Searcharxiv⌕ Search

arXiv subjects

Matthias Köppe

Publications and source records attributed to Matthias Köppe.

At least 37 records · Page 2Linked to original sources
arXiv2015

Light on the Infinite Group Relaxation

This is a survey on the infinite group problem, an infinite-dimensional relaxation of integer linear optimization problems introduced by Ralph Gomory and Ellis Johnson in their groundbreaking papers titled "Some continuous functions related to corner polyhedra I, II" [Math. Programming 3 (1972), 23-85, 359-389]. The survey presents the infinite group problem in the modern context of cut generating functions. It focuses on the recent developments, such as algorithms for testing extremality and breakthroughs for the k-row problem for general k >= 1 that extend previous work on the single-row and two-row problems. The survey also includes some previously unpublished results; among other things, it unveils piecewise linear extreme functions with more than four different slopes. An interactive companion program, implemented in the open-source computer algebra package Sage, provides an updated compendium of known extreme functions.

math.OC↗
arXiv2014

Coefficients of Sylvester's Denumerant

For a given sequence $\mathbfα = [α_1,α_2,\dots,α_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\mathbfα)(t)$ that counts the nonnegative integer solutions of the equation $α_1x_1+α_2 x_2+\cdots+α_{N} x_{N}+α_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(\mathbfα)(t)$ is a quasi-polynomial function in the variable $t$ of degree $N$. In combinatorial number theory this function is known as Sylvester's denumerant. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial time the highest $k+1$ coefficients of the quasi-polynomial $E(\mathbfα)(t)$ as step polynomials of $t$ (a simpler and more explicit representation). Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for $E(\mathbfα)(t)$ and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Our algorithm also uses Barvinok's fundamental fast decomposition of a polyhedral cone into unimodular cones. This paper also presents a simple algorithm to predict the first non-constant coefficient and concludes with a report of several computational experiments using an implementation of our algorithm in LattE integrale. We compare it with various Maple programs for partial or full computation of the denumerant.

math.NT↗
arXiv2014

Intermediate Sums on Polyhedra II: Bidegree and Poisson Formula

We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449-1466]. By well-known decompositions, it is sufficient to consider the case of affine cones s+c, where s is an arbitrary real vertex and c is a rational polyhedral cone. For a given rational subspace L, we integrate a given polynomial function h over all lattice slices of the affine cone s + c parallel to the subspace L and sum up the integrals. We study these intermediate sums by means of the intermediate generating functions $S^L(s+c)(ξ)$, and expose the bidegree structure in parameters s and $ξ$, which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra, Found. Comput. Math. 12 (2012), 435-469] and [Intermediate sums on polyhedra: Computation and real Ehrhart theory, Mathematika 59 (2013), 1-22]. The bidegree structure is key to a new proof for the Baldoni--Berline--Vergne approximation theorem for discrete generating functions [Local Euler--Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes, Contemp. Math. 452 (2008), 15-33], using the Fourier analysis with respect to the parameter s and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.

math.CO↗
arXiv2014

Unique Minimal Liftings for Simplicial Polytopes

For a minimal inequality derived from a maximal lattice-free simplicial polytope in $\R^n$, we investigate the region where minimal liftings are uniquely defined, and we characterize when this region covers $\R^n$. We then use this characterization to show that a minimal inequality derived from a maximal lattice-free simplex in $\R^n$ with exactly one lattice point in the relative interior of each facet has a unique minimal lifting if and only if all the vertices of the simplex are lattice points.

math.OC↗
arXiv2013

s-Lecture Hall Partitions, Self-Reciprocal Polynomials, and Gorenstein Cones

In 1997, Bousquet-Melou and Eriksson initiated the study of lecture hall partitions, a fascinating family of partitions that yield a finite version of Euler's celebrated odd/distinct partition theorem. In subsequent work on s-lecture hall partitions, they considered the self-reciprocal property for various associated generating functions, with the goal of characterizing those sequences s that give rise to generating functions of the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$. We continue this line of investigation, connecting their work to the more general context of Gorenstein cones. We focus on the Gorenstein condition for s-lecture hall cones when s is a positive integer sequence generated by a second-order homogeneous linear recurrence with initial values 0 and 1. Among such sequences s, we prove that the n-dimensional s-lecture hall cone is Gorenstein for all n greater than or equal to 1 if and only if s is an l-sequence. One consequence is that among such sequences s, unless s is an l-sequence, the generating function for the s-lecture hall partitions can have the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$ for at most finitely many n. We also apply the results to establish several conjectures by Pensyl and Savage regarding the symmetry of h*-vectors for s-lecture hall polytopes. We end with open questions and directions for further research.

math.CO↗
arXiv2013

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. I. The One-Dimensional Case

