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Generalized Chvatal-Gomory Closures for Integer Programswith Bounds on Variables
Dabeen Lee
Carnegie Mellon University
January 7, 2018
Joint work with Sanjeeb Dash and Oktay Gunluk
1/20
Integer linear programming
Integer linear programming (ILP)
Given A ∈ Qm×n, b ∈ Qm, and w ∈ Qn, we want to solve
min wxs.t. Ax ≥ b
x ∈ Zn P = {x : Ax ≥ b}
• General-purpose cutting planes are developed for solving ILP.
• Split cuts, Gomory’s GMI cuts, Chvatal-Gomory cuts, etc.
• We will study a generalization of Chvatal-Gomory cuts.
2/20
Integer linear programming
Integer linear programming (ILP)
Given A ∈ Qm×n, b ∈ Qm, and w ∈ Qn, we want to solve
min wxs.t. Ax ≥ b
x ∈ Zn P = {x : Ax ≥ b}
• General-purpose cutting planes are developed for solving ILP.
• Split cuts, Gomory’s GMI cuts, Chvatal-Gomory cuts, etc.
• We will study a generalization of Chvatal-Gomory cuts.
2/20
Integer linear programming
Integer linear programming (ILP)
Given A ∈ Qm×n, b ∈ Qm, and w ∈ Qn, we want to solve
min wxs.t. Ax ≥ b
x ∈ Zn P = {x : Ax ≥ b}
• General-purpose cutting planes are developed for solving ILP.
• Split cuts, Gomory’s GMI cuts, Chvatal-Gomory cuts, etc.
• We will study a generalization of Chvatal-Gomory cuts.
2/20
Integer linear programming
Integer linear programming (ILP)
Given A ∈ Qm×n, b ∈ Qm, and w ∈ Qn, we want to solve
min wxs.t. Ax ≥ b
x ∈ Zn P = {x : Ax ≥ b}
• General-purpose cutting planes are developed for solving ILP.
• Split cuts, Gomory’s GMI cuts, Chvatal-Gomory cuts, etc.
• We will study a generalization of Chvatal-Gomory cuts.
2/20
3/20
Chvatal-Gomory closure
• The Chvatal-Gomory closure of a rational polyhedron P ⊆ Rn is defined as
P ′ :=⋂c∈Zn
{x ∈ Rn : cx ≥ dmin{cy : y ∈ P}e}
P
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
3/20
Chvatal-Gomory closure
• The Chvatal-Gomory closure of a rational polyhedron P ⊆ Rn is defined as
P ′ :=⋂c∈Zn
{x ∈ Rn : cx ≥ dmin{cy : y ∈ P}e}
P
P ′
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
4/20
Chvatal-Gomory closure
• The Chvatal-Gomory closure of a rational polyhedron P ⊆ Rn is defined as
P ′ :=⋂c∈Zn
{x ∈ Rn : cx ≥ dmin{cy : y ∈ P}e}
Theorem [Chvatal, 1973, Schrijver, 1980]
Let P be a rational polyhedron. Then P ′ is also a rational polyhedron.
• We often face integer programs that involve constraints to bound thevalues of variables x (problems in practice and in combinatorialoptimization).
• Can we provide a generalization of Chvatal-Gomory closure closures forsuch integer programs?
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
4/20
Chvatal-Gomory closure
• The Chvatal-Gomory closure of a rational polyhedron P ⊆ Rn is defined as
P ′ :=⋂c∈Zn
{x ∈ Rn : cx ≥ dmin{cy : y ∈ P}e}
Theorem [Chvatal, 1973, Schrijver, 1980]
Let P be a rational polyhedron. Then P ′ is also a rational polyhedron.
• We often face integer programs that involve constraints to bound thevalues of variables x (problems in practice and in combinatorialoptimization).
• Can we provide a generalization of Chvatal-Gomory closure closures forsuch integer programs?
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
4/20
Chvatal-Gomory closure
• The Chvatal-Gomory closure of a rational polyhedron P ⊆ Rn is defined as
P ′ :=⋂c∈Zn
{x ∈ Rn : cx ≥ dmin{cy : y ∈ P}e}
Theorem [Chvatal, 1973, Schrijver, 1980]
Let P be a rational polyhedron. Then P ′ is also a rational polyhedron.
• We often face integer programs that involve constraints to bound thevalues of variables x (problems in practice and in combinatorialoptimization).
• Can we provide a generalization of Chvatal-Gomory closure closures forsuch integer programs?
