784 research outputs found
Meunier Conjecture
Fr\'ed\'eric Meunier's question about a multicolored Sperner lemma is
addressed, leaving the question of connectivity for the color hypergraphs of
such a multicolored simplex. Sperner's lemma asserts the existence of a simplex
using all the colors for any vertex coloring of a subdivision of a large
simplex with appropriate boundary conditions. Meunier's questions generalizes
this to the situation of having several such colorings and asserts the
existence of a simplex using enough different colors from each coloring
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
Random multi-index matching problems
The multi-index matching problem (MIMP) generalizes the well known matching
problem by going from pairs to d-uplets. We use the cavity method from
statistical physics to analyze its properties when the costs of the d-uplets
are random. At low temperatures we find for d>2 a frozen glassy phase with
vanishing entropy. We also investigate some properties of small samples by
enumerating the lowest cost matchings to compare with our theoretical
predictions.Comment: 22 pages, 16 figure
Discovering a junction tree behind a Markov network by a greedy algorithm
In an earlier paper we introduced a special kind of k-width junction tree,
called k-th order t-cherry junction tree in order to approximate a joint
probability distribution. The approximation is the best if the Kullback-Leibler
divergence between the true joint probability distribution and the
approximating one is minimal. Finding the best approximating k-width junction
tree is NP-complete if k>2. In our earlier paper we also proved that the best
approximating k-width junction tree can be embedded into a k-th order t-cherry
junction tree. We introduce a greedy algorithm resulting very good
approximations in reasonable computing time.
In this paper we prove that if the Markov network underlying fullfills some
requirements then our greedy algorithm is able to find the true probability
distribution or its best approximation in the family of the k-th order t-cherry
tree probability distributions. Our algorithm uses just the k-th order marginal
probability distributions as input.
We compare the results of the greedy algorithm proposed in this paper with
the greedy algorithm proposed by Malvestuto in 1991.Comment: The paper was presented at VOCAL 2010 in Veszprem, Hungar
Non-Uniform Robust Network Design in Planar Graphs
Robust optimization is concerned with constructing solutions that remain
feasible also when a limited number of resources is removed from the solution.
Most studies of robust combinatorial optimization to date made the assumption
that every resource is equally vulnerable, and that the set of scenarios is
implicitly given by a single budget constraint. This paper studies a robustness
model of a different kind. We focus on \textbf{bulk-robustness}, a model
recently introduced~\cite{bulk} for addressing the need to model non-uniform
failure patterns in systems.
We significantly extend the techniques used in~\cite{bulk} to design
approximation algorithm for bulk-robust network design problems in planar
graphs. Our techniques use an augmentation framework, combined with linear
programming (LP) rounding that depends on a planar embedding of the input
graph. A connection to cut covering problems and the dominating set problem in
circle graphs is established. Our methods use few of the specifics of
bulk-robust optimization, hence it is conceivable that they can be adapted to
solve other robust network design problems.Comment: 17 pages, 2 figure
Distributed Symmetry Breaking in Hypergraphs
Fundamental local symmetry breaking problems such as Maximal Independent Set
(MIS) and coloring have been recognized as important by the community, and
studied extensively in (standard) graphs. In particular, fast (i.e.,
logarithmic run time) randomized algorithms are well-established for MIS and
-coloring in both the LOCAL and CONGEST distributed computing
models. On the other hand, comparatively much less is known on the complexity
of distributed symmetry breaking in {\em hypergraphs}. In particular, a key
question is whether a fast (randomized) algorithm for MIS exists for
hypergraphs.
In this paper, we study the distributed complexity of symmetry breaking in
hypergraphs by presenting distributed randomized algorithms for a variety of
fundamental problems under a natural distributed computing model for
hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can
be solved in rounds ( is the number of nodes of the
hypergraph) in the LOCAL model. We then present a key result of this paper ---
an -round hypergraph MIS algorithm in
the CONGEST model where is the maximum node degree of the hypergraph
and is any arbitrarily small constant.
To demonstrate the usefulness of hypergraph MIS, we present applications of
our hypergraph algorithm to solving problems in (standard) graphs. In
particular, the hypergraph MIS yields fast distributed algorithms for the {\em
balanced minimal dominating set} problem (left open in Harris et al. [ICALP
2013]) and the {\em minimal connected dominating set problem}. We also present
distributed algorithms for coloring, maximal matching, and maximal clique in
hypergraphs.Comment: Changes from the previous version: More references adde
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