FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension

We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.Comment: 16 pages, 4 figures; to appear in Mathematical Programmin

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. 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.Comment: 19 pages, 1 figur

A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs

In this paper we generalize N-fold integer programs and two-stage integer programs with N scenarios to N-fold 4-block decomposable integer programs. We show that for fixed blocks but variable N, these integer programs are polynomial-time solvable for any linear objective. Moreover, we present a polynomial-time computable optimality certificate for the case of fixed blocks, variable N and any convex separable objective function. We conclude with two sample applications, stochastic integer programs with second-order dominance constraints and stochastic integer multi-commodity flows, which (for fixed blocks) can be solved in polynomial time in the number of scenarios and commodities and in the binary encoding length of the input data. In the proof of our main theorem we combine several non-trivial constructions from the theory of Graver bases. We are confident that our approach paves the way for further extensions

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Coded Parity Packet Transmission Method for Two Group Resource Allocation

Gap value control is investigated when the number of source and parity packets is adjusted in a concatenated coding scheme whilst keeping the overall coding rate fixed. Packet-based outer codes which are generated from bit-wise XOR combinations of the source packets are used to adjust the number of both source packets. Having the source packets, the number of parity packets, which are the bit-wise XOR combinations of the source packets can be adjusted such that the gap value, which measures the gap between the theoretical and the required signal-to-noise ratio (SNR), is controlled without changing the actual coding rate. Consequently, the required SNR reduces, yielding a lower required energy to realize the transmission data rate. Integrating this coding technique with a two-group resource allocation scheme renders efficient utilization of the total energy to further improve the data rates. With a relatively small-sized set of discrete data rates, the system throughput achieved by the proposed two-group loading scheme is observed to be approximately equal to that of the existing loading scheme, which is operated with a much larger set of discrete data rates. The gain obtained by the proposed scheme over the existing equal rate and equal energy loading scheme is approximately 5 dB. Furthermore, a successive interference cancellation scheme is also integrated with this coding technique, which can be used to decode and provide consecutive symbols for inter-symbol interference (ISI) and multiple access interference (MAI) mitigation. With this integrated scheme, the computational complexity is signi cantly reduced by eliminating matrix inversions. In the same manner, the proposed coding scheme is also incorporated into a novel fixed energy loading, which distributes packets over parallel channels, to control the gap value of the data rates although the SNR of each code channel varies from each other

A Generating Function for all Semi-Magic Squares and the Volume of the Birkhoff Polytope

We present a multivariate generating function for all n x n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B_n of n x n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B_n and any of its faces.Comment: 24 pages, 1 figure. To appear in Journal of Algebraic Combinatoric

An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

Prodsimplicial-Neighborly Polytopes

Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal and Ziegler's "projecting deformed products" construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of r simplices, when the dimensions of these simplices are all large compared to k.Comment: 28 pages, 9 figures; minor correction

Group theoretic dimension of stationary symmetric \alpha-stable random fields

The growth rate of the partial maximum of a stationary stable process was first studied in the works of Samorodnitsky (2004a,b), where it was established, based on the seminal works of Rosi\'nski (1995,2000), that the growth rate is connected to the ergodic theoretic properties of the flow that generates the process. The results were generalized to the case of stable random fields indexed by Z^d in Roy and Samorodnitsky (2008), where properties of the group of nonsingular transformations generating the stable process were studied as an attempt to understand the growth rate of the partial maximum process. This work generalizes this connection between stable random fields and group theory to the continuous parameter case, that is, to the fields indexed by R^d.Comment: To appear in Journal of Theoretical Probability. Affiliation of the authors are update