Related papers: Light on the Infinite Group Relaxation
In this note we announce the availability of an electronic compendium of extreme functions for Gomory--Johnson's infinite group problem. These functions serve as the strongest cut-generating functions for integer linear optimization…
For the one dimensional infinite group relaxation, we construct a sequence of extreme valid functions that are piecewise linear and such that for every natural number $k\geq 2$, there is a function in the sequence with $k$ slopes. This…
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…
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…
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,…
Integer Linear Programs (ILPs) are a flexible and ubiquitous model for discrete optimization problems. Solving ILPs is \textsf{NP-Hard} yet of great practical importance. Super-quadratic quantum speedups for ILPs have been difficult to…
We describe new computer-based search strategies for extreme functions for the Gomory--Johnson infinite group problem. They lead to the discovery of new extreme functions, whose existence settles several open questions.
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…
In a recent paper, Basu, Hildebrand, and Molinaro established that the set of continuous minimal functions for the 1-dimensional Gomory-Johnson infinite group relaxation possesses a dense subset of extreme functions. The $n$-dimensional…
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…
We present software for investigations with cut-generating functions in the Gomory--Johnson model and extensions, implemented in the computer algebra system SageMath.
This is an expository article on recent developments in the theory of group relaxations in integer programming from an algebraic perspective.
We give some connections between various functions defined on finitely presented groups (isoperimetric, isodiametric, Todd-Coxeter radius, filling length functions, etc.), and we study the relation between those functions and the…
We give an algorithm for testing the extremality of a large class of minimal valid functions for the two-dimensional infinite group problem.
We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in…
In Mathematical Programming 2003, Gomory and Johnson conjecture that the facets of the infinite group problem are always generated by piecewise linear functions. In this paper we give an example showing that the Gomory-Johnson conjecture is…
The cutting-plane approach to integer programming was initiated more that 40 years ago: Gomory introduced the corner polyhedron as a relaxation of a mixed integer set in tableau form and Balas introduced intersection cuts for the corner…
For the linear inverse problem with sparsity constraints, the $l_0$ regularized problem is NP-hard, and existing approaches either utilize greedy algorithms to find almost-optimal solutions or to approximate the $l_0$ regularization with…
An extension of the finite and infinite Lie groups properties of complex numbers and functions of complex variable is proposed. This extension is performed exploiting hypercomplex number systems that follow the elementary algebra rules. In…
Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. S{\'e}nizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is…