87,771 research outputs found
Partitioning a permutation graph: algorithms and an application.
In this paper we discuss the problem of partitioning a permutation graph into cliques of bounded size, and describe a real-life application of this problem encountered at a manufacturing company. We formulate the problem as an integer program, and present two exact algorithms for solving it. The first algorithm is a branch-and-price algorithm based on the integer programming formulation; the second one is an algorithm based on the concept of bounded clique-width. The latter algorithm was motivated by the structure present in the real-life instances. Test results are given, both for real-life instances and randomly generated instances. As far as we are aware, this is the first implementation of an algorithm based on bounded clique-width.Algorithms; Analysis of algorithms; Branch-and-price; Companies; Integer programming; Manufacturing; Real life; Size; Structure;
The parallel approximability of a subclass of quadratic programming
In this paper we deal with the parallel approximability of a special class of Quadratic Programming (QP), called Smooth Positive Quadratic Programming. This subclass of QP is obtained by imposing restrictions on the coefficients of the QP instance. The Smoothness condition restricts the magnitudes of the coefficients while the positiveness requires that all the coefficients be non-negative. Interestingly, even with these restrictions several combinatorial problems can be modeled by Smooth QP. We show NC Approximation Schemes for the instances of Smooth Positive QP. This is done by reducing the instance of QP to an instance of Positive Linear Programming, finding in NC an approximate fractional solution to the obtained program, and then rounding the fractional solution to an integer approximate solution for the original problem. Then we show how to extend the result for positive instances of bounded degree to Smooth Integer Programming problems. Finally, we formulate several important combinatorial problems as Positive Quadratic Programs (or Positive Integer Programs) in packing/covering form and show that the techniques presented can be used to obtain NC Approximation Schemes for "dense" instances of such problems.Peer ReviewedPostprint (published version
Exact Enumeration and Sampling of Matrices with Specified Margins
We describe a dynamic programming algorithm for exact counting and exact
uniform sampling of matrices with specified row and column sums. The algorithm
runs in polynomial time when the column sums are bounded. Binary or
non-negative integer matrices are handled. The method is distinguished by
applicability to non-regular margins, tractability on large matrices, and the
capacity for exact sampling
An Algorithmic Theory of Integer Programming
We study the general integer programming problem where the number of
variables is a variable part of the input. We consider two natural
parameters of the constraint matrix : its numeric measure and its
sparsity measure . We show that integer programming can be solved in time
, where is some computable function of the
parameters and , and is the binary encoding length of the input. In
particular, integer programming is fixed-parameter tractable parameterized by
and , and is solvable in polynomial time for every fixed and .
Our results also extend to nonlinear separable convex objective functions.
Moreover, for linear objectives, we derive a strongly-polynomial algorithm,
that is, with running time , independent of the rest of
the input data.
We obtain these results by developing an algorithmic framework based on the
idea of iterative augmentation: starting from an initial feasible solution, we
show how to quickly find augmenting steps which rapidly converge to an optimum.
A central notion in this framework is the Graver basis of the matrix , which
constitutes a set of fundamental augmenting steps. The iterative augmentation
idea is then enhanced via the use of other techniques such as new and improved
bounds on the Graver basis, rapid solution of integer programs with bounded
variables, proximity theorems and a new proximity-scaling algorithm, the notion
of a reduced objective function, and others.
As a consequence of our work, we advance the state of the art of solving
block-structured integer programs. In particular, we develop near-linear time
algorithms for -fold, tree-fold, and -stage stochastic integer programs.
We also discuss some of the many applications of these classes.Comment: Revision 2: - strengthened dual treedepth lower bound - simplified
proximity-scaling algorith
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