Robust Integer Programming
(and more)
Shmuel Onn
Technion – Israel Institute of Technology
Outline
1. Graver bases and integer programming
Shmuel Onn
Outline
1. Graver bases and integer programming
2. Robust integer programming
Shmuel Onn
Outline
1. Graver bases and integer programming
2. Robust integer programming
3. Huge multiway tables: P versus NP intersect coNP
Shmuel Onn
Graver Bases
and
Integer Programming
The Fundamental Theorem
Definition: The Graver basis of an integer matrix A is the finite set G(A)
of conformal-minimal nonzero integer vectors x satisfying Ax = 0.
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The Fundamental Theorem
Definition: 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|
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The Fundamental Theorem
Definition: 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|
Theorem: separable convex minimization in polytime with G(A):
min
 Zn }
{Σ fi(xi) : Ax = b, l ≤ x ≤ u, x in
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The Fundamental Theorem
Definition: 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|
Theorem: separable convex minimization in polytime with G(A):
min
 Zn }
{Σ fi(xi) : Ax = b, l ≤ x ≤ u, x in
Reference: A polynomial oracle-time algorithm for convex integer minimization
(Hemmecke, Onn, Weismantel) Mathematical Programming, 2011
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The Fundamental Algorithm
To solve min {f(x) : Ax = b, l ≤ x ≤ u, x 
in Zn }
n
R
with the Graver basis G(A)
Do:
f
R
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The Fundamental Algorithm
To solve min {f(x) : Ax = b, l ≤ x ≤ u, x 
in Zn }
n
R
f
with the Graver basis G(A)
Do:
1. Find initial point by auxiliary program
R
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The Fundamental Algorithm
To solve min {f(x) : Ax = b, l ≤ x ≤ u, x 
in Zn }
n
R
f
R
with the Graver basis G(A)
Do:
1. Find initial point by auxiliary program
2. Iteratively improve by Graver-best steps,
 Z and z in
 G(A).
that is, by best cz with c in
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The Fundamental Algorithm
To solve min {f(x) : Ax = b, l ≤ x ≤ u, x 
in Zn }
n
R
f
R
with the Graver basis G(A)
Do:
1. Find initial point by auxiliary program
2. Iteratively improve by Graver-best steps,
 Z and z in
 G(A).
that is, by best cz with c in
Using supermodality of f and integer Caratheodory
theorem (Cook-Fonlupt-Schrijver, Sebo) we can
show polytime convergence to some optimal solution
Shmuel Onn
Comprehensive development
of the theory of Graver bases
and integer programming
(and more) is in my monograph
Available electronically
from my homepage
(with kind permission of EMS)
Robust
Integer Programming
Robust Integer Programming
Player 1 (decision maker) chooses point x feasible in
X := {x Zn : Ax = b, l ≤ x ≤ u }
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Robust Integer Programming
Player 1 (decision maker) chooses point x feasible in
X := {x Zn : Ax = b, l ≤ x ≤ u }
Player 2 (nature) reveals cost c in a set of one of two forms:
Box:
C := {c  Zn : d ≤ c ≤ e }
List:
C := {c1, . . . , ck }
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Robust Integer Programming
Player 1 (decision maker) chooses point x feasible in
X := {x Zn : Ax = b, l ≤ x ≤ u }
Player 2 (nature) reveals cost c in a set of one of two forms:
Box:
C := {c  Zn : d ≤ c ≤ e }
List:
C := {c1, . . . , ck }
The problem is:
minx  X maxc C cx
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Robust Integer Programming
Player 1 (decision maker) chooses point x feasible in
X := {x Zn : Ax = b, l ≤ x ≤ u }
Player 2 (nature) reveals cost c in a set of one of two forms:
Box:
C := {c  Zn : d ≤ c ≤ e }
List:
C := {c1, . . . , ck }
The problem is:
minx  X maxc C cx
The dual problem is:
maxc  C minx  X cx
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Robust Integer Programming
Theorem: The following hold for Robust IP with G(A) given:
minx  X maxc  C cx
maxc C minx  X cx
Box C
List C
polynomial time
NP-hard
NP-hard
polynomial time
Reference: Robust integer programming
(Onn) Operations Research Letters, 2014
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Robust Integer Programming
Player 1 chooses point in
X := {x  Zn : Ax = a, 0 ≤ x ≤ 1 }
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Robust Integer Programming
Player 1 chooses point in
X := {x  Zn : Ax = a, 0 ≤ x ≤ 1 }
Player 2 chooses point in
Y := {y  Zn : By = b, 0 ≤ y ≤ 1 }
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Robust Integer Programming
