283 research outputs found
The Width and Integer Optimization on Simplices With Bounded Minors of the Constraint Matrices
In this paper, we will show that the width of simplices defined by systems of
linear inequalities can be computed in polynomial time if some minors of their
constraint matrices are bounded. Additionally, we present some
quasi-polynomial-time and polynomial-time algorithms to solve the integer
linear optimization problem defined on simplices minus all their integer
vertices assuming that some minors of the constraint matrices of the simplices
are bounded.Comment: 12 page
Hamiltonian structures of fermionic two-dimensional Toda lattice hierarchies
By exhibiting the corresponding Lax pair representations we propose a wide
class of integrable two-dimensional (2D) fermionic Toda lattice (TL)
hierarchies which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL
hierarchies as particular cases. We develop the generalized graded R-matrix
formalism using the generalized graded bracket on the space of graded operators
with involution generalizing the graded commutator in superalgebras, which
allows one to describe these hierarchies in the framework of the Hamiltonian
formalism and construct their first two Hamiltonian structures. The first
Hamiltonian structure is obtained for both bosonic and fermionic Lax operators
while the second Hamiltonian structure is established for bosonic Lax operators
only.Comment: 12 pages, LaTeX, the talks delivered at the International Workshop on
Classical and Quantum Integrable Systems (Dubna, January 24 - 28, 2005) and
International Conference on Theoretical Physics (Moscow, April 11 - 16, 2005
An FPTAS for the -modular multidimensional knapsack problem
It is known that there is no EPTAS for the -dimensional knapsack problem
unless . It is true already for the case, when . But, an
FPTAS still can exist for some other particular cases of the problem.
In this note, we show that the -dimensional knapsack problem with a
-modular constraints matrix admits an FPTAS, whose complexity bound
depends on linearly. More precisely, the proposed algorithm complexity
is
where is the linear programming complexity bound. In particular, for
fixed the arithmetical complexity bound becomes Our algorithm is actually a
generalisation of the classical FPTAS for the -dimensional case.
Strictly speaking, the considered problem can be solved by an exact
polynomial-time algorithm, when is fixed and grows as a polynomial
on . This fact can be observed combining previously known results. In this
paper, we give a slightly more accurate analysis to present an exact algorithm
with the complexity bound Note that
the last bound is non-linear by with respect to the given FPTAS
FPT-algorithms for some problems related to integer programming
In this paper, we present FPT-algorithms for special cases of the shortest
lattice vector, integer linear programming, and simplex width computation
problems, when matrices included in the problems' formulations are near square.
The parameter is the maximum absolute value of rank minors of the corresponding
matrices. Additionally, we present FPT-algorithms with respect to the same
parameter for the problems, when the matrices have no singular rank
sub-matrices.Comment: arXiv admin note: text overlap with arXiv:1710.00321 From author:
some minor corrections has been don
On lattice point counting in -modular polyhedra
Let a polyhedron be defined by one of the following ways:
(i) , where ,
and ;
(ii) , where , and .
And let all rank order minors of be bounded by in absolute
values. We show that the short rational generating function for the power
series can be computed with the
arithmetic complexity where and are fixed, , and
is the complexity to compute the Smith Normal Form for integer matrix. In particular, for the case (i) and for
the case (ii).
The simplest examples of polyhedra that meet conditions (i) or (ii) are the
simplicies, the subset sum polytope and the knapsack or multidimensional
knapsack polytopes.
We apply these results to parametric polytopes, and show that the step
polynomial representation of the function , where
is parametric polytope, can be computed by a polynomial time even in
varying dimension if has a close structure to the cases (i) or (ii). As
another consequence, we show that the coefficients of the Ehrhart
quasi-polynomial can be computed by a polynomial time algorithm for fixed and
A faster algorithm for counting the integer points number in -modular polyhedra
Let a polytope be defined by a system . We consider the
problem to count a number of integer points inside , assuming that is
-modular. The polytope is -modular if all the rank
sub-determinants of are bounded by in the absolute value. We
present a new FPT-algorithm, parameterized by and by the number of
simple cones in the normal fun triangulation of , which is more efficient
for -modular problems, than the approach of A.~Barvinok et al. To this
end, we do not directly compute the short rational generating function for , which is commonly used for the considered problem. We compute its
particular representation in the form of exponential series that depends on one
variable, using the dynamic programming principle. We completely do not use the
A.~Barvinok's unimodular sign decomposition technique. Using our new complexity
bound, we consider different special cases that may be of independent interest.
For example, we give FPT-algorithms for counting the integer points number in
-modular simplicies and similar polytopes that have facets.
For any fixed , we give an FPT-algorithm to count solutions of the unbounded
-dimensional -modular knapsack problem. For the case, when
grows slowly with respect to , we give a counting algorithm, which is more
effective, than the state of the art ILP feasibility algorithm
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