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Publication# Unambiguous DNFs and Alon-Saks-Seymour

Résumé

We exhibit an unambiguous k-DNF formula that requires CNF width (Omega) over tilde (k(2)), which is optimal up to logarithmic factors. As a consequence, we get a near-optimal solution to the Alon-Saks-Seymour problem in graph theory (posed in 1991), which asks: How large a gap can there be between the chromatic number of a graph and its biclique partition number? Our result is also known to imply several other improved separations in query and communication complexity.

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Théorie des graphes

vignette|Un tracé de graphe.
La théorie des graphes est la discipline mathématique et informatique qui étudie les graphes, lesquels sont des modèles abstraits de dessins de réseaux reliant des objets

Complexité de la communication

La complexité de la communication ou complexité de communication est une notion étudiée en informatique théorique. Le dispositif abstrait classique est le suivant : Alice et Bob ont chacun un message

Théorie de la complexité (informatique théorique)

vignette|Quelques classes de complexité étudiées dans le domaine de la théorie de la complexité. Par exemple, P est la classe des problèmes décidés en temps polynomial par une machine de Turing déterm

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Graph Coloring is a very active field of research in graph theory as well as in the domain of the design of efficient heuristics to solve problems which, due to their computational complexity, cannot be solved exactly (no guarantee that an optimal solution will be reported), see [Cul] for a list of over 450 references. The graph coloring problem involves coloring the vertices of a given graph in such a way that two adjacent vertices never share the same color. The goal is to find the smallest number of colors needed to color all vertices in a fashion that satisfies this requirement. This number is called chromatic number and is denoted by Χ. In the first chapter, we present our research on suboptimal colorings and graphs which can be colored in such a way that the number of different colors appearing on the closed neighborhood (a vertex plus its neighbors) of any vertex v is less than Χ. We call such graphs oligomatic. The most interesting result is the following: given a graph and a coloring using Χ + p colors, there exists a vertex v such that there are at least Χ different colors among all vertices which are at a distance of [p/2] + 1 or less from v. We also study the existence of oligomatic graphs in special classes. Additionally, we present results of research on universal graphs which are "generic" oligomatic graphs in the sense that most properties of oligomatic graphs can be analyzed by restricting ourselves to universal graphs. Chapters Two to Four deal with the development of heuristics for two types of graph coloring problems. The tabu search heuristic was a central focus of our research. A tabu search iteratively modifies a candidate solution (which becomes the new candidate solution) with the goal of improving it. In such a procedure it is forbidden (tabu) to undo a modification for a certain number iterations. This mechanism allows to escape from local minima. In Chapter Two, we propose general improvements for tabu search based on some new and simple mechanisms (called reactive tabu schemes) to adapt the tabu tenure (which corresponds to the number of iterations a modification stays tabu). We also introduce distance and similarity measures for graph coloring problems which are needed in iterative procedures such as those of the tabu search. In the third chapter, we present the PARTIALCOL heuristic for the graph coloring problem. This method obtains excellent results compared to other similar methods and is able to color the well known DIMACS [JT96] benchmark graph flat300_28_0 optimally with 28 colors. (The best colorings found to date by other researchers use at least 31 colors.) PARTIALCOL uses partial solutions in its search space, which is a little explored way of approaching the graph coloring problem. Most approaches either use colorings and try to minimize the number of colors, or they use improper colorings (having conflicts, i.e. adjacent vertices which may have the same color) and try to minimize the number of conflicts. In the final chapter, we present a weighted version of the graph coloring problem which has applications in batch scheduling and telecommunication problems. We present two different adaptations of tabu search to the weighted graph coloring problem and test several of the reactive tabu schemes presented in Chapter Two. Further, we devise an adaptive memory algorithm AMA inspired by genetic algorithms. A large set of benchmark graphs with different properties is presented. All benchmark graphs with known optima have been solved to optimality by AMA. A key element of this algorithm is its capacity to determine the number k of colors to be used. Most other graph coloring heuristics need this parameter to be supplied by the user. Considering the results obtained on various graphs, we are confident that the methods developed are very efficient.

