arXiv subjects

Johannes Middeke

Publications and source records attributed to Johannes Middeke.

8 recordsLinked to original sources

arXiv2026

When Algebraic Symmetry Breaking Meets Solvers: An Experimental Study

We present an experimental evaluation of automatically generated polynomial symmetry breaking constraints for integer linear programs. Starting from the method that we introduced at the International Symposium on Symbolic and Algebraic Computation (ISSAC) 2026, we compare solver native quadratic handling, solver-internal reformulation, and explicit linearization on near half-capacity bin-packing benchmarks. Experiments with several mathematical programming solvers and satisfiability modulo theory solvers show that the effectiveness of polynomial symmetry breaking is strongly solver-dependent. Compact quadratic breaker families can improve performance, whereas linearization, large breaker sets, or solver reformulations may offset these gains through increased model size or less favorable search behavior. These results suggest that automatically generated symmetry breakers should be evaluated in a solver-aware manner rather than treated as solver-independent additions to a model.

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arXiv2026

Automatic Generation of Polynomial Symmetry Breaking Constraints

Symmetry in integer programming causes redundant search and is often handled with symmetry breaking constraints that remove as many equivalent solutions as possible. We propose an algebraic method which allows to generate a random family of polynomial inequalities which can be used as symmetry breakers. The method requires as input an arbitrary base polynomial and a group of permutations which is specific to the integer program. The computations can be easily carried out in any major symbolic computation software. In order to test our approach, we describe a case study on near half-capacity 0-1 bin packing instances which exhibit substantial symmetries. We statically generate random quadratic breakers and add them to a baseline integer programming problem which we then solve with Gurobi. It turns out that simple symmetry breakers, especially combining few variables and permutations, most consistently reduce work time.

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arXiv2025

Hypergeometric Solutions of Linear Difference Systems

We extend Petkovšek's algorithm for computing hypergeometric solutions of scalar difference equations to the case of difference systems $τ(Y) = M Y$, with $M \in {\rm GL}_n(C(x))$, where $τ$ is the shift operator. Hypergeometric solutions are solutions of the form $γP$ where $P \in C(x)^n$ and $γ$ is a hypergeometric term over $C(x)$, i.e. ${τ(γ)}/γ \in C(x)$. Our contributions concern efficient computation of a set of candidates for ${τ(γ)}/γ$ which we write as $λ= c\frac{A}{B}$ with monic $A, B \in C[x]$, $c \in C^*$. Factors of the denominators of $M^{-1}$ and $M$ give candidates for $A$ and $B$, while another algorithm is needed for $c$. We use the super-reduction algorithm to compute candidates for $c$, as well as other ingredients to reduce the list of candidates for $A/B$. To further reduce the number of candidates $A/B$, we bound the so-called type of $A/B$ by bounding local types. Our algorithm has been implemented in Maple and experiments show that our implementation can handle systems of high dimension, which is useful for factoring operators.

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arXiv2020

A Family of Denominator Bounds for First Order Linear Recurrence Systems

For linear recurrence systems, the problem of finding rational solutions is reduced to the problem of computing polynomial solutions by computing a content bound or a denominator bound. There are several bounds in the literature. The sharpest bound leads to polynomial solutions of lower degrees, but this advantage need not compensate for the time spent on computing that bound. To strike the best balance between sharpness of the bound versus CPU time spent obtaining it, we will give a family of bounds. The $J$'th member of this family is similar to (Abramov, Barkatou, 1998) when $J=1$, similar to (van Hoeij, 1998) when $J$ is large, and novel for intermediate values of $J$, which give the best balance between sharpness and CPU time. The setting for our content bounds are systems $τ(Y) = MY$ where $τ$ is an automorphism of a UFD, and $M$ is an invertible matrix with entries in its field of fractions. This setting includes the shift case, the $q$-shift case, the multi-basic case and others. We give two versions, a global version, and a version that bounds each entry separately.

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arXiv2020

Common Factors in Fraction-Free Matrix Decompositions

We consider LU and QR matrix decompositions using exact computations. We show that fraction-free Gauss--Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors. We identify two types of common factors: systematic and statistical. Systematic factors depend on the reduction process, independent of the data, while statistical factors depend on the specific data. We relate the existence of row factors in the LU decomposition to factors appearing in the Smith--Jacobson normal form of the matrix. For statistical factors, we identify some of the mechanisms that create them and give estimates of the frequency of their occurrence. Similar observations apply to the common factors in a fraction-free QR decomposition. Our conclusions are tested experimentally.

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arXiv2017

Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K

We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of higher order q-recurrence equations with rational coefficients. We extend a method for finding a bound on the maximal power of t in the denominator of arbitrary rational solutions y(t) as well as a method for bounding the degree of polynomial solutions from the scalar case to the systems case. The approach is direct and does not rely on uncoupling or reduction to a first order system. Unlike in the scalar case this usually requires an initial transformation of the system.

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arXiv2017

Denominator Bounds for Systems of Recurrence Equations using $ΠΣ$-Extensions

We consider linear systems of recurrence equations whose coefficients are given in terms of indefinite nested sums and products covering, e.g., the harmonic numbers, hypergeometric products, $q$-hypergeometric products or their mixed versions. These linear systems are formulated in the setting of $ΠΣ$-extensions and our goal is to find a denominator bound (also known as universal denominator) for the solutions; i.e., a non-zero polynomial $d$ such that the denominator of every solution of the system divides $d$. This is the first step in computing all rational solutions of such a rather general recurrence system. Once the denominator bound is known, the problem of solving for rational solutions is reduced to the problem of solving for polynomial solutions.

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arXiv2016

Matrix factoring by fraction-free reduction

We consider exact matrix decomposition by Gauss-Bareiss reduction. We investigate two aspects of the process: common row and column factors and the influence of pivoting strategies. We identify two types of common factors: systematic and statistical. Systematic factors depend on the process, while statistical factors depend on the specific data. We show that existing fraction-free QR (Gram-Schmidt) algorithms create a common factor in the last column of Q. We relate the existence of row factors in LU decomposition to factors appearing in the Smith normal form of the matrix. For statistical factors, we identify mechanisms and give estimates of the frequency. Our conclusions are tested by experimental data. For pivoting strategies, we compare the sizes of output factors obtained by different strategies. We also comment on timing differences.

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