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
5/20
Generalization of Chvatal-Gomory inequalities
P
min{cx : x ∈ P}
• S := {x ∈ Zn : ` ≤ x ≤ u}.• P ⊆ {x ∈ Rn : ` ≤ x ≤ u}.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
5/20
Generalization of Chvatal-Gomory inequalities
P
min{cx : x ∈ P}
• S := {x ∈ Zn : ` ≤ x ≤ u}.• P ⊆ {x ∈ Rn : ` ≤ x ≤ u}.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
5/20
Generalization of Chvatal-Gomory inequalities
P
dmin{cx : x ∈ P}e
• S := {x ∈ Zn : ` ≤ x ≤ u}.• P ⊆ {x ∈ Rn : ` ≤ x ≤ u}.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
5/20
Generalization of Chvatal-Gomory inequalities
P
S
• S := {x ∈ Zn : ` ≤ x ≤ u}.• P ⊆ {x ∈ Rn : ` ≤ x ≤ u}.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
5/20
Generalization of Chvatal-Gomory inequalities
P
S
• S := {x ∈ Zn : ` ≤ x ≤ u}.• P ⊆ {x ∈ Rn : ` ≤ x ≤ u}.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
5/20
Generalization of Chvatal-Gomory inequalities
P
S
• S := {x ∈ Zn : ` ≤ x ≤ u}.• P ⊆ {x ∈ Rn : ` ≤ x ≤ u}.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
6/20
Generalization of Chvatal-Gomory inequalities
P
S
min{cx : x ∈ P}dmin{cx : x ∈ P}e
min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
7/20
Generalization of Chvatal-Gomory inequalities
P
S
min{cx : x ∈ P}min{cz : cz ≥ min{cx : x ∈ P}, z ∈ Zn}
min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
8/20
Generalization of Chvatal-Gomory closures
Let S ⊆ Zn be the set of integer points that satisfy all the bound constraints.
Given a valid inequality cx ≥ d for a polyhedron P,
• cx ≥ dde = min{cz : cz ≥ d , z ∈ Zn} is the Chvatal-Gomory inequalityobtained from it.
• However, cx ≥ min{cz : cz ≥ d , z ∈ S} is also valid for P ∩ Zn (⊆ S).
• We have min{cz : cz ≥ d , z ∈ S} ≥ dde = min{cz : cz ≥ d , z ∈ Zn}.
Given S such that P ⊆ conv(S) and S = conv(S) ∩ Zn, theS-Chvatal-Gomory-Gomory closure of a rational polyhedron P ⊆ Rn is definedas
P ′S :=⋂c∈Zn
x ∈ Rn : cx ≥ min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}︸ ︷︷ ︸S-Chvatal-Gomory inequality
• It was first studied by Dunkel and Schulz (2012) and Pokutta (2011).
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
8/20
Generalization of Chvatal-Gomory closures
Let S ⊆ Zn be the set of integer points that satisfy all the bound constraints.
Given a valid inequality cx ≥ d for a polyhedron P,
• cx ≥ dde = min{cz : cz ≥ d , z ∈ Zn} is the Chvatal-Gomory inequalityobtained from it.
• However, cx ≥ min{cz : cz ≥ d , z ∈ S} is also valid for P ∩ Zn (⊆ S).
• We have min{cz : cz ≥ d , z ∈ S} ≥ dde = min{cz : cz ≥ d , z ∈ Zn}.
Given S such that P ⊆ conv(S) and S = conv(S) ∩ Zn, theS-Chvatal-Gomory-Gomory closure of a rational polyhedron P ⊆ Rn is definedas
P ′S :=⋂c∈Zn
x ∈ Rn : cx ≥ min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}︸ ︷︷ ︸S-Chvatal-Gomory inequality
• It was first studied by Dunkel and Schulz (2012) and Pokutta (2011).
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
8/20
Generalization of Chvatal-Gomory closures
Let S ⊆ Zn be the set of integer points that satisfy all the bound constraints.
Given a valid inequality cx ≥ d for a polyhedron P,
• cx ≥ dde = min{cz : cz ≥ d , z ∈ Zn} is the Chvatal-Gomory inequalityobtained from it.
• However, cx ≥ min{cz : cz ≥ d , z ∈ S} is also valid for P ∩ Zn (⊆ S).
• We have min{cz : cz ≥ d , z ∈ S} ≥ dde = min{cz : cz ≥ d , z ∈ Zn}.
