Integer Programming in Strongly Polynomial Oracle Time
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Shmuel Onn Technion – Israel Institute of Technology http://ie.technion.ac.il/~onn Integer Programming in Strongly Polynomial Oracle Time joint work with Martin Koutecký and Asaf Levin, coming ICALP
Transcript of Integer Programming in Strongly Polynomial Oracle Time
Shmuel Onn
Technion – Israel Institute of Technology
http://ie.technion.ac.il/~onn
Integer Programming inStrongly Polynomial Oracle Time
joint work with Martin Koutecký and Asaf Levin, coming ICALP
Integer Programming
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IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
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l,u: lower/upper bounds in Zn w: objective in Zn
A: integer m x n matrix b: right-hand side in Zm
Integer Programming
IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
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l,u: lower/upper bounds in Zn w: objective in Zn
A: integer m x n matrix b: right-hand side in Zm
The bit size of the data is denoted by [A,w,b,l,u]
Integer Programming
IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
Outline
The Strongly Polynomial Oracle Algorithm
Outline
The Strongly Polynomial Oracle Algorithm
Effective Realization of Oracles
Outline
The Strongly Polynomial Oracle Algorithm
Programs with Block Structure
Effective Realization of Oracles
Outline
The Strongly Polynomial Oracle Algorithm
Programs with Block Structure
Effective Realization of Oracles
Programs with Bounded Tree-Depth
The Strongly Polynomial Oracle Algorithm
:Background in my Book
Theory of Graver bases for integer programming
(and more)
Available electronically from my homepage
(with kind permission of EMS)
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Graver Bases
x is conformal to y if xiyi ≥ 0 (same orthant) and |xi| ≤ |yi| for all i
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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Graver-Best OracleLet x be a feasible point of the integer program
IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
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Graver-Best OracleLet x be a feasible point of the integer program
A Graver-best step for x is a vector h such that x+h is feasibleand at least as good as any feasible x+cg with c in Z and g in G(A).
IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
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Graver-Best OracleLet x be a feasible point of the integer program
A Graver-best step for x is a vector h such that x+h is feasibleand at least as good as any feasible x+cg with c in Z and g in G(A).
A Graver-best oracle for an integer matrix A is one that queriedon w,b,l,u and feasible x returns a Graver-best step h for x.
IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
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Weakly Polynomial Solution
Lemma: IP is solvable with Graver-best oracle for A in time O(n [A,w,b,l,u])
IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
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(Hemmecke, Onn, Weismantel and my book Theorem 3.12)
Lemma: IP is solvable with Graver-best oracle for A in time O(n [A,w,b,l,u])
IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
Weakly Polynomial Solution
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Lemma: IP is solvable with Graver-best oracle for A in time O(n [A,w,b,l,u])
IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
Weakly Polynomial Solution
The algorithm iteratively adds Graver-best steps till optimum is reached
(Hemmecke, Onn, Weismantel and my book Theorem 3.12)
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Strongly Polynomial Solution
Lemma: IP is solvable with Graver-best oracle for A in time O(n [A,w,b,l,u])
IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
Theorem 1: IP is solvable with Graver-best oracle for A in time poly(n[A])
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Lemma: IP is solvable with Graver-best oracle for A in time O(n [A,w,b,l,u])
(our postdoc Koutecký, Levin, Onn, coming ICALP)
IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
Theorem 1: IP is solvable with Graver-best oracle for A in time poly(n[A])
Strongly Polynomial Solution
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Talk Wednesday morning by Martin Koutecký:breakthroughs in computational social choice
Lemma: IP is solvable with Graver-best oracle for A in time O(n [A,w,b,l,u])
Theorem 1: IP is solvable with Graver-best oracle for A in time poly(n[A])
IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
Strongly Polynomial Solution
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Lemma: IP is solvable with Graver-best oracle for A in time O(n [A,w,b,l,u])
Theorem 1: IP is solvable with Graver-best oracle for A in time poly(n[A])
IP: max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
Strongly Polynomial Solution
1. Solve the LP-relaxation and get real optimal y* (Tardos)
Proof:
2. By proximity lemma (HS,HKW) can search for integer optimal]A] polynomial in [u,l,bwith [u ,l,bto y* and reduce data to closex*
) Tardos-Frank] (A] polynomial in [wwith [w. Reduce data to 3
])A[n])=poly(u,l,b,wA,[nto solve the reduced program in O(lemma. Use 4
Effective Realization
of
Graver-Best Oracles
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Realization of Graver-Best Oracles
The primal graph of A has columns as vertices and two columnsform an edge if some row has nonzero entries in both columns.