Player 1 chooses point in
X := {x  Zn : Ax = a, 0 ≤ x ≤ 1 }
Player 2 chooses point in
Y := {y  Zn : By = b, 0 ≤ y ≤ 1 }
The cooperative problem is:
minx  X miny  Y xy
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Robust Integer Programming
Theorem: The following hold for Cooperative IP:
1. The following is NP-hard even if G(A) and G(B) are given:
min {xy : x, y  {0,1}n, Ax = a, By = b
}
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Robust Integer Programming
Theorem: The following hold for Cooperative IP:
1. The following is NP-hard even if G(A) and G(B) are given:
min {xy : x, y  {0,1}n, Ax = a, By = b
}
2. The following is NP-hard even if A=B is totally unimodular:
min {xy : x, y {0,1}n, Ax = a, Ay = b }
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Robust Integer Programming
Theorem: The following hold for Cooperative IP:
1. The following is NP-hard even if G(A) and G(B) are given:
min {xy : x, y  {0,1}n, Ax = a, By = b
}
2. The following is NP-hard even if A=B is totally unimodular:
min {xy : x, y {0,1}n, Ax = a, Ay = b }
3. The following is polynomial time solvable if A is totally unimodular :
min {x1x2… xp : xk  {0,1}n, Axk = a, k=1,…,p }
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Application: Minimizing Critical Edges
We wish to choose p paths from s to t in a digraph G, and then use
one of them randomly to make it Robust against an adversary who
tries to block us. An edge is critical if used by all paths. We protect
at high cost each critical edge against the adversary. So our problem
is to choose p paths with minimum number of critical edges. It is
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Application: Minimizing Critical Edges
We wish to choose p paths from s to t in a digraph G, and then use
one of them randomly to make it Robust against an adversary who
tries to block us. An edge is critical if used by all paths. We protect
at high cost each critical edge against the adversary. So our problem
is to choose p paths with minimum number of critical edges. It is
min {x1x2… xp : xk {0,1}n, Axk = a, k=1,…,p }
with A the vertex-arc matrix of G and a suitable integer vector.
So by our theorem, we can solve this problem in polynomial time.
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Application: Minimizing Critical Edges
We wish to choose p paths from s to t in a digraph G, and then use
one of them randomly to make it Robust against an adversary who
tries to block us. An edge is critical if used by all paths. We protect
at high cost each critical edge against the adversary. So our problem
is to choose p paths with minimum number of critical edges. It is
min {x1x2… xp : xk {0,1}n, Axk = a, k=1,…,p }
with A the vertex-arc matrix of G and a suitable integer vector.
So by our theorem, we can solve this problem in polynomial time.
In fact, we solve a harder problem of lexicographically minimizing
the number of edges used by at least k paths for k=p,p-1,…,2,1
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The Unimodular Intersection Problem
Theorem: For c nondecreasing and A totally unimodular, can solve in
polynomial time the following nonlinear IP with x the compression of x:
min {cx : x {0,1}np, Axk = a, k=1,…,p }
Reference: The unimodular intersection problem
(Kaibel, Onn, Pauline Sarrabezolles) Arxiv, 2015
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Huge Multiway Tables
Multiway Tables
The multiway table problem of Motzkin (1952)
concerns minimization over
m1 X . . . X mk X mk+1 tables with given margins.
6
3
1
3
2
0
4
0
8
1
5
0
1
0
9
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Multiway Tables
The multiway table problem of Motzkin (1952)
concerns minimization over
m1 X . . . X mk X mk+1 tables with given margins.
6
3
1
3
2
0
4
0
8
1
5
0
1
0
If all mi fixed then solvable by fixed dimension theory.
9
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Multiway Tables
The multiway table problem of Motzkin (1952)
6
concerns minimization over
m1 X . . . X mk X n tables with given margins.
3
n
1
3
2
0
4
0
8
1
5
0
1
0
9
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Multiway Tables
The multiway table problem of Motzkin (1952)
6
concerns minimization over
m1 X . . . X mk X n tables with given margins.
3
0
n
4
0
1
5
g
If one side n is variable then we have the following:
1
3
2
8
0
1
0
9
Lemma: For fixed m1,. . ., mk there is Graver complexity g:=g(m1,. . ., mk)
such that the relevant Graver basis consists of all O(ng)
lifts of the relevant Graver basis of m1 X . . . X mk X g tables.