Given S such that P ⊆ conv(S) and S = conv(S) ∩ Zn, theS-Chvatal-Gomory-Gomory closure of a rational polyhedron P ⊆ Rn is definedas
P ′S :=⋂c∈Zn
x ∈ Rn : cx ≥ min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}︸ ︷︷ ︸S-Chvatal-Gomory inequality
• It was first studied by Dunkel and Schulz (2012) and Pokutta (2011).
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
8/20
Generalization of Chvatal-Gomory closures
Let S ⊆ Zn be the set of integer points that satisfy all the bound constraints.
Given a valid inequality cx ≥ d for a polyhedron P,
• cx ≥ dde = min{cz : cz ≥ d , z ∈ Zn} is the Chvatal-Gomory inequalityobtained from it.
• However, cx ≥ min{cz : cz ≥ d , z ∈ S} is also valid for P ∩ Zn (⊆ S).
• We have min{cz : cz ≥ d , z ∈ S} ≥ dde = min{cz : cz ≥ d , z ∈ Zn}.
Given S such that P ⊆ conv(S) and S = conv(S) ∩ Zn, theS-Chvatal-Gomory-Gomory closure of a rational polyhedron P ⊆ Rn is definedas
P ′S :=⋂c∈Zn
x ∈ Rn : cx ≥ min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}︸ ︷︷ ︸S-Chvatal-Gomory inequality
• It was first studied by Dunkel and Schulz (2012) and Pokutta (2011).
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
8/20
Generalization of Chvatal-Gomory closures
Let S ⊆ Zn be the set of integer points that satisfy all the bound constraints.
Given a valid inequality cx ≥ d for a polyhedron P,
• cx ≥ dde = min{cz : cz ≥ d , z ∈ Zn} is the Chvatal-Gomory inequalityobtained from it.
• However, cx ≥ min{cz : cz ≥ d , z ∈ S} is also valid for P ∩ Zn (⊆ S).
• We have min{cz : cz ≥ d , z ∈ S} ≥ dde = min{cz : cz ≥ d , z ∈ Zn}.
Given S such that P ⊆ conv(S) and S = conv(S) ∩ Zn, theS-Chvatal-Gomory-Gomory closure of a rational polyhedron P ⊆ Rn is definedas
P ′S :=⋂c∈Zn
x ∈ Rn : cx ≥ min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}︸ ︷︷ ︸S-Chvatal-Gomory inequality
• It was first studied by Dunkel and Schulz (2012) and Pokutta (2011).
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
8/20
Generalization of Chvatal-Gomory closures
Let S ⊆ Zn be the set of integer points that satisfy all the bound constraints.
Given a valid inequality cx ≥ d for a polyhedron P,
• cx ≥ dde = min{cz : cz ≥ d , z ∈ Zn} is the Chvatal-Gomory inequalityobtained from it.
• However, cx ≥ min{cz : cz ≥ d , z ∈ S} is also valid for P ∩ Zn (⊆ S).
• We have min{cz : cz ≥ d , z ∈ S} ≥ dde = min{cz : cz ≥ d , z ∈ Zn}.
Given S such that P ⊆ conv(S) and S = conv(S) ∩ Zn, theS-Chvatal-Gomory-Gomory closure of a rational polyhedron P ⊆ Rn is definedas
P ′S :=⋂c∈Zn
x ∈ Rn : cx ≥ min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}︸ ︷︷ ︸S-Chvatal-Gomory inequality
• It was first studied by Dunkel and Schulz (2012) and Pokutta (2011).
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
8/20
Generalization of Chvatal-Gomory closures
Let S ⊆ Zn be the set of integer points that satisfy all the bound constraints.
Given a valid inequality cx ≥ d for a polyhedron P,
• cx ≥ dde = min{cz : cz ≥ d , z ∈ Zn} is the Chvatal-Gomory inequalityobtained from it.
• However, cx ≥ min{cz : cz ≥ d , z ∈ S} is also valid for P ∩ Zn (⊆ S).
• We have min{cz : cz ≥ d , z ∈ S} ≥ dde = min{cz : cz ≥ d , z ∈ Zn}.
Given S such that P ⊆ conv(S) and S = conv(S) ∩ Zn, theS-Chvatal-Gomory-Gomory closure of a rational polyhedron P ⊆ Rn is definedas
P ′S :=⋂c∈Zn
x ∈ Rn : cx ≥ min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}︸ ︷︷ ︸S-Chvatal-Gomory inequality
• It was first studied by Dunkel and Schulz (2012) and Pokutta (2011).
