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Shmuel Onn Technion – Israel Institute of Technology Nonlinear Integer Programming is Fixed-Parameter Tractable

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Page 1: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel OnnTechnion – Israel Institute of Technology

Nonlinear Integer Programmingis

Fixed-Parameter Tractable

Page 2: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

(with Berstein, De Loera, Hemmecke, Lee, Romanchuk, Rothblum, Weismantel)

Outline

Overview our theory of Graver bases for integer programming

Shmuel Onn

Page 3: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

(with Berstein, De Loera, Hemmecke, Lee, Romanchuk, Rothblum, Weismantel)

Outline

Overview our theory of Graver bases for integer programming

(with Hemmecke and my student Romanchuk, Mathematical Programming, 2013)

Explain the fixed-parameter tractability of integer programming

Shmuel Onn

Page 4: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

(with Berstein, De Loera, Hemmecke, Lee, Romanchuk, Rothblum, Weismantel)

Outline

Overview our theory of Graver bases for integer programming

Explain the fixed-parameter tractability of integer programming

Describe an approximation hierarchy for integer programming

Shmuel Onn

(with Hemmecke and my student Romanchuk, Mathematical Programming, 2013)

Page 5: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

(Non)-Linear Integer Programming

l,u: lower/upper bounds in Zn f: function from Zn to R

with data: A: integer m x n matrix b: right-hand side in Zm

Shmuel Onn

The problem is: min/max { f(x) : Ax ≤ b, l ≤ x ≤ u, x in Zn }

Page 6: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

(Non)-Linear Integer Programming

Shmuel Onn

Our theory enables polynomial time solution of broad naturaluniversal (non)-linear integer programs in variable dimension

The problem is: min/max { f(x) : Ax ≤ b, l ≤ x ≤ u, x in Zn }

Page 7: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

(Non)-Linear Integer Programming

Shmuel Onn

under which integer programming is fixed-parameter tractable.

The problem is: min/max { f(x) : Ax ≤ b, l ≤ x ≤ u, x in Zn }

It gives variable dimension parameterization of integer programming,

Our theory enables polynomial time solution of broad naturaluniversal (non)-linear integer programs in variable dimension

Page 8: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Graver Basesand

Nonlinear Integer Programming

Page 9: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Graver Bases

The Graver basis of an integer matrix A is the finite set G(A) of conformal-minimal nonzero integer vectors x satisfying Ax = 0.

x is conformal-minimal if no other y in same orthant has all |yi| ≤ |xi|

Page 10: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

circuits: ±(2 -1 0), ±(1 0 -1), ±(0 1 -2)

Example: Consider A=(1 2 1). Then G(A) consists of

non-circuits: ±(1 -1 1)

Graver Bases

The Graver basis of an integer matrix A is the finite set G(A) of conformal-minimal nonzero integer vectors x satisfying Ax = 0.

Page 11: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Reference: N-fold integer programming (De Loera, Hemmecke, Onn, Weismantel)Discrete Optimization (Volume in memory of George Dantzig), 2008

Shmuel Onn

Theorem 1: linear optimization in polytime with G(A):

Some Theorems on(Non)-Linear Integer Programming

max { wx : Ax = b, l ≤ x ≤ u, x in Zn }

Page 12: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Reference: A polynomial oracle-time algorithm for convex integer minimization (Hemmecke, Onn, Weismantel) Mathematical Programming, 2011

Theorem 2: separable convex minimization in polytime with G(A):

Some Theorems on(Non)-Linear Integer Programming

min {Σfi(xi) : Ax = b, l ≤ x ≤ u, x in Zn }

Page 13: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Theorem 3: integer point lp-nearest to x in polytime with G(A):

Reference: A polynomial oracle-time algorithm for convex integer minimization (Hemmecke, Onn, Weismantel) Mathematical Programming, 2011

Some Theorems on(Non)-Linear Integer Programming

min {|x - x|p : Ax = b, l ≤ x ≤ u, x in Zn}

Page 14: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Reference: Convex integer maximization via Graver bases (De Loera, Hemmecke, Onn, Rothblum, Weismantel) Journal of Pure and Applied Algebra, 2009

