4,251 research outputs found
Parameterized Algorithms for Modular-Width
It is known that a number of natural graph problems which are FPT
parameterized by treewidth become W-hard when parameterized by clique-width. It
is therefore desirable to find a different structural graph parameter which is
as general as possible, covers dense graphs but does not incur such a heavy
algorithmic penalty.
The main contribution of this paper is to consider a parameter called
modular-width, defined using the well-known notion of modular decompositions.
Using a combination of ILPs and dynamic programming we manage to design FPT
algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path
and Hamiltonian cycle), which are W-hard for both clique-width and its recently
introduced restriction, shrub-depth. We thus argue that modular-width occupies
a sweet spot as a graph parameter, generalizing several simpler notions on
dense graphs but still evading the "price of generality" paid by clique-width.Comment: to appear in IPEC 2013. arXiv admin note: text overlap with
arXiv:1304.5479 by other author
Counting Problems in Parameterized Complexity
This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity. After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large graphs, and counting perfect matchings and Hamiltonian cycles in well-structured graphs.
While this survey presupposes familiarity with parameterized algorithms and complexity, we aim at explaining all relevant notions from counting complexity in a self-contained way
Approximating the MaxCover Problem with Bounded Frequencies in FPT Time
We study approximation algorithms for several variants of the MaxCover
problem, with the focus on algorithms that run in FPT time. In the MaxCover
problem we are given a set N of elements, a family S of subsets of N, and an
integer K. The goal is to find up to K sets from S that jointly cover (i.e.,
include) as many elements as possible. This problem is well-known to be NP-hard
and, under standard complexity-theoretic assumptions, the best possible
polynomial-time approximation algorithm has approximation ratio (1 - 1/e). We
first consider a variant of MaxCover with bounded element frequencies, i.e., a
variant where there is a constant p such that each element belongs to at most p
sets in S. For this case we show that there is an FPT approximation scheme
(i.e., for each B there is a B-approximation algorithm running in FPT time) for
the problem of maximizing the number of covered elements, and a randomized FPT
approximation scheme for the problem of minimizing the number of elements left
uncovered (we take K to be the parameter). Then, for the case where there is a
constant p such that each element belongs to at least p sets from S, we show
that the standard greedy approximation algorithm achieves approximation ratio
exactly (1-e^{-max(pK/|S|, 1)}). We conclude by considering an unrestricted
variant of MaxCover, and show approximation algorithms that run in exponential
time and combine an exact algorithm with a greedy approximation. Some of our
results improve currently known results for MaxVertexCover
The Racist Algorithm?
Review of The Black Box Society: The Secret Algorithms That Control Money and Information by Frank Pasquale
Parameterized Rural Postman Problem
The Directed Rural Postman Problem (DRPP) can be formulated as follows: given
a strongly connected directed multigraph with nonnegative integral
weights on the arcs, a subset of and a nonnegative integer ,
decide whether has a closed directed walk containing every arc of and
of total weight at most . Let be the number of weakly connected
components in the the subgraph of induced by . Sorge et al. (2012) ask
whether the DRPP is fixed-parameter tractable (FPT) when parameterized by ,
i.e., whether there is an algorithm of running time where is a
function of only and the notation suppresses polynomial factors.
Sorge et al. (2012) note that this question is of significant practical
relevance and has been open for more than thirty years. Using an algebraic
approach, we prove that DRPP has a randomized algorithm of running time
when is bounded by a polynomial in the number of vertices in
. We also show that the same result holds for the undirected version of
DRPP, where is a connected undirected multigraph
Leibniz and Toulmin: Rationalism without Dogmas (Pluralism, Pragmatism, and Gradualism)
The aim of this paper is to connect Leibniz’s and Toulmin’s conceptions about practical and deliberative rationality. When trying to rationally justify contingent judgments Leibniz, like Toulmin, defends a weighing argumentative method. Thus, in Leibniz we can discern the balance between the legitimate demands of formal models of rationality and the lessons of a practice “situated” on a historical, social, and evaluative context (theoria cum praxi)
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