bridging the gap between fixed and variable dimension · 2016. 6. 21. · knapsack polytopes: k n 2...

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Robert Weismantel

Bridging the gap between fixed and variable dimension

ETH Zurich

Lecture 6

Robert Weismantel Lecture 6 1 / 27

Table of contents

Part 1

Affine TU decomposition of matrices(based on joint work with J. Bader, R. Hildebrand and R. Zenklusen)

Part 2

A polyhedral Frobenius problem(based on joint work with T. Oertel and D. Adjiashvili)

Return to ILP

goal: solve Integer LinearProgramming problem (IP)

max cTxs.t. Ax ≤ b

x ∈ Zn

x ∈ Rn, W x ∈ Zk

2x1+x2∈Z

c

••

•

NP-hard [Karp, 1971], polynomial if n is constant [Lenstra, 1983]

MILP relaxation is stronger than LP relaxation

MILP relaxation is polynomial time solvable if k = O( log nlog log n ) [Kannan,

1986], based on [Lenstra, Lenstra and Lovasz, 1982]

Return to ILP

��x ∈ Zn x ∈ Rn

, W x ∈ Zk

2x1+x2∈Z

c

•

Return to ILP

��x ∈ Zn x ∈ Rn, W x ∈ Zk

2x1+x2∈Zc

•••

Return to ILP

��x ∈ Zn x ∈ Rn, W x ∈ Zk

2x1+x2∈Zc

•••

W x ∈ Zk reformulation

goal: describe the integer hull of a polyhedron P

Desired property

For W ∈ Zk×n,

conv(P ∩ Zn) ⊆ conv({x ∈ P |W x ∈ Zk})

number of integrality constraints reduced from n to k

MILPs with few integer variables are solved efficiently, e.g. usingcutting planes and branch-and-bound techniques [Lodi, 2010]

Desired property

For W ∈ Zk×n,

conv(P ∩ Zn)= conv({x ∈ P |W x ∈ Zk})

Desired property

For W ∈ Zk×n,

conv(P ∩ Zn)= conv({x ∈ P |W x ∈ Zk})

A 0-1 knapsack polytope

conv({

x ∈ {0, 1}8 | x1 + x2 + x3 + x4 + 4x5 + 4x6 + 5x7 + 5x8 ≤ 11})

polyhedral description: 39 inequalities

replace x ∈ Z8 by x5 + x6 + x7 + x8 ∈ Z

for all knapsack polytopes possible: aggregate variables withneighboring weights

A 0-1 knapsack polytope

conv({

x ∈ {0, 1}8 | x1 + x2 + x3 + x4 + 4x5 + 4x6 + 5x7 + 5x8 ≤ 11})

polyhedral description: 39 inequalities

replace x ∈ Z8 by x5 + x6 + x7 + x8 ∈ Z

for all knapsack polytopes possible: aggregate variables withneighboring weights

The parity polytope

convex hull of all {0, 1}n vectors that have an even cardinality support

polyhedral description: 2n−1 inequalities [Jeroslow, 1975]

extended formulation with 4n − 1 inequalities [Carr and Konjevod, 2004]

described by

conv({x ∈ [0, 1]n | 1

2

n∑i=1

xi integral}),

projection of

conv({(x, z) ∈ [0, 1]n×R |n∑

i=1

xi−2z = 0, z ∈ Z})

(1, 1, 0)

(0, 0, 0)

(1, 0, 1)

(0, 1, 1)

The parity polytope

described by

conv({x ∈ [0, 1]n | 1

2

n∑i=1

xi integral}),

projection of

conv({(x, z) ∈ [0, 1]n×R |n∑

i=1

xi−2z = 0, z ∈ Z})

(1, 1, 0)

(0, 0, 0)

(1, 0, 1)

(0, 1, 1)

The parity polytope

described by

conv({x ∈ [0, 1]n | 1

2

n∑i=1

xi integral}),

projection of

conv({(x, z) ∈ [0, 1]n×R |n∑

i=1

xi−2z = 0, z ∈ Z})

(1, 1, 0)

(0, 0, 0)

(1, 0, 1)

(0, 1, 1)

The parity polytope

described by

conv({x ∈ [0, 1]n | 1

2

n∑i=1

xi integral}),

projection of

conv({(x, z) ∈ [0, 1]n×R |n∑

i=1

xi−2z = 0, z ∈ Z})

(1, 1, 0)

(0, 0, 0)

(1, 0, 1)

(0, 1, 1)

Knapsack polytopes

Let n ≥ 4, a ∈ Zn, b ∈ Z.

Theorem.

