52,337 research outputs found
Fast Evaluation of Generalized Todd Polynomials: Applications to MacMahon's Partition Analysis and Integer Programming
The Todd polynomials are defined by their
generating functions It appears as a basic block in Todd class of a toric
variety, which is important in the theory of lattice polytopes and in number
theory. We find generalized Todd polynomials arise naturally in MacMahon's
partition analysis, especially in Erhart series computation.We give fast
evaluation of generalized Todd polynomials for numerical 's. In order to
do so, we develop fast operations in the quotient ring
modulo for large prime . As applications, i) we recompute the Ehrhart
series of magic squares of order 6, which was first solved by the first named
author. The running time is reduced from 70 days to about 1 day; ii) we give a
polynomial time algorithm for Integer Linear Programming when the dimension is
fixed, with a good performance.Comment: 2 table
On the Optimality of Pseudo-polynomial Algorithms for Integer Programming
In the classic Integer Programming (IP) problem, the objective is to decide
whether, for a given matrix and an -vector , there is a non-negative integer -vector such that . Solving
(IP) is an important step in numerous algorithms and it is important to obtain
an understanding of the precise complexity of this problem as a function of
natural parameters of the input.
The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981]
for instances of (IP) with a constant number of constraints was only recently
improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder
[ArXiv 2018]. We continue this line of work and show that under the Exponential
Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal.
We also show that when the matrix is assumed to be non-negative, a
component of Papadimitriou's original algorithm is already nearly optimal under
ETH.
This motivates us to pick up the line of research initiated by Cunningham and
Geelen [IPCO 2007] who studied the complexity of solving (IP) with non-negative
matrices in which the number of constraints may be unbounded, but the
branch-width of the column-matroid corresponding to the constraint matrix is a
constant. We prove a lower bound on the complexity of solving (IP) for such
instances and obtain optimal results with respect to a closely related
parameter, path-width. Specifically, we prove matching upper and lower bounds
for (IP) when the path-width of the corresponding column-matroid is a constant.Comment: 29 pages, To appear in ESA 201
Box Drawings for Learning with Imbalanced Data
The vast majority of real world classification problems are imbalanced,
meaning there are far fewer data from the class of interest (the positive
class) than from other classes. We propose two machine learning algorithms to
handle highly imbalanced classification problems. The classifiers constructed
by both methods are created as unions of parallel axis rectangles around the
positive examples, and thus have the benefit of being interpretable. The first
algorithm uses mixed integer programming to optimize a weighted balance between
positive and negative class accuracies. Regularization is introduced to improve
generalization performance. The second method uses an approximation in order to
assist with scalability. Specifically, it follows a \textit{characterize then
discriminate} approach, where the positive class is characterized first by
boxes, and then each box boundary becomes a separate discriminative classifier.
This method has the computational advantages that it can be easily
parallelized, and considers only the relevant regions of feature space
An EPTAS for Scheduling on Unrelated Machines of Few Different Types
In the classical problem of scheduling on unrelated parallel machines, a set
of jobs has to be assigned to a set of machines. The jobs have a processing
time depending on the machine and the goal is to minimize the makespan, that is
the maximum machine load. It is well known that this problem is NP-hard and
does not allow polynomial time approximation algorithms with approximation
guarantees smaller than unless PNP. We consider the case that there
are only a constant number of machine types. Two machines have the same
type if all jobs have the same processing time for them. This variant of the
problem is strongly NP-hard already for . We present an efficient
polynomial time approximation scheme (EPTAS) for the problem, that is, for any
an assignment with makespan of length at most
times the optimum can be found in polynomial time in the
input length and the exponent is independent of . In particular
we achieve a running time of , where
denotes the input length. Furthermore, we study three other problem
variants and present an EPTAS for each of them: The Santa Claus problem, where
the minimum machine load has to be maximized; the case of scheduling on
unrelated parallel machines with a constant number of uniform types, where
machines of the same type behave like uniformly related machines; and the
multidimensional vector scheduling variant of the problem where both the
dimension and the number of machine types are constant. For the Santa Claus
problem we achieve the same running time. The results are achieved, using mixed
integer linear programming and rounding techniques
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