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
8/20
Generalization of Chvatal-Gomory closures
Let S ⊆ Zn be the set of integer points that satisfy all the bound constraints.
Given a valid inequality cx ≥ d for a polyhedron P,
• cx ≥ dde = min{cz : cz ≥ d , z ∈ Zn} is the Chvatal-Gomory inequalityobtained from it.
• However, cx ≥ min{cz : cz ≥ d , z ∈ S} is also valid for P ∩ Zn (⊆ S).
• We have min{cz : cz ≥ d , z ∈ S} ≥ dde = min{cz : cz ≥ d , z ∈ Zn}.
Given S such that P ⊆ conv(S) and S = conv(S) ∩ Zn, theS-Chvatal-Gomory-Gomory closure of a rational polyhedron P ⊆ Rn is definedas
P ′S :=⋂c∈Zn
x ∈ Rn : cx ≥ min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}︸ ︷︷ ︸S-Chvatal-Gomory inequality
• It was first studied by Dunkel and Schulz (2012) and Pokutta (2011).
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
9/20
Generalization of Chvatal-Gomory closures
• Remark that S-Chvatal-Gomory inequalities can be viewed as cuts from“wide split disjunctions” or “S-free split disjunctions”.
P
S
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
9/20
Generalization of Chvatal-Gomory closures
• Remark that S-Chvatal-Gomory inequalities can be viewed as cuts from“wide split disjunctions” or “S-free split disjunctions”.
P
S
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
10/20
Generalization of Chvatal-Gomory closures
We know that the Chvatal-Gomory closure of a rational polyhedron is arational polyhedron. What about the S-Chvatal-Gomory closure?
Theorem [Dash, Gunluk, L]
Let S =
z = (z1, z2, z3, z4) ∈ Zn :`1 ≤ z1 ≤ u1,`2 ≤ z2 ,
z3 ≤ u3
.
If P is a rational polyhedron contained in conv(S), then the S-Chvatal-Gomoryclosure of P is also a rational polyhedron.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
10/20
Generalization of Chvatal-Gomory closures
We know that the Chvatal-Gomory closure of a rational polyhedron is arational polyhedron. What about the S-Chvatal-Gomory closure?
Theorem [Dash, Gunluk, L]
Let S =
z = (z1, z2, z3, z4) ∈ Zn :`1 ≤ z1 ≤ u1,`2 ≤ z2 ,
z3 ≤ u3
.
If P is a rational polyhedron contained in conv(S), then the S-Chvatal-Gomoryclosure of P is also a rational polyhedron.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
11/20
A difference
P
P ′S
• Some facets are not defined by S-Chvatal-Gomory inequalities.
• In this example, the inequality defining a facet is the “limit” of a sequenceof S-Chvatal-Gomory inequalities.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
11/20
A difference
P
P ′S
• Some facets are not defined by S-Chvatal-Gomory inequalities.
• In this example, the inequality defining a facet is the “limit” of a sequenceof S-Chvatal-Gomory inequalities.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
11/20
A difference
P
P ′S
• Some facets are not defined by S-Chvatal-Gomory inequalities.
• In this example, the inequality defining a facet is the “limit” of a sequenceof S-Chvatal-Gomory inequalities.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
12/20
A difference
There is a more concrete example.
• Let S = {0, 1}4 and P be the convex hull of six points in [0, 1]4:
P = conv
{(1
2, 0, 0, 0
), (1, 0, 0, 0), (0, 1, 1, 0), (0, 1, 0, 1), (0, 0, 1, 1), (1, 1, 1, 1)
}.
• 2x1 + x2 + x3 + x4 ≥ 2 is a facet-defining inequality for the integer hull ofP.
• However, 2x1 + x2 + x3 + x4 ≥ 1 is supporting P at ( 12, 0, 0, 0) and
(0, 1, 0, 0) is on its corresponding hyperplane.
• 2x1 + (1− δ)x2 + (1− δ)x3 + (1− δ)x4 ≥ 2− 2δ is the S-Chvatal-Gomoryinequality obtained from 2x1 + (1− δ)x2 + (1− δ)x3 + (1− δ)x4 ≥ 1 forδ < 1
2.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
12/20
A difference
There is a more concrete example.
• Let S = {0, 1}4 and P be the convex hull of six points in [0, 1]4:
P = conv
{(1
2, 0, 0, 0
), (1, 0, 0, 0), (0, 1, 1, 0), (0, 1, 0, 1), (0, 0, 1, 1), (1, 1, 1, 1)
}.
