87 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
Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-width
The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback
Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for
NMC disconnects a given set of terminals, and for SFVS intersects all cycles
containing a vertex of a given set. We design a meta-algorithm that allows to
solve both problems in time , , and where is the rank-width, the
-rank-width, and the mim-width of a given decomposition. This
answers in the affirmative an open question raised by Jaffke et al.
(Algorithmica, 2019) concerning an XP algorithm for SFVS parameterized by
mim-width.
By a unified algorithm, this solves both problems in polynomial-time on the
following graph classes: Interval, Permutation, and Bi-Interval graphs,
Circular Arc and Circular Permutation graphs, Convex graphs, -Polygon,
Dilworth- and Co--Degenerate graphs for fixed ; and also on Leaf Power
graphs if a leaf root is given as input, on -Graphs for fixed if an
-representation is given as input, and on arbitrary powers of graphs in all
the above classes. Prior to our results, only SFVS was known to be tractable
restricted only on Interval and Permutation graphs, whereas all other results
are new
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