2,813 research outputs found
On Polynomial Kernels for Integer Linear Programs: Covering, Packing and Feasibility
We study the existence of polynomial kernels for the problem of deciding
feasibility of integer linear programs (ILPs), and for finding good solutions
for covering and packing ILPs. Our main results are as follows: First, we show
that the ILP Feasibility problem admits no polynomial kernelization when
parameterized by both the number of variables and the number of constraints,
unless NP \subseteq coNP/poly. This extends to the restricted cases of bounded
variable degree and bounded number of variables per constraint, and to covering
and packing ILPs. Second, we give a polynomial kernelization for the Cover ILP
problem, asking for a solution to Ax >= b with c^Tx <= k, parameterized by k,
when A is row-sparse; this generalizes a known polynomial kernelization for the
special case with 0/1-variables and coefficients (d-Hitting Set)
A structural approach to kernels for ILPs: Treewidth and Total Unimodularity
Kernelization is a theoretical formalization of efficient preprocessing for
NP-hard problems. Empirically, preprocessing is highly successful in practice,
for example in state-of-the-art ILP-solvers like CPLEX. Motivated by this,
previous work studied the existence of kernelizations for ILP related problems,
e.g., for testing feasibility of Ax <= b. In contrast to the observed success
of CPLEX, however, the results were largely negative. Intuitively, practical
instances have far more useful structure than the worst-case instances used to
prove these lower bounds.
In the present paper, we study the effect that subsystems with (Gaifman graph
of) bounded treewidth or totally unimodularity have on the kernelizability of
the ILP feasibility problem. We show that, on the positive side, if these
subsystems have a small number of variables on which they interact with the
remaining instance, then we can efficiently replace them by smaller subsystems
of size polynomial in the domain without changing feasibility. Thus, if large
parts of an instance consist of such subsystems, then this yields a substantial
size reduction. We complement this by proving that relaxations to the
considered structures, e.g., larger boundaries of the subsystems, allow
worst-case lower bounds against kernelization. Thus, these relaxed structures
can be used to build instance families that cannot be efficiently reduced, by
any approach.Comment: Extended abstract in the Proceedings of the 23rd European Symposium
on Algorithms (ESA 2015
Parameterized Complexity of Sparse Linear Complementarity Problems
In this paper, we study the parameterized complexity of the linear complementarity problem (LCP), which is one of the most fundamental mathematical optimization problems. The parameters we focus on are the sparsities of the input and the output of the LCP: the maximum numbers of nonzero entries per row/column in the coefficient matrix and the number of nonzero entries in a solution. Our main result is to present a fixed-parameter algorithm for the LCP with all the parameters. We also show that if we drop any of the three parameters, then the LCP is fixed-parameter intractable.
In addition, we discuss the nonexistence of a polynomial kernel for the LCP
Scattering Calculations with Wavelets
We show that the use of wavelet bases for solving the momentum-space
scattering integral equation leads to sparse matrices which can simplify the
solution. Wavelet bases are applied to calculate the K-matrix for
nucleon-nucleon scattering with the s-wave Malfliet-Tjon V potential. We
introduce a new method, which uses special properties of the wavelets, for
evaluating the singular part of the integral. Analysis of this test problem
indicates that a significant reduction in computational size can be achieved
for realistic few-body scattering problems.Comment: 26 pages, Latex, 6 eps figure
Hitting forbidden minors: Approximation and Kernelization
We study a general class of problems called F-deletion problems. In an
F-deletion problem, we are asked whether a subset of at most vertices can
be deleted from a graph such that the resulting graph does not contain as a
minor any graph from the family F of forbidden minors.
We obtain a number of algorithmic results on the F-deletion problem when F
contains a planar graph. We give (1) a linear vertex kernel on graphs excluding
-claw , the star with leves, as an induced subgraph, where
is a fixed integer. (2) an approximation algorithm achieving an approximation
ratio of , where is the size of an optimal solution on
general undirected graphs. Finally, we obtain polynomial kernels for the case
when F contains graph as a minor for a fixed integer . The graph
consists of two vertices connected by parallel edges. Even
though this may appear to be a very restricted class of problems it already
encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback
Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is
based on a non-trivial application of protrusion techniques, previously used
only for problems on topological graph classes
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