• 2x1 + x2 + x3 + x4 ≥ 2 is a facet-defining inequality for the integer hull ofP.
• However, 2x1 + x2 + x3 + x4 ≥ 1 is supporting P at ( 12, 0, 0, 0) and
(0, 1, 0, 0) is on its corresponding hyperplane.
• 2x1 + (1− δ)x2 + (1− δ)x3 + (1− δ)x4 ≥ 2− 2δ is the S-Chvatal-Gomoryinequality obtained from 2x1 + (1− δ)x2 + (1− δ)x3 + (1− δ)x4 ≥ 1 forδ < 1
2.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
12/20
A difference
There is a more concrete example.
• Let S = {0, 1}4 and P be the convex hull of six points in [0, 1]4:
P = conv
{(1
2, 0, 0, 0
), (1, 0, 0, 0), (0, 1, 1, 0), (0, 1, 0, 1), (0, 0, 1, 1), (1, 1, 1, 1)
}.
• 2x1 + x2 + x3 + x4 ≥ 2 is a facet-defining inequality for the integer hull ofP.
• However, 2x1 + x2 + x3 + x4 ≥ 1 is supporting P at ( 12, 0, 0, 0) and
(0, 1, 0, 0) is on its corresponding hyperplane.
• 2x1 + (1− δ)x2 + (1− δ)x3 + (1− δ)x4 ≥ 2− 2δ is the S-Chvatal-Gomoryinequality obtained from 2x1 + (1− δ)x2 + (1− δ)x3 + (1− δ)x4 ≥ 1 forδ < 1
2.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
12/20
A difference
There is a more concrete example.
• Let S = {0, 1}4 and P be the convex hull of six points in [0, 1]4:
P = conv
{(1
2, 0, 0, 0
), (1, 0, 0, 0), (0, 1, 1, 0), (0, 1, 0, 1), (0, 0, 1, 1), (1, 1, 1, 1)
}.
• 2x1 + x2 + x3 + x4 ≥ 2 is a facet-defining inequality for the integer hull ofP.
• However, 2x1 + x2 + x3 + x4 ≥ 1 is supporting P at ( 12, 0, 0, 0) and
(0, 1, 0, 0) is on its corresponding hyperplane.
• 2x1 + (1− δ)x2 + (1− δ)x3 + (1− δ)x4 ≥ 2− 2δ is the S-Chvatal-Gomoryinequality obtained from 2x1 + (1− δ)x2 + (1− δ)x3 + (1− δ)x4 ≥ 1 forδ < 1
2.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
13/20
Proof idea
Can we use techniques developed for proving polyhedrality of other closures?
• Chvatal-Gomory closure ([Chvatal, 1973], [Schrijver, 1980], [Dadush, Dey,Vielma, 2011], etc).
• Split closure ([Cook, Kannan, Schrijver, 1990], [Andersen, Cornuejols, Li,2005], [Dash, Gunluk, Lodi, 2007], etc).
• Closure from lattice-free sets with bounded max-facet-width ([Andersen,Louveaux, Weismantel, 2010], [Averkov, 2012], etc)
In fact, it seemed difficult to use the techniques from these papers.
• min{cz : cz ≥ miny∈P cy , z ∈ S} −miny∈P cy cannot be bounded by afixed constant.
• An S-free split strip (from disjunction) does not necessarily have boundedfacet-width.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
13/20
Proof idea
Can we use techniques developed for proving polyhedrality of other closures?
• Chvatal-Gomory closure ([Chvatal, 1973], [Schrijver, 1980], [Dadush, Dey,Vielma, 2011], etc).
• Split closure ([Cook, Kannan, Schrijver, 1990], [Andersen, Cornuejols, Li,2005], [Dash, Gunluk, Lodi, 2007], etc).
• Closure from lattice-free sets with bounded max-facet-width ([Andersen,Louveaux, Weismantel, 2010], [Averkov, 2012], etc)
In fact, it seemed difficult to use the techniques from these papers.
• min{cz : cz ≥ miny∈P cy , z ∈ S} −miny∈P cy cannot be bounded by afixed constant.
• An S-free split strip (from disjunction) does not necessarily have boundedfacet-width.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
13/20
Proof idea
Can we use techniques developed for proving polyhedrality of other closures?
• Chvatal-Gomory closure ([Chvatal, 1973], [Schrijver, 1980], [Dadush, Dey,Vielma, 2011], etc).
