41,324 research outputs found
Kernelization and Enumeration: New Approaches to Solving Hard Problems
NP-Hardness is a well-known theory to identify the hardness of computational problems.
It is believed that NP-Hard problems are unlikely to admit polynomial-time algorithms.
However since many NP-Hard problems are of practical significance, different approaches
are proposed to solve them: Approximation algorithms, randomized algorithms and heuristic
algorithms. None of the approaches meet the practical needs. Recently parameterized
computation and complexity has attracted a lot of attention and been a fruitful branch of
the study of efficient algorithms. By taking advantage of the moderate value of parameters
in many practical instances, we can design efficient algorithms for the NP-Hard problems in
practice.
In this dissertation, we discuss a new approach to design efficient parameterized algorithms,
kernelization. The motivation is that instances of small size are easier to solve.
Roughly speaking, kernelization is a preprocess on the input instances and is able to significantly reduce their sizes.
We present a 2k kernel for the cluster editing problem, which improves the previous
best kernel of size 4k; We also present a linear kernel of size 7k 2d for the d-cluster
editing problem, which is the first linear kernel for the problem. The kernelization algorithm
is simple and easy to implement.
We propose a quadratic kernel for the pseudo-achromatic number problem. This
implies that the problem is tractable in term of parameterized complexity. We also study
the general problem, the vertex grouping problem and prove it is intractable in term of
parameterized complexity.
In practice, many problems seek a set of good solutions instead of a good solution.
Motivated by this, we present the framework to study enumerability in term of parameterized
complexity. We study three popular techniques for the design of parameterized algorithms,
and show that combining with effective enumeration techniques, they could be transferred
to design efficient enumeration algorithms
Paradigms for Parameterized Enumeration
The aim of the paper is to examine the computational complexity and
algorithmics of enumeration, the task to output all solutions of a given
problem, from the point of view of parameterized complexity. First we define
formally different notions of efficient enumeration in the context of
parameterized complexity. Second we show how different algorithmic paradigms
can be used in order to get parameter-efficient enumeration algorithms in a
number of examples. These paradigms use well-known principles from the design
of parameterized decision as well as enumeration techniques, like for instance
kernelization and self-reducibility. The concept of kernelization, in
particular, leads to a characterization of fixed-parameter tractable
enumeration problems.Comment: Accepted for MFCS 2013; long version of the pape
Complexity classifications for different equivalence and audit problems for Boolean circuits
We study Boolean circuits as a representation of Boolean functions and
consider different equivalence, audit, and enumeration problems. For a number
of restricted sets of gate types (bases) we obtain efficient algorithms, while
for all other gate types we show these problems are at least NP-hard.Comment: 25 pages, 1 figur
Efficient enumeration of solutions produced by closure operations
In this paper we address the problem of generating all elements obtained by
the saturation of an initial set by some operations. More precisely, we prove
that we can generate the closure of a boolean relation (a set of boolean
vectors) by polymorphisms with a polynomial delay. Therefore we can compute
with polynomial delay the closure of a family of sets by any set of "set
operations": union, intersection, symmetric difference, subsets, supersets
). To do so, we study the problem: for a set
of operations , decide whether an element belongs to the closure
by of a family of elements. In the boolean case, we prove that
is in P for any set of boolean operations
. When the input vectors are over a domain larger than two
elements, we prove that the generic enumeration method fails, since
is NP-hard for some . We also study the
problem of generating minimal or maximal elements of closures and prove that
some of them are related to well known enumeration problems such as the
enumeration of the circuits of a matroid or the enumeration of maximal
independent sets of a hypergraph. This article improves on previous works of
the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of
the same name which appeared in STACS 2016. Final version for DMTCS journa
Solving the Shortest Vector Problem in Lattices Faster Using Quantum Search
By applying Grover's quantum search algorithm to the lattice algorithms of
Micciancio and Voulgaris, Nguyen and Vidick, Wang et al., and Pujol and
Stehl\'{e}, we obtain improved asymptotic quantum results for solving the
shortest vector problem. With quantum computers we can provably find a shortest
vector in time , improving upon the classical time
complexity of of Pujol and Stehl\'{e} and the of Micciancio and Voulgaris, while heuristically we expect to find a
shortest vector in time , improving upon the classical time
complexity of of Wang et al. These quantum complexities
will be an important guide for the selection of parameters for post-quantum
cryptosystems based on the hardness of the shortest vector problem.Comment: 19 page
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