Raymond Hemmecke
We show the existence of an FPTAS for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.
We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an…
We consider N-fold 4-block decomposable integer programs, which simultaneously generalize N-fold integer programs and two-stage stochastic integer programs with N scenarios. In previous work [R. Hemmecke, M. Koeppe, R. Weismantel, A…
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…
We settle the computational complexity of fundamental questions related to multicriteria integer linear programs, when the dimensions of the strategy space and of the outcome space are considered fixed constants. In particular we construct:…
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…
For given matrix $A\in\Z^{d\times n}$, the set $P_{b}=\{z:Az=b,z\in\Z^n_+\}$ describes the preimage or fiber of $b\in\Z^d$ under the $\Z$-linear map $f_A:\Z^n_+\to\Z^d$, $x\mapsto Ax$. The fiber $P_{b}$ is called atomic, if…
Motivated by Bland's linear-programming generalization of the renowned Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm, we discuss three closely-related natural augmentation rules for linear and integer-linear…
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…
We consider the connectivity of fiber graphs with respect to Gr\"obner basis and Graver basis moves. First, we present a sequence of fiber graphs using moves from a Gr\"obner basis and prove that their edge-connectivity is lowest possible…
Both the combinatorial and the circuit diameters of polyhedra are of interest to the theory of linear programming for their intimate connection to a best-case performance of linear programming algorithms. We study the diameters of dual…
In this paper we introduce the circuit diameter of polyhedra, which is always bounded from above by the combinatorial diameter. We consider dual transportation polyhedra defined on general bipartite graphs. For complete $M{\times}N$…
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…
In this paper, we present a construction that turns certain relations on Graver basis elements of an $M$-fold matrix $A^{(M)}$ into relations on Graver basis elements of an $(M+1)$-fold matrix $A^{(M+1)}$. In doing so, we strengthen the…
Associated to any vector configuration A is a toric ideal encoded by vectors in the kernel of A. Each toric ideal has two special generating sets: the universal Gr\"obner basis and the Graver basis. While the former is generally a proper…
Motivated by fundamental problems in chemistry and biology we study cluster graphs arising from a set of initial states $S\subseteq\Z^n_+$ and a set of transitions/reactions $M\subseteq\Z^n_+\times\Z^n_+$. The clusters are formed out of…
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…
The multicommodity flow problem is NP-hard already for two commodities over bipartite graphs. Nonetheless, using our recent theory of n-fold integer programming and extensions developed herein, we are able to establish the surprising…
Bayesian networks are basic graphical models, used widely both in statistics and artificial intelligence. These statistical models of conditional independence structure are described by acyclic directed graphs whose nodes correspond to…
In this paper we present two independent computational proofs that the monoid derived from $5\times 5\times 3$ contingency tables is normal, completing the classification by Hibi and Ohsugi. We show that Vlach's vector disproving normality…