• Split closure ([Cook, Kannan, Schrijver, 1990], [Andersen, Cornuejols, Li,2005], [Dash, Gunluk, Lodi, 2007], etc).
• Closure from lattice-free sets with bounded max-facet-width ([Andersen,Louveaux, Weismantel, 2010], [Averkov, 2012], etc)
In fact, it seemed difficult to use the techniques from these papers.
• min{cz : cz ≥ miny∈P cy , z ∈ S} −miny∈P cy cannot be bounded by afixed constant.
• An S-free split strip (from disjunction) does not necessarily have boundedfacet-width.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
13/20
Proof idea
Can we use techniques developed for proving polyhedrality of other closures?
• Chvatal-Gomory closure ([Chvatal, 1973], [Schrijver, 1980], [Dadush, Dey,Vielma, 2011], etc).
• Split closure ([Cook, Kannan, Schrijver, 1990], [Andersen, Cornuejols, Li,2005], [Dash, Gunluk, Lodi, 2007], etc).
• Closure from lattice-free sets with bounded max-facet-width ([Andersen,Louveaux, Weismantel, 2010], [Averkov, 2012], etc)
In fact, it seemed difficult to use the techniques from these papers.
• min{cz : cz ≥ miny∈P cy , z ∈ S} −miny∈P cy cannot be bounded by afixed constant.
• An S-free split strip (from disjunction) does not necessarily have boundedfacet-width.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
13/20
Proof idea
Can we use techniques developed for proving polyhedrality of other closures?
• Chvatal-Gomory closure ([Chvatal, 1973], [Schrijver, 1980], [Dadush, Dey,Vielma, 2011], etc).
• Split closure ([Cook, Kannan, Schrijver, 1990], [Andersen, Cornuejols, Li,2005], [Dash, Gunluk, Lodi, 2007], etc).
• Closure from lattice-free sets with bounded max-facet-width ([Andersen,Louveaux, Weismantel, 2010], [Averkov, 2012], etc)
In fact, it seemed difficult to use the techniques from these papers.
• min{cz : cz ≥ miny∈P cy , z ∈ S} −miny∈P cy cannot be bounded by afixed constant.
• An S-free split strip (from disjunction) does not necessarily have boundedfacet-width.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
13/20
Proof idea
Can we use techniques developed for proving polyhedrality of other closures?
• Chvatal-Gomory closure ([Chvatal, 1973], [Schrijver, 1980], [Dadush, Dey,Vielma, 2011], etc).
• Split closure ([Cook, Kannan, Schrijver, 1990], [Andersen, Cornuejols, Li,2005], [Dash, Gunluk, Lodi, 2007], etc).
• Closure from lattice-free sets with bounded max-facet-width ([Andersen,Louveaux, Weismantel, 2010], [Averkov, 2012], etc)
In fact, it seemed difficult to use the techniques from these papers.
• min{cz : cz ≥ miny∈P cy , z ∈ S} −miny∈P cy cannot be bounded by afixed constant.
• An S-free split strip (from disjunction) does not necessarily have boundedfacet-width.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
13/20
Proof idea
Can we use techniques developed for proving polyhedrality of other closures?
• Chvatal-Gomory closure ([Chvatal, 1973], [Schrijver, 1980], [Dadush, Dey,Vielma, 2011], etc).
• Split closure ([Cook, Kannan, Schrijver, 1990], [Andersen, Cornuejols, Li,2005], [Dash, Gunluk, Lodi, 2007], etc).
• Closure from lattice-free sets with bounded max-facet-width ([Andersen,Louveaux, Weismantel, 2010], [Averkov, 2012], etc)
In fact, it seemed difficult to use the techniques from these papers.
• min{cz : cz ≥ miny∈P cy , z ∈ S} −miny∈P cy cannot be bounded by afixed constant.
• An S-free split strip (from disjunction) does not necessarily have boundedfacet-width.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
14/20
Proof idea
Our proof is based on the following lemma, due to Dunkel and Schulz (2012):
Lemma [Dunkel, Schulz, 2012]
Let P ⊆ Rn be a rational polyhedron. Then⋂(α,β)∈H
{x ∈ P : αx ≥ β}
is a rational polyhedron if H ⊆ Rn+1 is a rational polyhedron.
We are going to write linear constraint on (α, β) ∈ Zn+1 for αx ≥ β to be anS-Chvatal-Gomory inequality.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
15/20
Proof idea
For example, consider the case when S is finite.