For

P ={

x ∈ [0, 1]n | aTx ≤ b}

there exists W ∈ Z(n−2)×n s.t.

conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zn−2}).

Knapsack polytopes

Let n ≥ 4, a ∈ Zn, b ∈ Z.

Theorem.

For P ={

x ∈ [0, 1]n | aTx ≤ b}

there exists W ∈ Z(n−2)×n s.t.

conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zn−2}).

Knapsack polytopes: k ≤ n − 2 sufficient

Proof idea.

consider conv(P ∩ {0, 1}n) and conv({0, 1}n \ P)

, there existsbn2c-dimensional face of either one not intersecting the respectiveother polytope

any 2-dimensional plane through three 0-1 points can be described bya system W x = d with TU matrix W ∈ Z(n−2)×n

conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zn−2})

Proof idea.

consider conv(P ∩ {0, 1}n) and conv({0, 1}n \ P) , there existsbn2c-dimensional face of either one not intersecting the respectiveother polytope

Proof idea.

Definition.

Affine TU decomposition: A = A + UW with U integral, [A;W ] TUAffine TU-dimension of A: minimal number of rows of W

matrices with affine TU-dimension 0: totally unimodular

Definition [Padberg, 1988].

A ∈ Zn×n is almost totally unimodular if A is not TUbut every proper submatrix of A is TU

1 0 11 1 00 1 1

building block for k-balanced matrices [Conforti, 2006]

almost TU matrices have affine TU-dimension 1:[a A] = [0n A] + a(1, 0, . . . , 0)

affine TU-dimension 1=nearly totally unimodular matrices [Gijswijt,

2005] (e.g. incidence matrix of nearly bipartite graphs)

Definition.

1 0 11 1 00 1 − 1

Definition.

1 0 11 1 00 1 1

Definition.

1 0 11 1 00 1 1

almost TU matrices have affine TU-dimension 1:[a A]

= [0n A] + a(1, 0, . . . , 0)

Definition.

1 0 11 1 00 1 1

Definition.

1 0 11 1 00 1 1

Theorem.

Affine TU decomposition: A = A + UW , U integral, [A;W ] TU.

Forall integral b, the polyhedron P = {x ∈ Rn | Ax ≤ b} satisfies

conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zk}).

Proof.

Pr ={

x ∈ Rn | Ax ≤ b− Ur,W x = r}

is an integral polyhedron for each r ∈ Zk ,

conv({

x ∈ Rn | Ax ≤ b,W x ∈ Zk})

= conv

⋃r∈Zk

Pr

is an integral set

still holds with additional bounds l ≤ x ≤ u (l,u ∈ Zn)

Theorem.

Affine TU decomposition: A = A + UW , U integral, [A;W ] TU. Forall integral b, the polyhedron P = {x ∈ Rn | Ax ≤ b} satisfies

Proof.

Pr ={

x ∈ Rn | Ax ≤ b− Ur,W x = r}

conv({

x ∈ Rn | Ax ≤ b,W x ∈ Zk})

= conv

⋃r∈Zk

Pr

is an integral set

Theorem.

Proof.

Pr ={

x ∈ Rn | Ax ≤ b− Ur,W x = r}

is an integral polyhedron for each r ∈ Zk

,

conv({

x ∈ Rn | Ax ≤ b,W x ∈ Zk})

= conv

⋃r∈Zk

Pr

is an integral set

Theorem.

Proof.

Pr ={

x ∈ Rn | Ax ≤ b− Ur,W x = r}

conv({

x ∈ Rn | Ax ≤ b,W x ∈ Zk})

= conv

⋃r∈Zk

Pr

is an integral set

Theorem.

Proof.

Pr ={

x ∈ Rn | Ax ≤ b− Ur,W x = r}

conv({

x ∈ Rn | Ax ≤ b,W x ∈ Zk})

= conv

⋃r∈Zk

Pr

is an integral set

Master knapsack polytopeP = conv({x ∈ {0, 1}n |

∑ni=1 ixi ≤ b})

use fewer than n integrality constraints by affine TU decomposition

aggregate neighboring coefficients:

(1, 2, . . . , n − 1, n) = (0, 1, 0, 1, . . . , 0, 1) + (1, 3, . . . , n − 1)

1 1

1 1. . .

1 1

affine TU decomposition with dn2e rows

there exists affine TU decomposition with 2d√ne − 2 rows

∑ni=1 ixi ≤ b})

(1, 2, . . . , n − 1, n) = (0, 1, 0, 1, . . . , 0, 1) + (1, 3, . . . , n − 1)

1 1

1 1. . .

