1,903 research outputs found
Polynomial Kernels for Weighted Problems
Kernelization is a formalization of efficient preprocessing for NP-hard
problems using the framework of parameterized complexity. Among open problems
in kernelization it has been asked many times whether there are deterministic
polynomial kernelizations for Subset Sum and Knapsack when parameterized by the
number of items.
We answer both questions affirmatively by using an algorithm for compressing
numbers due to Frank and Tardos (Combinatorica 1987). This result had been
first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We
further illustrate its applicability by giving polynomial kernels also for
weighted versions of several well-studied parameterized problems. Furthermore,
when parameterized by the different item sizes we obtain a polynomial
kernelization for Subset Sum and an exponential kernelization for Knapsack.
Finally, we also obtain kernelization results for polynomial integer programs
Parameterized complexity of machine scheduling: 15 open problems
Machine scheduling problems are a long-time key domain of algorithms and
complexity research. A novel approach to machine scheduling problems are
fixed-parameter algorithms. To stimulate this thriving research direction, we
propose 15 open questions in this area whose resolution we expect to lead to
the discovery of new approaches and techniques both in scheduling and
parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc
The Computational Complexity of the Game of Set and its Theoretical Applications
The game of SET is a popular card game in which the objective is to form Sets
using cards from a special deck. In this paper we study single- and multi-round
variations of this game from the computational complexity point of view and
establish interesting connections with other classical computational problems.
Specifically, we first show that a natural generalization of the problem of
finding a single Set, parameterized by the size of the sought Set is W-hard;
our reduction applies also to a natural parameterization of Perfect
Multi-Dimensional Matching, a result which may be of independent interest.
Second, we observe that a version of the game where one seeks to find the
largest possible number of disjoint Sets from a given set of cards is a special
case of 3-Set Packing; we establish that this restriction remains NP-complete.
Similarly, the version where one seeks to find the smallest number of disjoint
Sets that overlap all possible Sets is shown to be NP-complete, through a close
connection to the Independent Edge Dominating Set problem. Finally, we study a
2-player version of the game, for which we show a close connection to Arc
Kayles, as well as fixed-parameter tractability when parameterized by the
number of rounds played
A shortcut to (sun)flowers: Kernels in logarithmic space or linear time
We investigate whether kernelization results can be obtained if we restrict
kernelization algorithms to run in logarithmic space. This restriction for
kernelization is motivated by the question of what results are attainable for
preprocessing via simple and/or local reduction rules. We find kernelizations
for d-Hitting Set(k), d-Set Packing(k), Edge Dominating Set(k) and a number of
hitting and packing problems in graphs, each running in logspace. Additionally,
we return to the question of linear-time kernelization. For d-Hitting Set(k) a
linear-time kernelization was given by van Bevern [Algorithmica (2014)]. We
give a simpler procedure and save a large constant factor in the size bound.
Furthermore, we show that we can obtain a linear-time kernel for d-Set
Packing(k) as well.Comment: 18 page
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)
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