αx ≥ α0αx ≥ β
• If αx ≥ β is an S-Chvatal-Gomory inequality for some (α, β) ∈ Zn+1, thenit is obtained from a valid inequality αx ≥ α0 for P.
• S can be partitioned into G and L, where
G = {z ∈ S : αz ≥ β} and L = {z ∈ S : αz ≤ α0}
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
15/20
Proof idea
For example, consider the case when S is finite.
αx ≥ α0
G L
αx ≥ β
• If αx ≥ β is an S-Chvatal-Gomory inequality for some (α, β) ∈ Zn+1, thenit is obtained from a valid inequality αx ≥ α0 for P.
• S can be partitioned into G and L, where
G = {z ∈ S : αz ≥ β} and L = {z ∈ S : αz ≤ α0}
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
16/20
Proof idea
For example, consider the case when S is finite.
αx ≥ α0
G L
αx ≥ β
(α, β) satisfies the following.
(α, α0) = (λA, λb),αz ≥ β, ∀z ∈ Gαz ≤ α0 − 1
∆, ∀z ∈ L
β ≥ α0
• If αx ≥ β is an S-Chvatal-Gomory inequality for some (α, β) ∈ Zn+1, thenit is obtained from a valid inequality αx ≥ α0 for P.
• S can be partitioned into G and L, where
G = {z ∈ S : αz ≥ β} and L = {z ∈ S : αz ≤ α0}.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
17/20
Proof idea
In fact,
Lemma [Dunkel, Schulz, 2012]
Let P = {x ∈ Rn : Ax ≥ b} be rational polyhedron. The S-Chvatal-Gomoryclosure of P is obtained after applying αx ≥ β for all (α, β) ∈ Zn+1 satisfying
(α, α0) = (λA, λb),αz ≥ β, ∀z ∈ Gαz ≤ α0 − 1
∆, ∀z ∈ L
β ≥ α0
for some λ ∈ Rm+, α0 ∈ R, and some partition (G , L) of S . ∆ is the max size of
a sub-determinant of A.
The number of partitions (G , L) of S is finite. so the S-Chvatal-Gomory closureof P is a rational polyhedron when S is finite.
• However, if S is not finite, then the number of partitions of S is infinite.
• Besides, if S is not finite, either G or L is an infinite set.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
17/20
Proof idea
In fact,
Lemma [Dunkel, Schulz, 2012]
Let P = {x ∈ Rn : Ax ≥ b} be rational polyhedron. The S-Chvatal-Gomoryclosure of P is obtained after applying αx ≥ β for all (α, β) ∈ Zn+1 satisfying
(α, α0) = (λA, λb),αz ≥ β, ∀z ∈ Gαz ≤ α0 − 1
∆, ∀z ∈ L
β ≥ α0
for some λ ∈ Rm+, α0 ∈ R, and some partition (G , L) of S . ∆ is the max size of
a sub-determinant of A.
The number of partitions (G , L) of S is finite. so the S-Chvatal-Gomory closureof P is a rational polyhedron when S is finite.
• However, if S is not finite, then the number of partitions of S is infinite.
• Besides, if S is not finite, either G or L is an infinite set.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
17/20
Proof idea
In fact,
Lemma [Dunkel, Schulz, 2012]
Let P = {x ∈ Rn : Ax ≥ b} be rational polyhedron. The S-Chvatal-Gomoryclosure of P is obtained after applying αx ≥ β for all (α, β) ∈ Zn+1 satisfying
(α, α0) = (λA, λb),αz ≥ β, ∀z ∈ Gαz ≤ α0 − 1
∆, ∀z ∈ L
β ≥ α0
for some λ ∈ Rm+, α0 ∈ R, and some partition (G , L) of S . ∆ is the max size of
a sub-determinant of A.
The number of partitions (G , L) of S is finite. so the S-Chvatal-Gomory closureof P is a rational polyhedron when S is finite.
• However, if S is not finite, then the number of partitions of S is infinite.
• Besides, if S is not finite, either G or L is an infinite set.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
17/20
Proof idea
In fact,
Lemma [Dunkel, Schulz, 2012]
Let P = {x ∈ Rn : Ax ≥ b} be rational polyhedron. The S-Chvatal-Gomoryclosure of P is obtained after applying αx ≥ β for all (α, β) ∈ Zn+1 satisfying
(α, α0) = (λA, λb),αz ≥ β, ∀z ∈ Gαz ≤ α0 − 1
∆, ∀z ∈ L
β ≥ α0
for some λ ∈ Rm+, α0 ∈ R, and some partition (G , L) of S . ∆ is the max size of
a sub-determinant of A.
