12,491 research outputs found
On Network Coding Capacity - Matroidal Networks and Network Capacity Regions
One fundamental problem in the field of network coding is to determine the
network coding capacity of networks under various network coding schemes. In
this thesis, we address the problem with two approaches: matroidal networks and
capacity regions.
In our matroidal approach, we prove the converse of the theorem which states
that, if a network is scalar-linearly solvable then it is a matroidal network
associated with a representable matroid over a finite field. As a consequence,
we obtain a correspondence between scalar-linearly solvable networks and
representable matroids over finite fields in the framework of matroidal
networks. We prove a theorem about the scalar-linear solvability of networks
and field characteristics. We provide a method for generating scalar-linearly
solvable networks that are potentially different from the networks that we
already know are scalar-linearly solvable.
In our capacity region approach, we define a multi-dimensional object, called
the network capacity region, associated with networks that is analogous to the
rate regions in information theory. For the network routing capacity region, we
show that the region is a computable rational polytope and provide exact
algorithms and approximation heuristics for computing the region. For the
network linear coding capacity region, we construct a computable rational
polytope, with respect to a given finite field, that inner bounds the linear
coding capacity region and provide exact algorithms and approximation
heuristics for computing the polytope. The exact algorithms and approximation
heuristics we present are not polynomial time schemes and may depend on the
output size.Comment: Master of Engineering Thesis, MIT, September 2010, 70 pages, 10
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Constructive Heuristics for the Minimum Labelling Spanning Tree Problem: a preliminary comparison
This report studies constructive heuristics for the minimum labelling spanning tree
(MLST) problem. The purpose is to find a spanning tree that uses edges that are as similar as
possible. Given an undirected labeled connected graph (i.e., with a label or color for each edge),
the minimum labeling spanning tree problem seeks a spanning tree whose edges have the smallest
possible number of distinct labels. The model can represent many real-world problems in
telecommunication networks, electric networks, and multimodal transportation networks, among
others, and the problem has been shown to be NP-complete even for complete graphs. A primary
heuristic, named the maximum vertex covering algorithm has been proposed. Several versions of
this constructive heuristic have been proposed to improve its efficiency. Here we describe the
problem, review the literature and compare some variants of this algorithm
Heuristic average-case analysis of the backtrack resolution of random 3-Satisfiability instances
An analysis of the average-case complexity of solving random 3-Satisfiability
(SAT) instances with backtrack algorithms is presented. We first interpret
previous rigorous works in a unifying framework based on the statistical
physics notions of dynamical trajectories, phase diagram and growth process. It
is argued that, under the action of the Davis--Putnam--Loveland--Logemann
(DPLL) algorithm, 3-SAT instances are turned into 2+p-SAT instances whose
characteristic parameters (ratio alpha of clauses per variable, fraction p of
3-clauses) can be followed during the operation, and define resolution
trajectories. Depending on the location of trajectories in the phase diagram of
the 2+p-SAT model, easy (polynomial) or hard (exponential) resolutions are
generated. Three regimes are identified, depending on the ratio alpha of the
3-SAT instance to be solved. Lower sat phase: for small ratios, DPLL almost
surely finds a solution in a time growing linearly with the number N of
variables. Upper sat phase: for intermediate ratios, instances are almost
surely satisfiable but finding a solution requires exponential time (2 ^ (N
omega) with omega>0) with high probability. Unsat phase: for large ratios,
there is almost always no solution and proofs of refutation are exponential. An
analysis of the growth of the search tree in both upper sat and unsat regimes
is presented, and allows us to estimate omega as a function of alpha. This
analysis is based on an exact relationship between the average size of the
search tree and the powers of the evolution operator encoding the elementary
steps of the search heuristic.Comment: to appear in Theoretical Computer Scienc
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Combinatorial optimization and metaheuristics
Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (âefficientâ) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find âquicklyâ (reasonable run-times), with âhighâ probability, provable âgoodâ solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods
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