Theorem 4: weighted convex maximization in polytime with G(A):

where W is d x n matrix and f convex function on Zd

(balancing d linear criteria or player utilities Wix)

Some Theorems on(Non)-Linear Integer Programming

max {f(Wx) : Ax = b, l ≤ x ≤ u, x in Zn }

Page 15: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Reference: The quadratic Graver cone, quadratic integer minimization & extensions(Lee, Onn, Romanchuk, Weismantel), Mathematical Programming, 2012

Theorem 5: quadratic minimization in polytime with G(A):

where V lies in cone K2(A) of possibly indefinite matrices, enabling minimization of some convex and some non-convex quadratics.

Some Theorems on(Non)-Linear Integer Programming

min {xTVx : Ax = b, l ≤ x ≤ u, x in Zn }

Page 16: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Theorem 6: polynomial minimization in polytime with G(A):

Reference: The quadratic Graver cone, quadratic integer minimization & extensions(Lee, Onn, Romanchuk, Weismantel), Mathematical Programming, 2012

where p is possibly indefinite polynomial of degree d in cone Kd(A),enabling minimization of some (non)-convex degree d polynomials.

Some Theorems on(Non)-Linear Integer Programming

min {p(x) : Ax = b, l ≤ x ≤ u, x in Zn }

Page 17: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

The Main Iterative Algorithm

Page 18: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Rn

Shmuel Onn

Proof of Theorems 1 and 2(linear and separable convex minimization)

R

f

with the Graver basis G(A)

To solve min {Σ fi(xi) : Ax = b, l ≤ x ≤ u, x in Zn }

Do:

Page 19: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Rn

Shmuel Onn

R

f1. Find initial point by auxiliary program

Proof of Theorems 1 and 2(linear and separable convex minimization)

with the Graver basis G(A)

To solve min {Σ fi(xi) : Ax = b, l ≤ x ≤ u, x in Zn }

Do:

Page 20: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Rn

Shmuel Onn

R

ff1. Find initial point by auxiliary program

Proof of Theorems 1 and 2(linear and separable convex minimization)

2. Iteratively improve by Graver-best steps,that is, by best cz with c in Z and z in G(A).

with the Graver basis G(A)

To solve min {Σ fi(xi) : Ax = b, l ≤ x ≤ u, x in Zn }

Do:

Page 21: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Rn

Shmuel Onn

1. Find initial point by auxiliary program

R

ff

Using supermodality of f and integer Caratheodorytheorem (Cook-Fonlupt-Schrijver, Sebo) we can show polytime convergence to some optimal solution

Proof of Theorems 1 and 2(linear and separable convex minimization)

2. Iteratively improve by Graver-best steps,that is, by best cz with c in Z and z in G(A).

with the Graver basis G(A)

To solve min {Σ fi(xi) : Ax = b, l ≤ x ≤ u, x in Zn }

Do:

Page 22: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

N-Fold Integer Programming

Page 23: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

N-Fold Products

A(n) =

n

The n-fold product of an (r,s) x t bimatrix A =is the (r+ns) x nt matrix

Page 24: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

N-Fold Products

The n-fold product of an (r,s) x t bimatrix A =is the (r+ns) x nt matrix

A(n) =

n

Lemma 1: For fixed A we can compute the Graver basis G(A(n)) in polynomial time O(ng(A)) with g(A) the Graver complexity of A.

Page 25: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

(Non)-Linear N-Fold Integer Programming

Shmuel Onn

A(n) =

n

References: see Nonlinear Discrete Optimization (Onn), Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2010

min{f(x) : A(n)x = b, l ≤ x ≤ u, x in Znt}

Theorem 7: for various f can solve in polynomial time O(ng(A) L):

Page 26: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Some Applications

Page 27: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

1

1

0

33

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14

50

86

9

Shmuel Onn

The multi-index transportation problem of Motzkin (1952) concerns minimization over m1 X . . . X mk X mk+1 tables with given margins:

1. Multiway Tables

Page 28: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

1

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0

33

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2 0

14

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9

Shmuel Onn

The multi-index transportation problem of Motzkin (1952) concerns minimization over m1 X . . . X mk X mk+1 tables with given margins.

