85 research outputs found
Approximating Bin Packing within O(log OPT * log log OPT) bins
For bin packing, the input consists of n items with sizes s_1,...,s_n in
[0,1] which have to be assigned to a minimum number of bins of size 1. The
seminal Karmarkar-Karp algorithm from '82 produces a solution with at most OPT
+ O(log^2 OPT) bins.
We provide the first improvement in now 3 decades and show that one can find
a solution of cost OPT + O(log OPT * log log OPT) in polynomial time. This is
achieved by rounding a fractional solution to the Gilmore-Gomory LP relaxation
using the Entropy Method from discrepancy theory. The result is constructive
via algorithms of Bansal and Lovett-Meka
New computer-based search strategies for extreme functions of the Gomory--Johnson infinite group problem
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.Comment: 54 pages, many figure
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.Comment: 18 pages, 2 figure
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 The survey also includes some previously unpublished results;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.Comment: 45 page
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.Comment: 38 pages, 10 figure
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.Comment: 67 pages, 21 figures; v2: changes to sections 10.2-10.3, improved
figures; v3: additional figures and minor updates, add reference to IPCO
abstract. CC-BY-S
The distributions of functions related to parametric integer optimization
We consider the asymptotic distribution of the IP sparsity function, which
measures the minimal support of optimal IP solutions, and the IP to LP distance
function, which measures the distance between optimal IP and LP solutions. We
create a framework for studying the asymptotic distribution of general
functions related to integer optimization. There has been a significant amount
of research focused around the extreme values that these functions can attain,
however less is known about their typical values. Each of these functions is
defined for a fixed constraint matrix and objective vector while the right hand
sides are treated as input. We show that the typical values of these functions
are smaller than the known worst case bounds by providing a spectrum of
probability-like results that govern their overall asymptotic distributions.Comment: Accepted for journal publicatio
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