The number of partitions (G , L) of S is finite. so the S-Chvatal-Gomory closureof P is a rational polyhedron when S is finite.
• However, if S is not finite, then the number of partitions of S is infinite.
• Besides, if S is not finite, either G or L is an infinite set.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
18/20
Proof idea
For the general case, let us first consider S = Zn+, which is infinite.
• First, take the Chvatal-Gomory closure of P: P ′.
• Apply S-Chvatal-Gomory inequalities cutting off some part of P ′.
Lemma [Dash, Gunluk, L]
If αx ≥ β cuts off some part of P ′, then either (α, β) ≥ 0 or (α, β) ≤ 0.
αx ≥ α0
αx ≥ β = dα0e
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
18/20
Proof idea
For the general case, let us first consider S = Zn+, which is infinite.
• First, take the Chvatal-Gomory closure of P: P ′.
• Apply S-Chvatal-Gomory inequalities cutting off some part of P ′.
Lemma [Dash, Gunluk, L]
If αx ≥ β cuts off some part of P ′, then either (α, β) ≥ 0 or (α, β) ≤ 0.
αx ≥ α0
αx ≥ β = dα0e
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
18/20
Proof idea
For the general case, let us first consider S = Zn+, which is infinite.
• First, take the Chvatal-Gomory closure of P: P ′.
• Apply S-Chvatal-Gomory inequalities cutting off some part of P ′.
Lemma [Dash, Gunluk, L]
If αx ≥ β cuts off some part of P ′, then either (α, β) ≥ 0 or (α, β) ≤ 0.
αx ≥ α0
αx ≥ β = dα0e
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
18/20
Proof idea
For the general case, let us first consider S = Zn+, which is infinite.
• First, take the Chvatal-Gomory closure of P: P ′.
• Apply S-Chvatal-Gomory inequalities cutting off some part of P ′.
Lemma [Dash, Gunluk, L]
If αx ≥ β cuts off some part of P ′, then either (α, β) ≥ 0 or (α, β) ≤ 0.
αx ≥ α0
αx ≥ β = dα0e
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
19/20
Proof idea
Lemma [Dash, Gunluk, L]
There exists a large constant M such that if αx ≥ β cuts off some part of P ′
and is non-redundant, then 0 ≤ 1βα ≤ M1.
• The intercepts of {x ∈ Rn : αx = β} with coordinate axes are allcontained in [0,M]n.
• Then, we can focus on the set of integer points contained in [0,M]n,which is finite.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
19/20
Proof idea
Lemma [Dash, Gunluk, L]
There exists a large constant M such that if αx ≥ β cuts off some part of P ′
and is non-redundant, then 0 ≤ 1βα ≤ M1.
• The intercepts of {x ∈ Rn : αx = β} with coordinate axes are allcontained in [0,M]n.
• Then, we can focus on the set of integer points contained in [0,M]n,which is finite.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
19/20
Proof idea
Lemma [Dash, Gunluk, L]
There exists a large constant M such that if αx ≥ β cuts off some part of P ′
and is non-redundant, then 0 ≤ 1βα ≤ M1.
• The intercepts of {x ∈ Rn : αx = β} with coordinate axes are allcontained in [0,M]n.
• Then, we can focus on the set of integer points contained in [0,M]n,which is finite.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
19/20
Proof idea
Lemma [Dash, Gunluk, L]
There exists a large constant M such that if αx ≥ β cuts off some part of P ′
and is non-redundant, then 0 ≤ 1βα ≤ M1.
• The intercepts of {x ∈ Rn : αx = β} with coordinate axes are allcontained in [0,M]n.
• Then, we can focus on the set of integer points contained in [0,M]n,which is finite.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
20/20
Theorem [Dash, Gunluk, L]
Let S =
z = (z1, z2, z3, z4) ∈ Zn :`1 ≤ z1 ≤ u1,`2 ≤ z2 ,
z3 ≤ u3
.
If P is a rational polyhedron contained in conv(S), then the S-Chvatal-Gomoryclosure of P is also a rational polyhedron.
A draft will be posted soon!
Thank you!
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables
20/20
Chvatal, V. (1973).Edmonds polytopes and a hierarchy of combinatorial problems.Discrete Mathematics, 4(4):305 – 337.
Schrijver, A. (1980).On cutting planes.Annals of Discrete Mathematics, 9:291 – 296.
Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables