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Matthias Köppe

Publications and source records attributed to Matthias Köppe.

At least 19 recordsLinked to original sources
arXiv2025

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. IV. The General Unimodular Two-Dimensional Case

We study an abstract setting for cutting planes for integer programming called the infinite group problem. In this abstraction, cutting planes are computed via cut generating function that act on the simplex tableau. In this function space, cut generating functions are classified as minimal, extreme, and facets as a proxy for understanding the strength or potential importance of these functions. Prior work developed algorithms for testing minimality, extremality, and facetness for cut generating functions applied to 1-row tableau and to some 2-row tableau in a restricted setting. We complement and generalize this work by giving an algorithm for testing the extremality of a large class of minimal valid functions for the two-dimensional infinite group problem. Along the way, we develop results of independent interest on functional equations and infinite systems of linear equations.

math.OC↗
arXiv2022

Fine-Grained Liquid Democracy for Cumulative Ballots

We investigate efficient ways for the incorporation of liquid democracy into election settings in which voters submit cumulative ballots, i.e., when each voter is assigned a virtual coin that she can then distribute as she wishes among the available election options. In particular, we are interested in fine-grained liquid democracy, meaning that voters are able to designate a partial coin to a set of election options and delegate the decision on how to further split this partial coin among those election options to another voter of her choice. The fact that we wish such delegations to be transitive -- combined with our aim at fully respecting such delegations -- means that inconsistencies and cycles can occur, thus we set to find computationally-efficient ways of resolving voter delegations. To this aim we develop a theory based fixed-point theorems and mathematical programming techniques and we show that for various variants of definitions regarding how to resolve such transitive delegations, there is always a feasible resolution; and we identify under which conditions such solutions are efficiently computable.

cs.GT↗
arXiv2020

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations

In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for the Gomory-Johnson model, in particular for testing extremality of piecewise linear functions whose breakpoints are rational numbers with huge denominators.

math.OC↗
arXiv2019

Facets, weak facets, and extreme functions of the Gomory-Johnson infinite group problem

We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing the hypothesis regarding rational breakpoints. We prove an if-and-only-if version of the Gomory-Johnson Facet Theorem. Finally, we separate the three notions using discontinuous examples.

math.OC↗
arXiv2019

All Cyclic Group Facets Inject

We give a variant of Basu-Hildebrand-Molinaro's approximation theorem for continuous minimal valid functions for Gomory-Johnson's infinite group problem by piecewise linear two-slope extreme functions [Minimal cut-generating functions are nearly extreme, IPCO 2016]. Our theorem is for piecewise linear minimal valid functions that have only rational breakpoints (in $1/q\,\mathbb{Z}$ for some $q\in \mathbb{N}$) and that take rational values at the breakpoints. In contrast to Basu et al.'s construction, our construction preserves all function values on $1/q\,\mathbb{Z}$. As a corollary, we obtain that every extreme function for the finite group problem on $1/q\,\mathbb{Z}$ is the restriction of a continuous piecewise linear two-slope extreme function for the infinite group problem with breakpoints on a refinement $1/(Mq)\,\mathbb{Z}$ for some $M\in \mathbb{N}$. In combination with Gomory's master theorem [Some Polyhedra related to Combinatorial Problems, Lin. Alg. Appl. 2 (1969), 451-558], this shows that the infinite group problem is the correct master problem for facets (extreme functions) of 1-row group relaxations.