1 1

∑ni=1 ixi ≤ b})

(1, 2, . . . , n − 1, n) = (0, 1, 0, 1, . . . , 0, 1) + (1, 3, . . . , n − 1)

1 1

1 1. . .

1 1

Excursion: A generalization of total dual integrality

Affine TU decompositions A = A + UW ∈ Zm×n:conv({x ∈ Rn | Ax ≤ b,W x ∈ Zk}) is integral for all integral b

Now we ask integrality for specific fixed b ∈ Rm

Observation

Let W ∈ Zk×n. If the system Ax ≤ b,W x = d is total dual integral foreach integral vector d, then

conv({x ∈ Rn | Ax ≤ b,W x ∈ Zk})

is integral.

Excursion: A generalization of total dual integrality

Affine TU decompositions A = A + UW ∈ Zm×n:conv({x ∈ Rn | Ax ≤ b,W x ∈ Zk}) is integral for all integral b

Now we ask integrality for specific fixed b ∈ Rm

Observation

Let W ∈ Zk×n. If the system Ax ≤ b,W x = d is total dual integral foreach integral vector d, then

conv({x ∈ Rn | Ax ≤ b,W x ∈ Zk})

is integral.

A Knapsack Example

Let b, k ∈ Z+, k ≥ 3. Assume k−1k+1b ≤ ai ≤ b for i ∈ B, and

bk+1 < ai ≤ b

k for i ∈ S .

Theorem

The system∑i ′∈S:ai′>ai

xi ′ +∑

j∈B:aj>b−ai

xj + (k − 1) ·∑

j∈B xj ≤ k for all i ∈ S ,

xi ≤ 1 for all i ∈ S ,xi ≥ 0 for all i ∈ S ∪ B∑

j∈B xj ≤ 1∑j∈B xj = d

is integral for all d ∈ Z.

A Knapsack Example

Let b, k ∈ Z+, k ≥ 3. Assume k−1k+1b ≤ ai ≤ b for i ∈ B, and

bk+1 < ai ≤ b

k for i ∈ S .

Theorem

The system∑i ′∈S:ai′>ai

xi ′ +∑

j∈B:aj>b−ai

xj + (k − 1) ·∑

j∈B xj ≤ k for all i ∈ S ,

xi ≤ 1 for all i ∈ S ,xi ≥ 0 for all i ∈ S ∪ B∑

j∈B xj ≤ 1∑j∈B xj = d

is integral for all d ∈ Z.

Determining the affine TU-dimension

affine TU-dimension 0: decomposition procedure to decide if a matrixis TU [Seymour, 1980] in polynomial time [Truemper, 1990]

Theorem.

Given A ∈ Zm×n, one can decide in polynomial time if A has affineTU-dimension k , provided that m and k are fixed.

Proof.

W.l.o.g., A has distinct columns.Then the (unknown) TU matrix [A;W ] has distinct columns, there arepolynomially many such matrices.For each of these [A;W ], decide in polynomial time if there is an integralU such that A = A + UW (every row of U needs to be an integral solutionvector to a system of m linear equations).

Theorem.

Proof.

Theorem.

Proof.

W.l.o.g., A has distinct columns.Then the (unknown) TU matrix [A;W ] has distinct columns, there arepolynomially many such matrices.

For each of these [A;W ], decide in polynomial time if there is an integralU such that A = A + UW (every row of U needs to be an integral solutionvector to a system of m linear equations).

Theorem.

Proof.

Theorem.

One can decide in polynomial time if A ∈ Zm×n has affine TU-dimensionk , provided that m and k are fixed.

Theorem.

To decide if A ∈ Zm×n has affine TU-dimension n is an NP-completeproblem.

Conjecture.

One can decide in polynomial time if A ∈ Zm×n has affine TU-dimensionk , provided that k is fixed.

Theorem.

Conjecture.

Theorem.

Conjecture.

Table of contents

Part 2

A polyhedral Frobenius problem

Application of parametric integer optimization to nonlinear discretesystems. (based on joint work with T. Oertel and D. Adjiashvili)

Motivation: bridge the gap

Nonlinear optimization in mixed integer variables

min{f (x , y) : Ax + By ≤ b, y ∈ Rn, x ∈ Zd}

Many results in fixed dimension d

d fixed, f convex (Grotschel, Lovasz, Schrijver ’1988)

d fixed, f and constraints quasi-convex polynomials (Khachiyan,Porkolab ’2000)

d fixed, f concave (Cook, Hartmann, Kannan, McDiarmid ’1992)

d = 2 and f polynomial of degree two (Del Pia, Weismantel ’2014)

Few results in variable dimension d

Separable-convex IP (Hochbaum, Shanthikumar ’1990)

N-fold IP (De Loera, Hemmecke, Onn, Weismantel ’2008)

Dynamic programming

Under which conditions on the input is the followingproblem tractable?