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Multiway Tables
The multiway table problem of Motzkin (1952)
6
concerns minimization over
3
m1 X . . . X mk X n tables with given margins.
0
n
4
0
1
5
g
If one side n is variable then we have the following:
1
3
2
8
0
1
0
9
Lemma: For fixed m1,. . ., mk there is Graver complexity g:=g(m1,. . ., mk)
such that the relevant Graver basis consists of all O(ng)
lifts of the relevant Graver basis of m1 X . . . X mk X g tables.
g(3,3)=9, g(3,4) = 27, g(3,5) = ?
Ω(2m) = g(3,m) = Ο(6m)
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Multiway Tables
The multiway table problem of Motzkin (1952)
6
concerns minimization over
m1 X . . . X mk X n tables with given margins.
3
1
3
2
0
n
4
0
8
1
5
g
0
1
0
9
Corollary: (Non)-linear optimization over m1 X . . . X mk X n tables
with given margins is doable in polynomial time O(ng(m1,…,mk) L) .
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Multiway Tables
The multiway table problem of Motzkin (1952)
6
concerns minimization over
m1 X . . . X mk X n tables with given margins.
3
n
1
3
2
0
4
0
8
1
5
0
1
0
9
Corollary: (Non)-linear optimization over m1 X . . . X mk X n tables
with given margins is doable in polynomial time O(ng(m1,…,mk) L) .
Corollary: Robust optimization with Box C over m1 X . . . X mk X n tables
with given margins is doable in polynomial time O(ng(m1,…,mk) L) .
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Multiway Tables
The multiway table problem of Motzkin (1952)
6
concerns minimization over
m1 X . . . X mk X n tables with given margins.
3
n
1
3
2
0
4
0
8
1
5
0
1
0
9
Corollary: (Non)-linear optimization over m1 X . . . X mk X n tables
with given margins is doable in polynomial time O(ng(m1,…,mk) L) .
However, if two sides are variable then have the Universality Theorem:
Every integer program is one over 3 X m X n tables with given line-sums.
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Multiway Tables
Theorem: Linear optimization over m1 X . . . X mk X n tables with
given margins is doable in fixed-parameter cubic time O(n3 L).
Reference: N-fold integer programming in cubic time
(Hemmecke, Onn, Lyubov Romanchuk) Mathematical Programming, 2013
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Huge Multiway Tables
Consider the problem of deciding existence
of an l X m X n huge table with t given types
of row-sums and column-sums having nk layers
of each type k with the nk encoded in binary.
6
3
n
1
3
2
0
4
0
8
1
5
0
1
0
9
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Huge Multiway Tables
Consider the problem of deciding existence
of an l X m X n huge table with t given types
of row-sums and column-sums having nk layers
of each type k with the nk encoded in binary.
6
3
n
1
3
2
0
4
0
8
1
5
0
1
0
9
Theorem: 1. For 3-way tables and variable t the problem is in P.
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Huge Multiway Tables
Consider the problem of deciding existence
of an l X m X n huge table with t given types
of row-sums and column-sums having nk layers
of each type k with the nk encoded in binary.
6
3
n
1
3
2
0
4
0
8
1
5
0
1
0
9
Theorem: 1. For 3-way tables and variable t the problem is in P.
2. For 4-way tables and fixed t the problem is in P.
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Huge Multiway Tables
Consider the problem of deciding existence
of an l X m X n huge table with t given types
of row-sums and column-sums having nk layers
of each type k with the nk encoded in binary.
6
3
n
1
3
2
0
4
0
8
1
5
0
1
0
9
Theorem: 1. For 3-way tables and variable t the problem is in P.
2. For 4-way tables and fixed t the problem is in P.
3. For 4-way tables and variable t it is in NP and coNP.
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Huge Multiway Tables
Consider the problem of deciding existence
of an l X m X n huge table with t given types
of row-sums and column-sums having nk layers
of each type k with the nk encoded in binary.
6
3
n
1
3
2
0
4
0
8
1
5
0
1
0
9
Theorem: 1. For 3-way tables and variable t the problem is in P.
2. For 4-way tables and fixed t the problem is in P.
3. For 4-way tables and variable t it is in NP and coNP.
Reference: Huge Multiway Table Problems (Onn) Discrete Optimization, 2014
Huge Unimodular N-Fold Programs (Onn, Sarrabezolles) Arxiv, 2015
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