math.GR↗
arXiv2018

Three Ehrhart Quasi-polynomials

Let $P(b)\subset R^d$ be a semi-rational parametric polytope, where $b=(b_j)\in R^N$ is a real multi-parameter. We study intermediate sums of polynomial functions $h(x)$ on $P(b)$, $$ S^L (P(b),h)=\sum_{y}\int_{P(b)\cap (y+L)} h(x) \mathrm dx, $$ where we integrate over the intersections of $P(b)$ with the subspaces parallel to a fixed rational subspace $L$ through all lattice points, and sum the integrals. The purely discrete sum is of course a particular case ($L=0$), so $S^0(P(b), 1)$ counts the integer points in the parametric polytopes. The chambers are the open conical subsets of $R^N$ such that the shape of $P(b)$ does not change when $b$ runs over a chamber. We first prove that on every chamber of $R^N$, $S^L (P(b),h)$ is given by a quasi-polynomial function of $b\in R^N$. A key point of our paper is an analysis of the interplay between two notions of degree on quasi-polynomials: the usual polynomial degree and a filtration, called the local degree. Then, for a fixed $k\leq d$, we consider a particular linear combination of such intermediate weighted sums, which was introduced by Barvinok in order to compute efficiently the $k+1$ highest coefficients of the Ehrhart quasi-polynomial which gives the number of points of a dilated rational polytope. Thus, for each chamber, we obtain a quasi-polynomial function of $b$, which we call Barvinok's patched quasi-polynomial (at codimension level $k$). Finally, for each chamber, we introduce a new quasi-polynomial function of $b$, the cone-by-cone patched quasi-polynomial (at codimension level $k$), defined in a refined way by linear combinations of intermediate generating functions for the cones at vertices of $P(b)$. We prove that both patched quasi-polynomials agree with the discrete weighted sum $b\mapsto S^0(P(b),h)$ in the terms corresponding to the $k+1$ highest polynomial degrees.

math.CO↗
arXiv2018

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VI. The curious case of two-sided discontinuous minimal valid functions

We construct a two-sided discontinuous piecewise linear minimal valid function for the 1-row Gomory--Johnson model which is not extreme, but which is not a convex combination of other piecewise linear minimal valid functions. The new function only admits piecewise microperiodic perturbations. We present an algorithm for computations with a restricted class of such perturbations.

math.OC↗
arXiv2018

Characterization and Approximation of Strong General Dual Feasible Functions

Dual feasible functions (DFFs) have been used to provide bounds for standard packing problems and valid inequalities for integer optimization problems. In this paper, the connection between general DFFs and a particular family of cut-generating functions is explored. We find the characterization of (restricted/strongly) maximal general DFFs and prove a 2-slope theorem for extreme general DFFs. We show that any restricted maximal general DFF can be well approximated by an extreme general DFF.

math.OC↗
arXiv2017

Generating functions and triangulations for lecture hall cones

We investigate the arithmetic-geometric structure of the lecture hall cone \[ L_n \ := \ \left\{λ\in \mathbb{R}^n: \, 0\leq \frac{λ_1}{1}\leq \frac{λ_2}{2}\leq \frac{λ_3}{3}\leq \cdots \leq \frac{λ_n}{n}\right\} . \] We show that $L_n$ is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart $h^*$-polynomial is given by the $(n-1)$st Eulerian polynomial, and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for $L_n$, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of $L_n$, including connections between enumerative and algebraic properties of $L_n$ and cones over unit cubes.

math.CO↗
arXiv2017

Structure and Interpretation of Dual-Feasible Functions

We study two techniques to obtain new families of classical and general Dual-Feasible Functions: A conversion from minimal Gomory--Johnson functions; and computer-based search using polyhedral computation and an automatic maximality and extremality test.

math.OC↗
arXiv2016

On the notions of facets, weak facets, and extreme functions of the Gomory-Johnson infinite group problem

We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing the hypothesis regarding rational breakpoints. We then separate the three notions using discontinuous examples.

math.OC↗
arXiv2016

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. III. Foundations for the k-Dimensional Case with Applications to k=2

We develop foundational tools for classifying the extreme valid functions for the k-dimensional infinite group problem. In particular, (1) we present the general regular solution to Cauchy's additive functional equation on bounded convex domains. This provides a k-dimensional generalization of the so-called interval lemma, allowing us to deduce affine properties of the function from certain additivity relations. (2) We study the discrete geometry of additivity domains of piecewise linear functions, providing a framework for finite tests of minimality and extremality. (3) We give a theory of non-extremality certificates in the form of perturbation functions. We apply these tools in the context of minimal valid functions for the two-dimensional infinite group problem that are piecewise linear on a standard triangulation of the plane, under the assumption of a regularity condition called diagonal constrainedness. We show that the extremality of a minimal valid function is equivalent to the extremality of its restriction to a certain finite two-dimensional group problem. This gives an algorithm for testing the extremality of a given minimal valid function.

math.OC↗