1,006 research outputs found
Broadcasting in weighted-vertex graphs
In this thesis, we present a new model for information dissemination in communication networks. The model is defined on networks in which nodes are assigned some weights representing the internal delay they should pass before sending data to their neighbors. The new model, called weighted-vertex model, comes to have real world applications in parallel computation and satellite terrestrial networks. Broadcasting in weighted-vertex model is a generalized version of classical broadcasting problem, which is NP_Complete. The problem remains NP_Complete in some classes of weighed-vertex graphs. We show existence of approximation algorithms for the broadcasting problem in weighted vertex model, as well as better approximations for specific subclasses of weighted graphs. In addition we study broadcasting in complete weighted graphs and present an algorithm for finding the optimum solution in this case. Broadcasting in complete graphs with uniform weights is studied separately. Finally we introduce some heuristics for the problem and compare their performance using computer simulations
On the probabilistic min spanning tree Problem
We study a probabilistic optimization model for min spanning tree, where any vertex vi of the input-graph G(V,E) has some presence probability pi in the final instance GâČ â G that will effectively be optimized. Suppose that when this ârealâ instance GâČ becomes known, a spanning tree T, called anticipatory or a priori spanning tree, has already been computed in G and one can run a quick algorithm (quicker than one that recomputes from scratch), called modification strategy, that modifies the anticipatory tree T in order to fit G âČ. The goal is to compute an anticipatory spanning tree of G such that, its modification for any G âČ â G is optimal for G âČ. This is what we call probabilistic min spanning tree problem. In this paper we study complexity and approximation of probabilistic min spanning tree in complete graphs under two distinct modification strategies leading to different complexity results for the problem. For the first of the strategies developed, we also study two natural subproblems of probabilistic min spanning tree, namely, the probabilistic metric min spanning tree and the probabilistic min spanning tree 1,2 that deal with metric complete graphs and complete graphs with edge-weights either 1, or 2, respectively
On Approximating the Sum-Rate for Multiple-Unicasts
We study upper bounds on the sum-rate of multiple-unicasts. We approximate
the Generalized Network Sharing Bound (GNS cut) of the multiple-unicasts
network coding problem with independent sources. Our approximation
algorithm runs in polynomial time and yields an upper bound on the joint source
entropy rate, which is within an factor from the GNS cut. It
further yields a vector-linear network code that achieves joint source entropy
rate within an factor from the GNS cut, but \emph{not} with
independent sources: the code induces a correlation pattern among the sources.
Our second contribution is establishing a separation result for vector-linear
network codes: for any given field there exist networks for which
the optimum sum-rate supported by vector-linear codes over for
independent sources can be multiplicatively separated by a factor of
, for any constant , from the optimum joint entropy
rate supported by a code that allows correlation between sources. Finally, we
establish a similar separation result for the asymmetric optimum vector-linear
sum-rates achieved over two distinct fields and
for independent sources, revealing that the choice of field
can heavily impact the performance of a linear network code.Comment: 10 pages; Shorter version appeared at ISIT (International Symposium
on Information Theory) 2015; some typos correcte
A note on the data-driven capacity of P2P networks
We consider two capacity problems in P2P networks. In the first one, the
nodes have an infinite amount of data to send and the goal is to optimally
allocate their uplink bandwidths such that the demands of every peer in terms
of receiving data rate are met. We solve this problem through a mapping from a
node-weighted graph featuring two labels per node to a max flow problem on an
edge-weighted bipartite graph. In the second problem under consideration, the
resource allocation is driven by the availability of the data resource that the
peers are interested in sharing. That is a node cannot allocate its uplink
resources unless it has data to transmit first. The problem of uplink bandwidth
allocation is then equivalent to constructing a set of directed trees in the
overlay such that the number of nodes receiving the data is maximized while the
uplink capacities of the peers are not exceeded. We show that the problem is
NP-complete, and provide a linear programming decomposition decoupling it into
a master problem and multiple slave subproblems that can be resolved in
polynomial time. We also design a heuristic algorithm in order to compute a
suboptimal solution in a reasonable time. This algorithm requires only a local
knowledge from nodes, so it should support distributed implementations.
We analyze both problems through a series of simulation experiments featuring
different network sizes and network densities. On large networks, we compare
our heuristic and its variants with a genetic algorithm and show that our
heuristic computes the better resource allocation. On smaller networks, we
contrast these performances to that of the exact algorithm and show that
resource allocation fulfilling a large part of the peer can be found, even for
hard configuration where no resources are in excess.Comment: 10 pages, technical report assisting a submissio
The complexity of the Pk partition problem and related problems in bipartite graphs
In this paper, we continue the investigation made in [MT05] about the approximability of Pk partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the Pk partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum Pk packing problem. Moreover, we propose some simple approximation algorithms for those problems
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