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Realization of Graver-Best Oracles
The primal graph of A has columns as vertices and two columnsform an edge if some row has nonzero entries in both columns.
The dual graph of A is the primal graph of its transpose AT
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Realization of Graver-Best Oracles
The primal graph of A has columns as vertices and two columnsform an edge if some row has nonzero entries in both columns.
The dual graph of A is the primal graph of its transpose AT
Primal Lemma (KLO): If the primal graph of A has bounded tree-width and
.is bounded then get an effective Graver-best oracle for A by DP
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Realization of Graver-Best Oracles
The primal graph of A has columns as vertices and two columnsform an edge if some row has nonzero entries in both columns.
The dual graph of A is the primal graph of its transpose AT
Primal Lemma (KLO): If the primal graph of A has bounded tree-width and
.is bounded then get an effective Graver-best oracle for A by DP
Dual Lemma (KLO): If the dual graph of A has bounded tree-width and
.is bounded then get an effective Graver-best oracle for A by DP
Programs with
Block Structure
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N-Fold Integer Programming
A =
n
The n-fold product of rxt block A1 and sxt block A2 is
max {wx : Ax = b, l ≤ x ≤ u, x in Znt}.
Parameterization of N-Fold IP
Consider n-fold integer programming over r x t block A1 and s xt block A2
A =
n
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Parameterization of N-Fold IP
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max {wx : Ax = b, l ≤ x ≤ u, x in Znt}.
A =
n
The parameters are r,s,t,a=max A
Consider n-fold integer programming over r x t block A1 and s xt block A2
The input is n and the bit size [w,b,l,u]
Parameterization of N-Fold IP
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The parameters are r,s,t,a=max A
max {wx : Ax = b, l ≤ x ≤ u, x in Znt}.
A =
n
Consider n-fold integer programming over r x t block A1 and s xt block A2
Complexity of N-Fold Integer Programming
-- Polynomial time nf(r,s,t,a)[w,b,l,u] (De Loera, Hemmecke, Onn, Weismantel)
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Complexity of N-Fold Integer Programming
-- Polynomial time nf(r,s,t,a)[w,b,l,u] (De Loera, Hemmecke, Onn, Weismantel)
-- Parameter tractable f(r,s,t,a)n3[w,b,l,u] (Hemmecke, Onn, Romanchuk)
Led to several recent breakthroughs in the theory of parameterized complexity
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Complexity of N-Fold Integer Programming
-- Strongly polynomial nf(r,s,t,a) (De Loera, Hemmecke, Lee)
-- Polynomial time nf(r,s,t,a)[w,b,l,u] (De Loera, Hemmecke, Onn, Weismantel)
-- Parameter tractable f(r,s,t,a)n3[w,b,l,u] (Hemmecke, Onn, Romanchuk)
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Complexity of N-Fold Integer Programming
Theorem 2 (Koutecký, Levin, Onn) :
Parameter tractable and strongly polynomial f(r,s,a) poly(n,t)
-- Strongly polynomial nf(r,s,t,a) (De Loera, Hemmecke, Lee)
-- Polynomial time nf(r,s,t,a)[w,b,l,u] (De Loera, Hemmecke, Onn, Weismantel)
-- Parameter tractable f(r,s,t,a)n3[w,b,l,u] (Hemmecke, Onn, Romanchuk)
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Complexity of N-Fold Integer Programming
Theorem 2 (Koutecký, Levin, Onn) :
Parameter tractable and strongly polynomial f(r,s,a) poly(n,t)
-- Strongly polynomial nf(r,s,t,a) (De Loera, Hemmecke, Lee)
-- Polynomial time nf(r,s,t,a)[w,b,l,u] (De Loera, Hemmecke, Onn, Weismantel)
-- Parameter tractable f(r,s,t,a)n3[w,b,l,u] (Hemmecke, Onn, Romanchuk)
Proof: Use dual lemma to get parameter tractable Graver-best oracle
and use Theorem 1 to get parameter tractable strongly polynomial time.
