the CUTn Extreme GAPs Separation Open Questions

Example

t t t ✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

S V \ S

2 1 3 b2 b1 b3 = 1 = 1 = 1

P

i∈S bi = (σ(b) + γ(b))/2

P

i∈V \S bi = (σ(b) − γ(b))/2

γ(b) := min ˘ |zT b| : z ∈ {±1}n¯ σ(b) = Pi=3

i=1 bi = 3

P

1≤i<j≤3 xij ≤ 2 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...

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...

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 ∀b ∈ Zn

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...

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 ∀b ∈ Zn

X

1≤i<j≤n

bibjxij ≤ ⌊σ(b)2/4⌋ and σ(b) odd ∀ b ∈ Zn

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...

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 ∀b ∈ Zn

X

1≤i<j≤n

bibjxij ≤ σ(b)2/4 ∀b ∈ Rn

gap ✲ rounded psd ✲ 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...

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 ∀b ∈ Zn

X

1≤i<j≤n

bibjxij ≤ σ(b)2/4 and γ(b) = 0 ∀ b ∈ Zn

gap ✲ rounded psd ✲ psd ✲ gap-0

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...

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 ∀b ∈ Zn

X

1≤i<j≤n

bibjxij ≤ 0 and σ(b) = 0 ∀ b ∈ Zn

gap ✲ rounded psd ✲ psd ✲ gap-0 ✲negative-type

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...

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 ∀b ∈ Zn

X

1≤i<j≤n

bibjxij ≤ ` σ(b)2 − γ(b)2´ /4 and γ(b) = 1 ∀ b ∈ Zn

gap ✲ rounded psd ✲ psd ✲ gap-0 ✲negative-type 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...

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 ∀b ∈ Zn

X

1≤i<j≤n

bibjxij ≤ ` σ(b)2 − γ(b)2´ /4 and γ(b) = 1 ∀ b ∈ Zn

gap ✲ rounded psd ✲ psd ✲ gap-0 ✲negative-type 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...

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 ∀b ∈ Zn

X

1≤i<j≤n

bibjxij ≤ 0 and σ(b) = 1 ∀ b ∈ Zn

gap ✲ rounded psd ✲ psd ✲ gap-0 ✲negative-type gap-1 ❆ ❆ ❆ ❆ ❆ ❆ ❯ ✑✑✑✑✑✑✑✑ ✑ ✸ ✲ hypermetric

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...

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 ∀b ∈ Zn

X

1≤i<j≤n

bibjxij ≤ 0 and σ(b) = 1 ∀ b ∈ Zn

gap ✲ rounded psd ✲ psd ✲ gap-0 ✲negative-type gap-1 ❆ ❆ ❆ ❆ ❆ ❆ ❯ ✑✑✑✑✑✑✑✑ ✑ ✸ ✲ hypermetric ✑✑✑✑✑✑✑✑ ✑ ✸

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...

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 ∀b ∈ Zn

X

1≤i<j≤n

bibjxij ≤ ⌊σ(b)2/4⌋ and σ(b) odd ∀ b ∈ {0, |1|}n

gap ✲ rounded psd ✲ psd ✲ gap-0 ✲negative-type gap-1 ❆ ❆ ❆ ❆ ❆ ❆ ❯ ✑✑✑✑✑✑✑✑ ✑ ✸ ✲ hypermetric ✑✑✑✑✑✑✑✑ ✑ ✸ ❏ ❏ ❫odd 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...

• 1≤i<j≤n

bibjxij ≤

• σ(b)2 − γ(b)2

/4 ∀b ∈ Zn

xij + xik + xjk ≤ 2 (1 ≤ i < j < k ≤ n) xij − xik − xjk ≤ 0 (1 ≤ i < j ≤ n; k = i, j)

gap ✲ rounded psd ✲ psd ✲ gap-0 ✲negative-type gap-1 ❆ ❆ ❆ ❆ ❆ ❆ ❯ ✑✑✑✑✑✑✑✑ ✑ ✸ ✲ hypermetric ✑✑✑✑✑✑✑✑ ✑ ✸ ❏ ❏ ❫odd clique ✲ triangle

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 NP-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 NP-complete: Given positive integers n and k and a vector b ∈ Zn, is γ(b) < k?’

1 A set S such that i∈S bi − i∈V \S bi < k is a short

certificate.

