A polynomial oracle-time algorithm for convex integer minimization

In this paper we consider the solution of certain convex integer minimization problems via greedy augmentation procedures. We show that a greedy augmentation procedure that employs only directions from certain Graver bases needs only polynomially many augmentation steps to solve the given problem. W...

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Bibliographic Details
Published in:Mathematical programming Vol. 126; no. 1; pp. 97 - 117
Main Authors: Hemmecke, Raymond, Onn, Shmuel, Weismantel, Robert
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.01.2011
Springer
Springer Nature B.V
Subjects:
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Augmentation
Calculus of Variations and Optimal Control; Optimization
Circuits
Combinatorics
Computer science; control theory; systems
Convex analysis
Convex and discrete geometry
Exact sciences and technology
Full Length Paper
Geometry
Information retrieval. Graph
Integer programming
Integers
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Minimization
Multivariate analysis
Numerical Analysis
Optimization
Polynomials
Probability and statistics
Sciences and techniques of general use
Statistics
Stochasticity
Studies
Theorems
Theoretical
Theoretical computing
68R
68U
52B
68W
52C
90B
62H
90C
68Q
Polytope
Statistical analysis
Probabilistic approach
Minimization
Multivariate analysis
Convex programming
Polynomial time
Integer programming
Discrete geometry
Oracle
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Description
Summary:In this paper we consider the solution of certain convex integer minimization problems via greedy augmentation procedures. We show that a greedy augmentation procedure that employs only directions from certain Graver bases needs only polynomially many augmentation steps to solve the given problem. We extend these results to convex N -fold integer minimization problems and to convex 2-stage stochastic integer minimization problems. Finally, we present some applications of convex N -fold integer minimization problems for which our approach provides polynomial time solution algorithms.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-009-0276-7