Notes on {a, b, c}-Modular Matrices

  title={Notes on \{a, b, c\}-Modular Matrices},
  author={Christoph Glanzer and Ingo Stallknecht and Robert Weismantel},
<jats:p>Let <jats:inline-formula><jats:alternatives><jats:tex-math>$A \in \mathbb {Z}^{m \times n}$</jats:tex-math><mml:math xmlns:mml=""> <mml:mi>A</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>ℤ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>×</mml:mo> <mml… 
On $\Delta$-Modular Integer Linear Problems In The Canonical Form And Equivalent Problems
This paper considers ILP problems in the canonical form max, where all the entries of A, b, c are integer, parameterized by the number of rows of A and ‖A‖max.


The stable set problem in graphs with bounded genus and bounded odd cycle packing number
It is shown that $2-sided odd cycles satisfy the Erdős-Posa property in graphs embedded in a fixed surface, which extends the fact that odd cycles satisfies the Erd ős -Posaproperty in graphs embed in afixed orientable surface.
Submodular Minimization Under Congruency Constraints
It is shown that efficient SFM is possible even for a significantly larger class than parity constraints, by introducing a new approach that combines techniques from Combinatorial Optimization, Combinatorics, and Number Theory.
Algorithms for matrix canonical forms
Computing canonical forms of matrices over rings is a classical math¬ ematical problem with many applications to computational linear alge¬ bra. These forms include the Frobenius form over a field,
Theory of linear and integer programming
  • A. Schrijver
  • Mathematics, Computer Science
    Wiley-Interscience series in discrete mathematics and optimization
  • 1999
Introduction and Preliminaries. Problems, Algorithms, and Complexity. LINEAR ALGEBRA. Linear Algebra and Complexity. LATTICES AND LINEAR DIOPHANTINE EQUATIONS. Theory of Lattices and Linear
A note on non-degenerate integer programs with small sub-determinants
The intention of this note is to find an algorithm to solve integer optimization problems in standard form defined by AZmn in polynomial-time provided that both the largest absolute value of an entry in A and m are constant.
On the Recognition of a, b, c-Modular Matrices
A note on the parametric integer programming in the average case: sparsity, proximity, and FPT-algorithms
It can be shown that, for almost all $b \in Z^m$, the original problem of the square ILP problem can be solved by an algorithm of the arithmetic complexity O(n \cdot \delta \CDot \log \Delta)$, where $\Delta$ is the maximum absolute value of $n \times n$ minors of $A$.
Extended Formulations for Stable Set Polytopes of Graphs Without Two Disjoint Odd Cycles
Building on structural results characterizing sufficiently connected graphs without two disjoint odd cycles, a size-$O(n^2)$ extended formulation for the stable set polytope of G is constructed.
The Integrality Number of an Integer Program
From the results it follows that IPs defined by only $n$ constraints can be solved via a MIP relaxation with $O(\sqrt{\Delta})$ many integer constraints.
On the Number of Distinct Rows of a Matrix with Bounded Subdeterminants
Let A be a matrix with $\mathrm{rank}(A) = n$, whose $(n \times n)-submatrices have a determinant of at most ${\mathop{\vartriangle}}$ in absolute value.