N-fold integer programming is a fundamental problem with a variety of natural applications in operations research and statistics. Moreover, it is universal and provides a new, variable-dimension, parametrization of all of integer programming. The fastest algorithm for n-fold integer programming predating the present article runs in time O n g(A) L with L the binary length of the numerical part of the input and g(A) the so-called Graver complexity of the bimatrix A defining the system. In this article we provide a drastic improvement and establish an algorithm which runs in time O n 3 L having cubic dependency on n regardless of the bimatrix A. Our algorithm can be extended to separable convex piecewise affine objectives as well, and also to systems defined by bimatrices with variable entries. Moreover, it can be used to define a hierarchy of approximations for any integer programming problem.

This monograph develops an algorithmic theory of nonlinear discrete optimization. It introduces a simple and useful setup which enables the polynomial time solution of broad fundamental classes of nonlinear combinatorial optimization and integer programming problems in variable dimension. An important part of this theory is enhanced by recent developments in the algebra of Graver bases. The power of the theory is demonstrated by deriving the first polynomial time algorithms in a variety of application areas within operations research and statistics, including vector partitioning, matroid optimization, experimental design, multicommodity flows, multiindex transportation and privacy in statistical databases. The monograph is based on twelve lectures which I gave within the Nachdiplom Lectures series at ETH Zürich in Spring 2009, broadly extending and updating five lectures on convex discrete optimization which I gave within the Séminaire de Mathématiques Supérieures series at Université de Montréal in June 2006. I thank the support of the Israel Science Foundation and the support and hospitality of ETH Zürich, Université de Montréal, and Mathematisches Forschungsinstitut Oberwolfach, where parts of the research underlying the monograph took place. I am indebted to Jesus De Loera, Raymond Hemmecke, Jon Lee, Uriel G. Rothblum and Robert Weismantel for their collaboration in developing the theory described herein. I am also indebted to Komei Fukuda for his interest and very helpful suggestions, and to many other colleagues including my co-authors on articles cited herein and my audience at the Nachdiplom Lectures at ETH. Finally, I am especially grateful to my wife Ruth, our son Amos, and our daughter Naomi, for their support throughout the writing of this monograph. Haifa, May 2010 Shmuel Onn n i=1 w i x i is linear, which is the case considered in most literature on discrete optimization and is already hard and often intractable. The matrix W is given explicitly, and the computational complexity of the problem depends on the encoding of its entries (binary versus unary). Nonlinear matroid optimization. Given a matroid M = (N, B) on ground set N := {1,. .. , n}, an integer d × n matrix W , and a function f : Z d → R, solve max{f (W x) : x ∈ S} with S ⊆ {0, 1} n the set of (indicators of) bases or independent sets of M :

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.

We consider the class of shaped partition problems of partitioning n given vectors in d-dimensional criteria space into p parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. This class has broad expressive power and captures NP-hard problems even if either d or p is fixed. In contrast, we show that when both d and p are fixed, the problem can be solved in strongly polynomial time. Our solution method relies on studying the corresponding class of shaped partition polytopes. Such polytopes may have exponentially many vertices and facets even when one of d or p is fixed; however, we show that when both d and p are fixed, the number of vertices of any shaped partition polytope is O(n d p 2) and all vertices can be produced in strongly polynomial time.

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Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colourable, Hamiltonian, etc.) if and only if a related system of polynomial equations has a solution.For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows.In the first part of the paper, we show that the minimum degree of a Nullstellensatz certificate for the non-existence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non-3-colourability, we proved that the minimum degree of a Nullstellensatz certificate is at least four. Our efforts so far have only yielded graphs with Nullstellensatz certificates of precisely that degree.In the second part of this paper, for the purpose of computation, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colourable subgraph. We include some applications to graph theory.

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The goal of this paper is to introduce the notion of certificates, which verify the accuracy of solutions of computational problems with convex structure. Such problems include minimizing convex functions, variational inequalities with monotone operators, computing saddle points of convex-concave functions, and solving convex Nash equilibrium problems. We demonstrate how the implementation of the ellipsoid method and other cutting plane algorithms can be augmented with the computation of such certificates without significant increase of the computational effort. Further, we show that (computable) certificates exist for any algorithm that is capable of producing solutions of guaranteed accuracy.