2 Reduction from the ‘partition’ problem.

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 NP-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 NP-complete: Given positive integers n and k and a vector b ∈ Zn, is γ(b) < k?’ Lemma 3: The following decision problem is Co NP-complete: Given n, k ∈ N, b ∈ Zn, is

1≤i<j≤n bibjxij ≤ σ(b)2−k2 4

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 NP-complete: Given positive integers n and k and a vector b ∈ Zn, is γ(b) < k?’ Lemma 3: The following decision problem is Co NP-complete: Given n, k ∈ N, b ∈ Zn, is

1≤i<j≤n bibjxij ≤ σ(b)2−k2 4

valid for CUTn? Theorem 1 : Suppose that every gap inequality is either a rounded psd, or implied by rounded psd’s. Then NP = Co NP.

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 NP-complete: Given positive integers n and k and a vector b ∈ Zn, is γ(b) < k?’ Lemma 3: The following decision problem is Co NP-complete: Given n, k ∈ N, b ∈ Zn, is

1≤i<j≤n bibjxij ≤ σ(b)2−k2 4

valid for CUTn? Theorem 1 : Suppose that every gap inequality is either a rounded psd, or implied by rounded psd’s. Then NP = Co NP. Corollary 1 : Suppose that every gap inequality is either a gap-1 inequality, or implied by gap-1 inequalities. Then NP = Co NP.

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 NP-complete: Given positive integers n and k and a vector b ∈ Zn, is γ(b) < k?’ Lemma 3: The following decision problem is Co NP-complete: Given n, k ∈ N, b ∈ Zn, is

1≤i<j≤n bibjxij ≤ σ(b)2−k2 4

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 NP-complete: Given positive integers n and k and a vector b ∈ Zn, is γ(b) < k?’ Lemma 3: The following decision problem is Co NP-complete: Given n, k ∈ N, b ∈ Zn, is

1≤i<j≤n bibjxij ≤ σ(b)2−k2 4

valid for CUTn? Lemma 4 : γ(b) can be computed in O(n||b||1) time. γ(b) = ||b||1−2

• max
• i∈V

|bi|yi :

• i∈V

|bi|yi ≤ ||b||1/2, y ∈ {0, 1}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 2: The following decision problem is NP-complete: Given positive integers n and k and a vector b ∈ Zn, is γ(b) < k?’ Lemma 3: The following decision problem is Co NP-complete: Given n, k ∈ N, b ∈ Zn, is

1≤i<j≤n bibjxij ≤ σ(b)2−k2 4

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 NP = Co NP.

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 NP: ‘Given an integer n ≥ 2 and x∗ ∈ [0, 1](n

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 NP: ‘Given an integer n ≥ 2 and x∗ ∈ [0, 1](n

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

• i=1

b2

i +

• 1≤i<j≤n

(2 − 4x∗

ij)bibj : γ(b) = 1, b ∈ [−U, U]n ∩ Zn

  

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

• i=1

b2

i +

• 1≤i<j≤n

(2 − 4x∗

ij)bibj : γ(b) = 1, b ∈ [−U, U]n ∩ Zn

  

• i∈S

bi −

• i∈V \S

bi = 1

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

• i=1

b2

i +

• 1≤i<j≤n

(2 − 4x∗

ij)bibj : γ(b) = 1, b ∈ [−U, U]n ∩ Zn

  

n

• i=1

bisi −

n

• i=1

bi(1 − si) =

n

• i=1

bi(2si − 1) = 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

• i=1

b2

i +

• 1≤i<j≤n

(2 − 4x∗

ij)bibj : γ(b) = 1, b ∈ [−U, U]n ∩ Zn

   n

1(2pi − 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 pi ∈ Zn, bi ∈ {0, 1}n, 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: General Gap Inequalities

A violated gap inequality exists iff : min   

n

• i=1

b2

i +

• 1≤i<j≤n

(2 − 4x∗

ij)bibj : γ(b) = 1, b ∈ Qn

   < 1

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: General Gap Inequalities

A violated gap inequality exists iff : min   

n

• i=1

b2

i +

• 1≤i<j≤n

(2 − 4x∗

ij)bibj : γ(b) ≥ 1, b ∈ Qn

   < 1  

i∈S

bi −

• i∈V \S

bi ≥ 1  ∨  

i∈S

bi −

• i∈V \S

bi ≤ −1   (∀S ⊂ V )

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: General Gap Inequalities

A violated gap inequality exists iff : min   

n

• i=1

b2

i +

• 1≤i<j≤n

(2 − 4x∗

ij)bibj : γ(b) ≥ 1, b ∈ Qn

   < 1 Solve the following Convex Quadratic Program for all subsets F′ ⊆ F, where F = 2V min n

i=1 b2 i + 1≤i<j≤n(2 − 4x∗ ij)bibj

s.t.

• i∈S bi −

i∈V \S bi ≥ 1

(∀S ∈ F′)

• i∈S bi −

i∈V \S bi ≤ −1

(∀S ∈ F \ F′) 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 NP = Co NP:

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