(Parametric integer optimization problem)

min {f (Wx) : Ax ≤ b, x ∈ Zn}

The input

Matrices A ∈ Zm×n and W ∈ Zd×n, a vector b ∈ Zm

We assume to have access to a fiber oracle.

Given a y ∈ Zd returns x ∈ {z ∈ Zn : Az ≤ b},such that Wx = y , or states that no such x exists.

A function f : Qd → Q presented by an integer minimization oracle.

(Query: y∗ ← arg min{f (y) : By ≤ c , y ∈ Λ})

The input

Some Clear Boundaries of Tractability

We assume W to be given in unary representation.

Partition problem. w1, · · · ,wn ∈ Z+, D = 12

∑ni=1 wi .

Solvemin{(wT x − D)2 : x ∈ {0, 1}n}.

With fiber oracle - solvable with a single oracle call.

We assume d to be fixed

Leverage algorithms for minimization in fixed d .

Solvemin{f (y) : y ∈ Zd}

where f is convex.

min {f (Wx) : Ax ≤ b, x ∈ Zn}We assume W to be given in unary representation.

∑ni=1 wi .

Solvemin{(wT x − D)2 : x ∈ {0, 1}n}.

where f is convex.

∑ni=1 wi .

Solvemin{(wT x − D)2 : x ∈ {0, 1}n}.

where f is convex.

∑ni=1 wi .

Solvemin{(wT x − D)2 : x ∈ {0, 1}n}.

where f is convex.

∑ni=1 wi .

Solvemin{(wT x − D)2 : x ∈ {0, 1}n}.

where f is convex.

∑ni=1 wi .

Solvemin{(wT x − D)2 : x ∈ {0, 1}n}.

where f is convex.Robert Weismantel Lecture 6 19 / 27

Our Main Result (from a optimization point of view)

The assumptions

Let A ∈ Zm×n, W ∈ Zd×n, b ∈ Zm and f : Rd → R.

Let d be a fixed constant.

Let ∆ denote the maximum sub-determinant of A and W.

Theorem.

There is an algorithm that solves the non-linear optimization problem

min {f (Wx) : Ax ≤ b, x ∈ Zn}.

The number of oracle calls it performs (to the optimization and fiberoracles) is polynomial in n and ∆.

Our Main Result (from a optimization point of view)

The assumptions

Let A ∈ Zm×n, W ∈ Zd×n, b ∈ Zm and f : Rd → R.

Let d be a fixed constant.

Let ∆ denote the maximum sub-determinant of A and W.

Theorem.

There is an algorithm that solves the non-linear optimization problem

min {f (Wx) : Ax ≤ b, x ∈ Zn}.

The number of oracle calls it performs (to the optimization and fiberoracles) is polynomial in n and ∆.

Our Approach

Understand the set

R = {Wx : x ∈ Zn, Ax ≤ b} ⊂ Zd

Typically,R ⊂ Q = {Wx : Ax ≤ b} ∩ Zd but R 6= Q.

[0, 3]3 ∩ Z3

W =

(1 2 1−2 0 1

)

The Main Tool - (Diagonal) Frobenius Number

Definition (Frobenius Number)

Given integers a1, · · · , an with gcd(a1, · · · , an) = 1, the Frobenius numberF (a1, · · · , an) is the largest integer k that can not be expressed as apositive integer combination of a1, · · · , an.

Some important facts about the Frobenius number

Finding F (a1, · · · , an) is also known as the coin problem.

F (a1, a2) = a1a2 − (a1 + a2) (Sylvester ’1884).

NP-hard to compute F (a1, · · · , an) (Kannan ’1992).

F (a1, · · · , an) ≤ cn‖(a1, · · · , an)‖22 (e.g. Brauer ’1942).

Definition (Diagonal Frobenius Number)

Let M ∈ Zd×m (d ≤ m) and v = M1 such that

M has HNF Identity, and

C (M) = {λ : Mλ ≥ 0} is a pointed cone.

F (M) is the smallest integer t st (tv + C (M)) ∩ Zd ⊂ {Mx : x ∈ Zm+}.