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Optimization over l X m X n tables with given line sums:
Corollary: Parameter tractable and strongly polynomial f(l,m) poly(n)
Example - Multiway Tables
1
1
0
33
0
2 0
14
50
86
9
n
Tree-Fold Integer Programming(Chen, Marx, SODA 2018): Tree-fold integer programs have a matrix with several blocks in tree structure, parameterized by si,t,a=max A
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Tree-Fold Integer Programming
Theorem 3 (Koutecký, Levin, Onn) :
Parameter tractable and strongly polynomial f(r,si,a) poly(n,t)
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Tree-Fold Integer Programming
Theorem 3 (Koutecký, Levin, Onn) :
Parameter tractable and strongly polynomial f(r,si,a) poly(n,t)
Proof: Use dual lemma to get parameter tractable Graver-best oracle
and use Theorem 1 to get parameter tractable strongly polynomial time.
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Tree-Fold Integer Programming
Improves on Chen, Marx, SODA 2018 - strongly polynomial and t variable
Theorem 3 (Koutecký, Levin, Onn) :
Parameter tractable and strongly polynomial f(r,si,a) poly(n,t)
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Proof: Use dual lemma to get parameter tractable Graver-best oracle
and use Theorem 1 to get parameter tractable strongly polynomial time.
Multistage Stochastic Integer ProgrammingThese programs have a matrix the transpose AT of a tree-fold matrix with several blocks in tree structure, parameterized by si,t,a=max Ai
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Multistage Stochastic Integer Programming
Theorem 4 (Koutecký, Levin, Onn) :
Parameter tractable and strongly polynomial f(r,si,a) poly(n,t)
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Multistage Stochastic Integer Programming
Theorem 4 (Koutecký, Levin, Onn) :
Parameter tractable and strongly polynomial f(r,si,a) poly(n,t)
Proof: Use primal lemma to get parameter tractable Graver-best oracle
and use Theorem 1 to get parameter tractable strongly polynomial time.
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Programs with
Bounded Tree-Depth
Programs with Bounded Tree-Depth
Theorem 5 (Koutecký, Levin, Onn) :Integer programs with matrix A parameterized by a=max Aand by the tree-depth d of the dual graph of A can be solved in parameter tractable and strongly polynomial time f(a,d) poly(n)
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Programs with Bounded Tree-Depth
Proof: Embed such programs in tree-fold programs and use Theorem 3
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Theorem 5 (Koutecký, Levin, Onn) :Integer programs with matrix A parameterized by a=max Aand by the tree-depth d of the dual graph of A can be solved in parameter tractable and strongly polynomial time f(a,d) poly(n)
Theorem 6 (Koutecký, Levin, Onn) :Integer programs with matrix A parameterized by a=max Aand by the tree-depth d of the primal graph of A can be solved in parameter tractable and strongly polynomial time f(a,d) poly(n)
Programs with Bounded Tree-Depth
Proof: Embed such programs in tree-fold programs and use Theorem 3
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Theorem 5 (Koutecký, Levin, Onn) :Integer programs with matrix A parameterized by a=max Aand by the tree-depth d of the dual graph of A can be solved in parameter tractable and strongly polynomial time f(a,d) poly(n)
Proof: Embed them in multistage stochastic programs and use Theorem 4
Programs with Bounded Tree-Depth
Proof: Embed such programs in tree-fold programs and use Theorem 3
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Theorem 5 (Koutecký, Levin, Onn) :Integer programs with matrix A parameterized by a=max Aand by the tree-depth d of the dual graph of A can be solved in parameter tractable and strongly polynomial time f(a,d) poly(n)
Theorem 6 (Koutecký, Levin, Onn) :Integer programs with matrix A parameterized by a=max Aand by the tree-depth d of the primal graph of A can be solved in parameter tractable and strongly polynomial time f(a,d) poly(n)
also available at http://ie.technion.ac.il/~onn
Martin Koutecký, Asaf Levin, Shmuel Onn, coming ICALP
A Parameterized Strongly Polynomial Algorithmfor Block Structured Integer Programs
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