If all mi fixed then solvable by fixed dimension theory.

1. Multiway Tables

Page 29: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

1

1

0

33

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2 0

14

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n

Shmuel Onn

The multi-index transportation problem of Motzkin (1952) concerns minimization over m1 X . . . X mk X n tables with given margins.

1. Multiway Tables

Page 30: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

1

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n

Shmuel Onn

The multi-index transportation problem of Motzkin (1952) concerns minimization over m1 X . . . X mk X n tables with given margins.

Corollary 1: (Non)-linear optimization over m1 X . . . X mk X n tables with given margins is doable in polynomial time O(ng(m1,…,mk) L) .

If one side n is variable then it is an n-fold program.

1. Multiway Tables

Page 31: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

1

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0

33

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2 0

14

50

86

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n

Shmuel Onn

However, if two sides are variable then have universality:Every integer program is one over 3 X m X n tables with given line-sums.

The multi-index transportation problem of Motzkin (1952) concerns minimization over m1 X . . . X mk X n tables with given margins.

1. Multiway Tables

Corollary 1: (Non)-linear optimization over m1 X . . . X mk X n tables with given margins is doable in polynomial time O(ng(m1,…,mk) L) .

Page 32: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

suppliers

Km,n

consumers

s1

sm

c1

cn

cost: Σijcij(Σk xijk) aij

capacities: uij

2. Multicommodity Flows

flow

Corollary 2: For any fixed l commodities and m suppliers, can find min-congestion multicommodity-flow for n consumers in polytime.

Page 33: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Better Way of Finding Graver-Best Steps

Reference: N-fold integer programming in cubic time(Hemmecke, Onn, Romanchuk) Mathematical Programming, 2013

Page 34: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

min {wx : A(n)x = b, l ≤ x ≤ u, x in Znt}.

Better Way of Finding Graver-Best StepsLet x be feasible at some iteration of solving the n-fold program

A(n) =

n

Page 35: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

S0={0} Si-1=V(A) Si=V(A) Sn={v inV(A) : A1v=0}

vi-1 vi

hi := vi–vi-1 inV(A)li ≤ xi+chi ≤ ui

length wihi

V(A) := {v inZt : v is the sum of at most g(A) elements of G(A2) }

Lemma 2: A Graver-best step for x can be computed in O(n2) time, solvingO(|G(A2)|

g(A)n) dynamic programs each in time O(|G(A2)|2g(A)n).

Better Way of Finding Graver-Best Steps

A =

Page 36: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Present Complexity of N-fold Integer Programming

Theorem 9: For any fixed bimatrix A, the following linear n-fold integer program is solvable in time O(n3 log(w,b,l,u)):

max{wx : A(n)x = b, l ≤ x ≤ u, x in Znt}

Reference: N-fold integer programming in cubic time(Hemmecke, Onn, Romanchuk) Mathematical Programming, 2013

Combining Lemma 2 with our bound on number of iterations gives:

Page 37: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Present Complexity of N-fold Integer Programming

Theorem 9: For any fixed bimatrix A, the following linear n-fold integer program is solvable in time O(n3 log(w,b,l,u)):

Reference: N-fold integer programming in cubic time(Hemmecke, Onn, Romanchuk) Mathematical Programming, 2013

Combining Lemma 2 with our bound on number of iterations gives:

Instead of O(ng(A) log(w,b,l,u))

max{wx : A(n)x = b, l ≤ x ≤ u, x in Znt}

Page 38: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

With more work we can obtain the following:

Theorem 10: For any fixed r,s,t and p, the following p-piecewise affine separable convex n-fold integer program with variable (r,s) x t bimatrixA is solvable in time cubic in n, linear in log(f,b,l,u), and polynomial in A:

min{Σ fi(xi) : A(n)x = b, l ≤ x ≤ u, x in Znt}

Reference: N-fold integer programming in cubic time(Hemmecke, Onn, Romanchuk) Mathematical Programming, 2013

Present Complexity of N-fold Integer Programming

Page 39: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Fixed-Parameter Tractabilityand

Graver Approximation Hierarchy

Page 40: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Integer Programming is Fixed-Parameter Tractable

Corollary 3: (Non)-linear optimization over m1 X . . . X mk X n tables with given margins is doable in fixed-parameter cubic time O(n3 L).