The advent of wireless communication technologies has created a paradigm shift in the accessibility of communication. With it has come an increased demand for throughput, a trend that is likely to increase further in the future. A key aspect of these challenges is to develop low complexity algorithms and architectures that can take advantage of the nature of the wireless medium like broadcasting and physical layer cooperation. In this thesis, we consider several problems in the domain of low complexity coding, relaying and scheduling for wireless networks. We formulate the Pliable Index Coding problem that models a server trying to send one or more new messages over a noiseless broadcast channel to a set of clients that already have a subset of messages as side information. We show through theoretical bounds and algorithms, that it is possible to design short length codes, poly-logarithmic in the number of clients, to solve this problem. The length of the codes are exponentially better than those possible in a traditional index coding setup. Next, we consider several aspects of low complexity relaying in half-duplex diamond networks. In such networks, the source transmits information to the destination through $n$ half-duplex intermediate relays arranged in a single layer. The half-duplex nature of the relays implies that they can either be in a listening or transmitting state at any point of time. To achieve high rates, there is an additional complexity of optimizing the schedule (i.e. the relative time fractions) of the relaying states, which can be $2^n$ in number. Using approximate capacity expressions derived from the quantize-map-forward scheme for physical layer cooperation, we show that for networks with $n\leq 6$ relays, the optimal schedule has atmost $n+1$ active states. This is an exponential improvement over the possible $2^n$ active states in a schedule. We also show that it is possible to achieve at least half the capacity of such networks (approximately) by employing simple routing strategies that use only two relays and two scheduling states. These results imply that the complexity of relaying in half-duplex diamond networks can be significantly reduced by using fewer scheduling states or fewer relays without adversely affecting throughput. Both these results assume centralized processing of the channel state information of all the relays. We take the first steps in analyzing the performance of relaying schemes where each relay switches between listening and transmitting states randomly and optimizes their relative fractions using only local channel state information. We show that even with such simple scheduling, we can achieve a significant fraction of the capacity of the network. Next, we look at the dual problem of selecting the subset of relays of a given size that has the highest capacity for a general layered full-duplex relay network. We formulate this as an optimization problem and derive efficient approximation algorithms to solve them. We end the thesis with the design and implementation of a practical relaying scheme called QUILT. In it the relay opportunistically decodes or quantizes its received signal and transmits the resulting sequence in cooperation with the source. To keep the complexity of the system low, we use LDPC codes at the source, interleaving at the relays and belief propagation decoding at the destination. We evaluate our system through testbed experiments over WiFi.

Dario Floreano, Giovanni Iacca, Andrea Maesani

The performance of evolutionary algorithms can be heavily undermined when constraints limit the feasible areas of the search space. For instance, while Covariance Matrix Adapta- tion Evolution Strategy is one of the most efficient algorithms for unconstrained optimization problems, it cannot be readily applied to constrained ones. Here, we used concepts from Memetic Computing, i.e. the harmonious combination of multiple units of algorithmic information, and Viability Evolution, an alternative abstraction of artificial evolution, to devise a novel approach for solving optimization problems with inequality constraints. Viability Evolution emphasizes elimination of solutions not satis- fying viability criteria, defined as boundaries on objectives and constraints. These boundaries are adapted during the search to drive a population of local search units, based on Covariance Matrix Adaptation Evolution Strategy, towards feasible regions. These units can be recombined by means of Differential Evolution operators. Of crucial importance for the performance of our method, an adaptive scheduler toggles between exploitation and exploration by selecting to advance one of the local search units and/or recombine them. The proposed algorithm can outperform several state-of-the-art methods on a diverse set of benchmark and engineering problems, both for quality of solutions and computational resources needed.