90 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
Finding Diverse Trees, Paths, and More
Mathematical modeling is a standard approach to solve many real-world
problems and {\em diversity} of solutions is an important issue, emerging in
applying solutions obtained from mathematical models to real-world problems.
Many studies have been devoted to finding diverse solutions. Baste et al.
(Algorithms 2019, IJCAI 2020) recently initiated the study of computing diverse
solutions of combinatorial problems from the perspective of fixed-parameter
tractability. They considered problems of finding solutions that maximize
some diversity measures (the minimum or sum of the pairwise Hamming distances
among them) and gave some fixed-parameter tractable algorithms for the diverse
version of several well-known problems, such as {\sc Vertex Cover}, {\sc
Feedback Vertex Set}, {\sc -Hitting Set}, and problems on bounded-treewidth
graphs. In this work, we investigate the (fixed-parameter) tractability of
problems of finding diverse spanning trees, paths, and several subgraphs. In
particular, we show that, given a graph and an integer , the problem of
computing spanning trees of maximizing the sum of the pairwise Hamming
distances among them can be solved in polynomial time. To the best of the
authors' knowledge, this is the first polynomial-time solvable case for finding
diverse solutions of unbounded size.Comment: 15 page
Extensor-coding
We devise an algorithm that approximately computes the number of paths of
length in a given directed graph with vertices up to a multiplicative
error of . Our algorithm runs in time . The algorithm is based on associating with
each vertex an element in the exterior (or, Grassmann) algebra, called an
extensor, and then performing computations in this algebra. This connection to
exterior algebra generalizes a number of previous approaches for the longest
path problem and is of independent conceptual interest. Using this approach, we
also obtain a deterministic time algorithm
to find a -path in a given directed graph that is promised to have few of
them. Our results and techniques generalize to the subgraph isomorphism problem
when the subgraphs we are looking for have bounded pathwidth. Finally, we also
obtain a randomized algorithm to detect -multilinear terms in a multivariate
polynomial given as a general algebraic circuit. To the best of our knowledge,
this was previously only known for algebraic circuits not involving negative
constants.Comment: To appear at STOC 2018: Symposium on Theory of Computing, June 23-27,
2018, Los Angeles, CA, US
Hierarchical Cut Labelling -- Scaling Up Distance Queries on Road Networks
Answering the shortest-path distance between two arbitrary locations is a
fundamental problem in road networks. Labelling-based solutions are the current
state-of-the-arts to render fast response time, which can generally be
categorised into hub-based labellings, highway-based labellings, and tree
decomposition labellings. Hub-based and highway-based labellings exploit
hierarchical structures of road networks with the aim to reduce labelling size
for improving query efficiency. However, these solutions still result in large
search spaces on distance labels at query time, particularly when road networks
are large. Tree decomposition labellings leverage a hierarchy of vertices to
reduce search spaces over distance labels at query time, but such a hierarchy
is generated using tree decomposition techniques, which may yield very large
labelling sizes and slow querying. In this paper, we propose a novel solution
\emph{hierarchical cut 2-hop labelling (HC2L)} to address the drawbacks of the
existing works. Our solution combines the benefits of hierarchical structures
from both perspectives - reduce the size of a distance labelling at
preprocessing time and further reduce the search space on a distance labelling
at query time. At its core, we propose a new hierarchy, \emph{balanced tree
hierarchy}, which enables a fast, efficient data structure to reduce the size
of distance labelling and to select a very small subset of labels to compute
the shortest-path distance at query time. To speed up the construction process
of HC2L, we further propose a parallel variant of our method, namely HC2L.
We have evaluated our solution on 10 large real-world road networks through
extensive experiments
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