Page 41: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Integer Programming is Fixed-Parameter Tractable

Corollary 3: (Non)-linear optimization over m1 X . . . X mk X n tables with given margins is doable in fixed-parameter cubic time O(n3 L).

Instead of O(ng(m1,…,mk) L)

Page 42: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Integer Programming is Fixed-Parameter Tractable

Now, have parameterization of all integer programming by universality:Every integer program is one over 3 X m X n tables with given line-sums.

Corollary 3: (Non)-linear optimization over m1 X . . . X mk X n tables with given margins is doable in fixed-parameter cubic time O(n3 L).

Instead of O(ng(m1,…,mk) L)

Page 43: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Integer Programming is Fixed-Parameter Tractable

Now, have parameterization of all integer programming by universality:Every integer program is one over 3 X m X n tables with given line-sums.

Hence (nonlinear) integer programming is fixed-parameter tractable:for each fixed m it is solvable in time O(n3 L) instead of O(ng(m) L).

Corollary 3: (Non)-linear optimization over m1 X . . . X mk X n tables with given margins is doable in fixed-parameter cubic time O(n3 L).

Instead of O(ng(m1,…,mk) L)

Page 44: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Our algorithm naturally enables a hierarchy of approximations of the universal integer program over 3 X m X n tables with variable m,n.

Graver Approximation Hierarchy

Page 45: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Graver Approximation Hierarchy

Vd(m) := { v in Z3m : v is the sum of at most d circuits of K3,m }

Our algorithm naturally enables a hierarchy of approximations of the universal integer program over 3 X m X n tables with variable m,n.

At level d of this hierarchy, we find approximated Graver-best stepsin time O(m9d n2), approximating V(A) in the dynamic programs by

Page 46: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Shmuel Onn

Graver Approximation Hierarchy

Vd(m) := { v in Z3m : v is the sum of at most d circuits of K3,m }

At level d of this hierarchy, we find approximated Graver-best stepsin time O(m9d n2), approximating V(A) in the dynamic programs by

We apply Graver-best steps while possible. If the approximation is satisfactory then we stop, else we proceed to the next level.

Our algorithm naturally enables a hierarchy of approximations of the universal integer program over 3 X m X n tables with variable m,n.

Page 47: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Some Bibliography (available at http://ie.technion.ac.il/~onn)

- The complexity of 3-way tables (SIAM J. Comp.)- Convex combinatorial optimization (Disc. Comp. Geom.)- Markov bases of 3-way tables (J. Symb. Comp.)- All linear and integer programs are slim 3-way programs (SIAM J. Opt.)- Graver complexity of integer programming (Annals Combin.)- N-fold integer programming (Disc. Opt. in memory of Dantzig) - Convex integer maximization via Graver bases (J. Pure App. Algebra)- Polynomial oracle-time convex integer minimization (Math. Prog.)- The quadratic Graver cone, quadratic integer minimization & extensions (Math Prog.)- N-fold integer programming in cubic time (Math. Prog.)

Shmuel Onn

Page 48: Nonlinear Integer Programming - Technionie.technion.ac.il/~onn/Talks/paris.pdf · 2015-06-08 · (Non)-Linear Integer Programming Shmuel Onn under which integer programming is fixed-parameter

Comprehensive development is in my monograph

available electronically from my homepage

(with kind permission of EMS)

The cubic time algorithm and thefixed-parameter tractability are in

N-fold integer programming in cubic timeMathematical Programming, 2013