1,950 research outputs found
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
The role of twins in computing planar supports of hypergraphs
A support or realization of a hypergraph is a graph on the same
vertex as such that for each hyperedge of it holds that its vertices
induce a connected subgraph of . The NP-hard problem of finding a planar}
support has applications in hypergraph drawing and network design. Previous
algorithms for the problem assume that twins}---pairs of vertices that are in
precisely the same hyperedges---can safely be removed from the input
hypergraph. We prove that this assumption is generally wrong, yet that the
number of twins necessary for a hypergraph to have a planar support only
depends on its number of hyperedges. We give an explicit upper bound on the
number of twins necessary for a hypergraph with hyperedges to have an
-outerplanar support, which depends only on and . Since all
additional twins can be safely removed, we obtain a linear-time algorithm for
computing -outerplanar supports for hypergraphs with hyperedges if
and are constant; in other words, the problem is fixed-parameter
linear-time solvable with respect to the parameters and
Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings
Consider a directed or an undirected graph with integral edge weights from
the set [-W, W], that does not contain negative weight cycles. In this paper,
we introduce a general framework for solving problems on such graphs using
matrix multiplication. The framework is based on the usage of Baur-Strassen's
theorem and of Strojohann's determinant algorithm. It allows us to give new and
simple solutions to the following problems:
* Finding Shortest Cycles -- We give a simple \tilde{O}(Wn^{\omega}) time
algorithm for finding shortest cycles in undirected and directed graphs. For
directed graphs (and undirected graphs with non-negative weights) this matches
the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the
other hand, no algorithm working in \tilde{O}(Wn^{\omega}) time was previously
known for undirected graphs with negative weights. Furthermore our algorithm
for a given directed or undirected graph detects whether it contains a negative
weight cycle within the same running time.
* Computing Diameter and Radius -- We give a simple \tilde{O}(Wn^{\omega})
time algorithm for computing a diameter and radius of an undirected or directed
graphs. To the best of our knowledge no algorithm with this running time was
known for undirected graphs with negative weights.
* Finding Minimum Weight Perfect Matchings -- We present an
\tilde{O}(Wn^{\omega}) time algorithm for finding minimum weight perfect
matchings in undirected graphs. This resolves an open problem posted by
Sankowski in 2006, who presented such an algorithm but only in the case of
bipartite graphs.
In order to solve minimum weight perfect matching problem we develop a novel
combinatorial interpretation of the dual solution which sheds new light on this
problem. Such a combinatorial interpretation was not know previously, and is of
independent interest.Comment: To appear in FOCS 201
Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials
This paper is our third step towards developing a theory of testing monomials
in multivariate polynomials and concentrates on two problems: (1) How to
compute the coefficients of multilinear monomials; and (2) how to find a
maximum multilinear monomial when the input is a polynomial. We
first prove that the first problem is \#P-hard and then devise a
upper bound for this problem for any polynomial represented by an arithmetic
circuit of size . Later, this upper bound is improved to for
polynomials. We then design fully polynomial-time randomized
approximation schemes for this problem for polynomials. On the
negative side, we prove that, even for polynomials with terms of
degree , the first problem cannot be approximated at all for any
approximation factor , nor {\em "weakly approximated"} in a much relaxed
setting, unless P=NP. For the second problem, we first give a polynomial time
-approximation algorithm for polynomials with terms of
degrees no more a constant . On the inapproximability side, we
give a lower bound, for any on the
approximation factor for polynomials. When terms in these
polynomials are constrained to degrees , we prove a lower
bound, assuming ; and a higher lower bound, assuming the
Unique Games Conjecture
Random non-crossing plane configurations: A conditioned Galton-Watson tree approach
We study various models of random non-crossing configurations consisting of
diagonals of convex polygons, and focus in particular on uniform dissections
and non-crossing trees. For both these models, we prove convergence in
distribution towards Aldous' Brownian triangulation of the disk. In the case of
dissections, we also refine the study of the maximal vertex degree and validate
a conjecture of Bernasconi, Panagiotou and Steger. Our main tool is the use of
an underlying Galton-Watson tree structure.Comment: 24 pages, 9 figure
A Survey of Alternating Permutations
This survey of alternating permutations and Euler numbers includes
refinements of Euler numbers, other occurrences of Euler numbers, longest
alternating subsequences, umbral enumeration of classes of alternating
permutations, and the cd-index of the symmetric group.Comment: 32 pages, 7 figure
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