We give an algorithm for testing the extremality of minimal valid functions for Gomory and Johnson's infinite group problem that are piecewise linear (possibly discontinuous) with rational breakpoints. This is the first set of necessary and sufficient conditions that can be tested algorithmically for deciding extremality in this important class of minimal valid functions. We also present an extreme function that is a piecewise linear function with some irrational breakpoints, whose extremality follows from a new principle.

math.OC↗
arXiv2013

The Triangle Closure is a Polyhedron

Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, Kannan and Schrijver who showed that the split closure is a polyhedron. Although some fairly general results were obtained by Andersen, Louveaux and Weismantel [ An analysis of mixed integer linear sets based on lattice point free convex sets, Math. Oper. Res. 35 (2010), 233--256] and Averkov [On finitely generated closures in the theory of cutting planes, Discrete Optimization 9 (2012), no. 4, 209--215], some basic questions have remained unresolved. For example, maximal lattice-free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron. In this paper, we show that when the number of integer variables $m=2$ the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data. The techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet-defining due to Cornuéjols and Margot [On the facets of mixed integer programs with two integer variables and two constraints, Mathematical Programming 120 (2009), 429--456] and obtain polynomial complexity results about the mixed integer hull.

math.OC↗
arXiv2012

Graver basis and proximity techniques for block-structured separable convex integer minimization problems

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 polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs, Proc. IPCO 2010, Lecture Notes in Computer Science, vol. 6080, Springer, 2010, pp. 219--229], it was proved that for fixed blocks but variable N, these integer programs are polynomial-time solvable for any linear objective. We extend this result to the minimization of separable convex objective functions. Our algorithm combines Graver basis techniques with a proximity result [D.S. Hochbaum and J.G. Shanthikumar, Convex separable optimization is not much harder than linear optimization, J. ACM 37 (1990), 843--862], which allows us to use convex continuous optimization as a subroutine.

math.OC↗
arXiv2012

A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)

We study the integer minimization of a quasiconvex polynomial with quasiconvex polynomial constraints. We propose a new algorithm that is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). This improvement is achieved by applying a new modern Lenstra-type algorithm, finding optimal ellipsoid roundings, and considering sparse encodings of polynomials. For the bounded case, our algorithm attains a time-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a bound on the number of monomials in each polynomial and r is the binary encoding length of a bound on the feasible region. In the general case, s l^{O(1)} d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the total degree of the polynomials and l bounds the maximum binary encoding size of the input.

math.OC↗
arXiv2011

A (k+1)-Slope Theorem for the k-Dimensional Infinite Group Relaxation

We prove that any minimal valid function for the k-dimensional infinite group relaxation that is piecewise linear with at most k+1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a theorem of Gomory and Johnson for k=1, and Cornuejols and Molinaro for k=2.

math.OC↗
arXiv2011

Algorithmic and Complexity Results for Cutting Planes Derived from Maximal Lattice-Free Convex Sets

We study a mixed integer linear program with m integer variables and k non-negative continuous variables in the form of the relaxation of the corner polyhedron that was introduced by Andersen, Louveaux, Weismantel and Wolsey [Inequalities from two rows of a simplex tableau, Proc. IPCO 2007, LNCS, vol. 4513, Springer, pp. 1--15]. We describe the facets of this mixed integer linear program via the extreme points of a well-defined polyhedron. We then utilize this description to give polynomial time algorithms to derive valid inequalities with optimal l_p norm for arbitrary, but fixed m. For the case of m=2, we give a refinement and a new proof of a characterization of the facets by Cornuejols and Margot [On the facets of mixed integer programs with two integer variables and two constraints, Math. Programming 120 (2009), 429--456]. The key point of our approach is that the conditions are much more explicit and can be tested in a more direct manner, removing the need for a reduction algorithm. These results allow us to show that the relaxed corner polyhedron has only polynomially many facets.

math.OC↗
arXiv2010

Intermediate Sums on Polyhedra: Computation and Real Ehrhart Theory

We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449--1466]. For a given semi-rational polytope P and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope P parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step polynomials. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory.

math.CO↗
arXiv2010

Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra

This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h. Our technique is based on a refinement of an algorithm of A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), pp. 1449--1466] in the unweighted case (i.e., h = 1). In contrast to Barvinok's method, our method is local, obtains an approximation on the level of generating functions, handles the general weighted case, and provides the coefficients in closed form as step polynomials of the dilation. To demonstrate the practicality of our approach we report on computational experiments which show even our simple implementation can compete with state of the art software.

math.CO↗
arXiv2010

On the complexity of nonlinear mixed-integer optimization

This is a survey on the computational complexity of nonlinear mixed-integer optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained fully polynomial time approximation schemes in fixed dimension, and to strongly polynomial-time algorithms for special cases.

math.OC↗
arXiv2009

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.

math.OC↗
arXiv2009

A parametric integer programming algorithm for bilevel mixed integer programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.

math.OC↗
arXiv2009

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.

math.OC↗