M =

(2 3 40 1 −1

)

M =

(2 3 40 1 −1

)

M =

(2 3 40 1 −1

)

v =

(90

)

M =

(2 3 40 1 −1

)

v =

(90

)F (M) = 1

The diagonal Frobenius number F (M) is the smallest integer t such that

(tv + C (M)) ∩ Zd ⊂ {Mx : x ∈ Zm+}.

Theorem (Aliev, Henk 2010).

F (M) ≤ (m − d)√m

2

√det(MMT ).

For fixed d , the bound is polynomial in the unary encoding of M.

{Wx : x ∈ P ∩ Zn} vs. {Wx : x ∈ P}∩Zd

P = Rn+ w ∈ Zn

P = Rn+ W ∈ Zd×n

P = {x ∈ Rn : Ax ≤ b}

W ∈ Zd×n

{Wx : x ∈ P ∩ Zn} vs. {Wx : x ∈ P}∩Zd

P = Rn+ w ∈ Zn

P = Rn+ W ∈ Zd×n

P = {x ∈ Rn : Ax ≤ b}

W ∈ Zd×n

{Wx : x ∈ P ∩ Zn} vs. {Wx : x ∈ P}∩Zd

P = Rn+ w ∈ Zn

P = Rn+ W ∈ Zd×n

P = {x ∈ Rn : Ax ≤ b}

W ∈ Zd×n

{Wx : x ∈ P ∩ Zn} vs. {Wx : x ∈ P}∩Zd

P = Rn+ w ∈ Zn

P = Rn+ W ∈ Zd×n

P = {x ∈ Rn : Ax ≤ b}

W ∈ Zd×n

δ-Regular Sets

Definition (δ-regular set).

We call a set S ⊂ Zd δ-regular, with respect to a region B ⊂ Rd , if thereexists a family of full-dimensional affine sub-lattices Λ1, · · · ,Λk of Zd withdeterminants det(Λi ) ≤δ such that S ∩ B =

⋃i Λi ∩ B.

δ-Regular Sets

⋃i Λi ∩ B.

δ-Regular Sets

⋃i Λi ∩ B.

Main Result (from a geometric point of view)

The notation

Let P = {x ∈ Rn : Ax ≤ b}, with A ∈ Zm×n and b ∈ Zm.

Let ∆ denote the maximum absolute sub-determinant of A.

Let Q = WP and let R = W (P ∩ Zn) with W ∈ Zd×n.

Define Qγ := {x ∈ Rd : x + Bγ ⊂ Q}.

Qγ

Theorem.

The set R is δ-regular with respect to the polyhedron Qγ , where γ and δare bounded polynomially in ∆, ‖W ‖max and n.

The notation

Qγ

Theorem.

The notation

Qγ

Theorem.

The notation

Qγ

Theorem.

The notation

Qγ

Theorem.

The Algorithm

For all affine lattices Λ ⊂ Zd with det(Λ) ≤ δ, solvemin{f (y) : y ∈ Qγ ∩ Λ} ⇒ yΛ.

Determinex∗ = argmin{f (Wx) : x ∈ P ∩ Zn such that Wx = yΛ for some Λ}.For each affine subspace L, sufficiently close to boundary

Recursively find a solution to min{f (Wx) : x ∈W−1L ∩ P} ⇒ x ′.Replace x∗ with x ′ if its objective value is smaller.

Return x∗.

The Algorithm

For all affine lattices Λ ⊂ Zd with det(Λ) ≤ δ, solvemin{f (y) : y ∈ Qγ ∩ Λ} ⇒ yΛ.Determinex∗ = argmin{f (Wx) : x ∈ P ∩ Zn such that Wx = yΛ for some Λ}.

For each affine subspace L, sufficiently close to boundaryRecursively find a solution to min{f (Wx) : x ∈W−1L ∩ P} ⇒ x ′.Replace x∗ with x ′ if its objective value is smaller.

Return x∗.

The Algorithm

For all affine lattices Λ ⊂ Zd with det(Λ) ≤ δ, solvemin{f (y) : y ∈ Qγ ∩ Λ} ⇒ yΛ.Determinex∗ = argmin{f (Wx) : x ∈ P ∩ Zn such that Wx = yΛ for some Λ}.For each affine subspace L, sufficiently close to boundary

Recursively find a solution to min{f (Wx) : x ∈W−1L ∩ P} ⇒ x ′.

Replace x∗ with x ′ if its objective value is smaller.

Return x∗.

The Algorithm

Return x∗.Robert Weismantel Lecture 6 28 / 27

The Algorithm

Return x∗.Robert Weismantel Lecture 6 28 / 27

bridging the gap between fixed and variable dimension · 2016. 6. 21. · knapsack polytopes: k n 2...

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