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We present exact characterizations of structures on which the greedy algorithm produces optimal solutions. Our characterization, which we call matroid embeddings, complete the partial characterizations of Rado, Gale, and Edmonds (matroids), and of Korte and Lovasz (greedoids). We show that the greedy algorithm optimizes all linear objective functions if and only if the problem structure (phrased in terms of either accessible set systems or hereditary languages) is a matroid embedding. We also present an exact characterization of the objective functions optimized by the greedy algorithm on matroid embeddings. Finally, we present an exact characterization of the structures on which the greedy algorithm optimizes all bottleneck functions, structures which are less constrained than matroid embeddings. 1 Introduction Obtaining an exact characterization of the class of problems for which the greedy algorithm returns an optimal solution has been an open problem. Rado [9], Gale [3], a...

The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming style for generating feasible solutions, rather than the fold and unfold operators of the functional programming style. The relationship between fold operators and loop operators is explored, and it is shown how to convert from the former to the latter. This fresh approach provides additional insights into the relationship between dynamic programming and greedy algorithms, and helps to unify previously distinct approaches to solving combinatorial optimization problems. Some of the solutions discovered are new and solve problems which had previously proved difficult. The material is illustrated with a selection of problems and solutions that is a mixture of old and new. Another contribution is the invention of a new calculus, called the graph calculus, which is a useful tool for reasoning in the relational calculus and other non-relational calculi. The graph

Borodin, Nielsen and Rackoff [13] introduced the class of priority algorithms as a framework for modeling deterministic greedy-like algorithms. In this paper we address the effect of randomization in greedy-like algorithms. More specifically, we consider approximation ratios within the context of randomized priority algorithms. As case studies, we prove inapproximation results for two well-studied optimization problems, namely facility location and makespan scheduling.

We will review a few key algorithmic and analysis concepts with application to physical design problems. We argue that design and detailed analysis of algorithms is of fundamental importance in developing better physical design tools and to cope with the complexity of present-day designs.

A graph is chordal if it does not contain any induced cycle of size greater than three. An alternative characterization of chordal graphs is via a perfect elimination ordering, which is an ordering of the vertices such that, for each vertex v, the neighbors of v that occur later than v in the ordering form a clique. Akcoglu et al [2] define an extension of chordal graphs whereby the neighbors of v that occur later than v in the elimination order have at most k independent vertices. We refer to such graphs as sequentially k-independent graphs. Clearly this extension of chordal graphs also extends the class of (k+1)-claw-free graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are sequentially k-independent for small k. In particular, any intersection graph of translates of a convex object in a two dimensional plane is a sequentially 3-independent graph; furthermore, any planar graph is a sequentially 3-independent graph. For any fixed constant k, we develop simple, polynomial time approximation algorithms for sequentially k-independent graphs with respect to several well-studied NPcomplete problems based on this k-sequentially independent ordering. Our generalized formulation unifies and extends several previously known results. We also consider other classes of sequential elimination graphs.

We present a straightforward linear algebraic model of greed, based only on extensions of classical majorization and convexity theory. This gives an alternative to other models of greedy-solvable problems such as matroids, greedoids, submodular functions, etc., and it is able to express established examples of greedy-solvable optimization problems that they cannot. The linear algebraic approach is also much closer in spirit to established practice in operations research and numerical optimization. The essence of the approach is to model `exchanges' with certain linear transformations. Modeling solutions as vectors also, we then exploit the fact that these exchanges define an ordering on solutions. When the exchanges are doubly-stochastic matrices, this ordering is the majorization ordering developed by Hardy, Littlewood, and P'olya in their pioneering work on inequality theory. We generalize majorization to permit any matrix semigroup of exchanges, but find that several kinds of stocha...

On the first hand, this report studies the class of Greedy Algorithms in order to find an as systematic as possible strategy that could be applied to the specification of some problems to lead to a correct program solving that problem. On the other hand, the standard formalisms underlying the Greedy Algorithms (matroid, greedoid and matroid embedding) which are dependent on the particular type set are generalized to a formalism independent of any data type based on an algebraic specification setting.

We explore antimatroids, also known as shelling structures, a construct used to formalize when greedy (local) algorithms are optimal, as well as their relation to the strong measure of progress P introduced in (Parmar 2002b). We begin with an example from the map coloring domain to spark the reader's intuitions, and then move towards a more general application of shelling to the strong measure of progress. We also introduce some extensions of shelling to planning on a different level. Macro-operators are another kind of mathematical structure that help give efficient and easy-to-understand plans, but we must be careful how we use them when defining strong measures of progress.

Korte and Lovász [12, 13] founded the theory of greedoids. These combinatorial structures characterize a class of optimization problems that can be solved by greedy algorithms. In particular, greedoids generalize matroids, introduced earlier by Whitney [16]. Antimatroids, introduced by Dilworth [3]