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Modal Kleene Algebra And Applications  A Survey
, 2004
"... Modal Kleene algebras are Kleene algebras with forward and backward modal operators defined via domain and codomain operations. They provide a concise and convenient algebraic framework that subsumes various other calculi and allows treating quite a variety of areas. We survey ..."
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Modal Kleene algebras are Kleene algebras with forward and backward modal operators defined via domain and codomain operations. They provide a concise and convenient algebraic framework that subsumes various other calculi and allows treating quite a variety of areas. We survey
A Relational Approach To Optimization Problems
, 1996
"... 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 s ..."
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Cited by 7 (0 self)
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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 nonrelational calculi. The graph
Predicting the Geometry of Metal Binding Sites from Protein Sequence
"... Metal binding is important for the structural and functional characterization of proteins. Previous prediction efforts have only focused on bonding state, i.e. deciding which protein residues act as metal ligands in some binding site. Identifying the geometry of metalbinding sites, i.e. deciding wh ..."
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Metal binding is important for the structural and functional characterization of proteins. Previous prediction efforts have only focused on bonding state, i.e. deciding which protein residues act as metal ligands in some binding site. Identifying the geometry of metalbinding sites, i.e. deciding which residues are jointly involved in the coordination of a metal ion is a new prediction problem that has been never attempted before from protein sequence alone. In this paper, we formulate it in the framework of learning with structured outputs. Our solution relies on the fact that, from a graph theoretical perspective, metal binding has the algebraic properties of a matroid, enabling the application of greedy algorithms for learning structured outputs. On a data set of 199 nonredundant metalloproteins, we obtained precision/recall levels of 75%/46 % correct ligandion assignments, which improves to 88%/88 % in the setting where the metal binding state is known. 1
Canonical greedy algorithms and dynamic programming by
"... Abstract: There has been little work on how to construct greedy algorithms to solve new optimization problems efficiently. Instead, greedy algorithms have generally been designed on an ad hoc basis. On the other hand, dynamic programming has a long history of being a useful tool for solving optimiza ..."
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Abstract: There has been little work on how to construct greedy algorithms to solve new optimization problems efficiently. Instead, greedy algorithms have generally been designed on an ad hoc basis. On the other hand, dynamic programming has a long history of being a useful tool for solving optimization problems, but is often inefficient. We show how dynamic programming can be used to derive efficient greedy algorithms that are optimal for a wide variety of problems. This approach also provides a way to obtain less efficient but optimal solutions to problems where derived greedy algorithms are nonoptimal.
Greedylike algorithms in Kleene algebra
 PARTICIPANTS’ PROCEEDINGS 7TH RELMICS/2ND KLEENE WORKSHOP, MALENTE, MAY 12–17, 2003
, 2003
"... This paper provides an algebraic background for the formal derivation of greedylike algorithms. Such derivations have previously been done in various frameworks including relation algebra. We propose Kleene algebra as a particularly simple alternative. Instead of converse and residuation we use mo ..."
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This paper provides an algebraic background for the formal derivation of greedylike algorithms. Such derivations have previously been done in various frameworks including relation algebra. We propose Kleene algebra as a particularly simple alternative. Instead of converse and residuation we use modal operators that are definable in a wide class of algebras, based on domain/codomain or image/preimage operations. By abstracting from earlier approaches we arrive at a very general theorem about the correctness of loops that covers particular forms of greedy algorithms as special cases.
Predicting metalbinding sites from protein sequence
 IEEE/ACM Trans. Comput. Biology Bioinform
"... Abstract—Prediction of binding sites from sequence can significantly help toward determining the function of uncharacterized proteins on a genomic scale. The task is highly challenging due to the enormous amount of alternative candidate configurations. Previous research has only considered this pred ..."
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Abstract—Prediction of binding sites from sequence can significantly help toward determining the function of uncharacterized proteins on a genomic scale. The task is highly challenging due to the enormous amount of alternative candidate configurations. Previous research has only considered this prediction problem starting from 3D information. When starting from sequence alone, only methods that predict the bonding state of selected residues are available. The sole exception consists of patternbased approaches, which rely on very specific motifs and cannot be applied to discover truly novel sites. We develop new algorithmic ideas based on structuredoutput learning for determining transitionmetalbinding sites coordinated by cysteines and histidines. The inference step (retrieving the best scoring output) is intractable for general output types (i.e., general graphs). However, under the assumption that no residue can coordinate more than one metal ion, we prove that metal binding has the algebraic structure of a matroid, allowing us to employ a very efficient greedy algorithm. We test our predictor in a highly stringent setting where the training set consists of protein chains belonging to SCOP folds different from the ones used for accuracy estimation. In this setting, our predictor achieves 56 percent precision and 60 percent recall in the identification of ligandion bonds. Index Terms—Metalbinding prediction, machine learning, structuredoutput learning, greedy algorithms. Ç
The Greedy Algorithms Class: Formalization, Synthesis and Generalization
, 1995
"... 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 G ..."
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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.
Synthesis Of Greedy Algorithms Using Dominance Relations
"... Greedy algorithms exploit problem structure and constraints to achieve lineartime performance. Yet there is still no completely satisfactory way of constructing greedy algorithms. For example, the Greedy Algorithm of Edmonds depends upon translating a problem into an algebraic structure called a ma ..."
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Greedy algorithms exploit problem structure and constraints to achieve lineartime performance. Yet there is still no completely satisfactory way of constructing greedy algorithms. For example, the Greedy Algorithm of Edmonds depends upon translating a problem into an algebraic structure called a matroid, but the existence of such a translation can be as hard to determine as the existence of a greedy algorithm itself. An alternative characterization of greedy algorithms is in terms of dominance relations, a wellknown algorithmic technique used to prune search spaces. We demonstrate a process by which dominance relations can be methodically derived for a number of greedy algorithms, including activity selection, and prefixfree codes. By incorporating our approach into an existing framework for algorithm synthesis, we demonstrate that it could be the basis for an effective engineering method for greedy algorithms. We also compare our approach with other characterizations of greedy algorithms. 1
The Application of Automated Reasoning to Formal Models of Combinatorial Optimization
 Applied Mathematics and Computation
"... Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynamic programming, branch and bound, and greedy. In 1989 Helman presented a common formalism that captures dynamic programming and branch and bound type algorithms. The formalism was late ..."
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Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynamic programming, branch and bound, and greedy. In 1989 Helman presented a common formalism that captures dynamic programming and branch and bound type algorithms. The formalism was later extended to include greedy algorithms. In this paper, we describe the application of automated reasoning techniques to the domain of our model, in particular considering some representational issues and demonstrating that proofs about the model can be obtained by an automated reasoning program. The longterm objective of this research is to develop a methodology for using automated reasoning to establish new results within the theory, including the derivation of new lower bounds and the discovery (and verification) of new combinatorial search strategies. 1 Introduction Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynami...
TRIOID: A GENERALIZATION OF MATROID AND THE ASSOCIATED POLYTOPE
"... We consider a generalization of the well known greedy algorithm, called mstep greedy algorithm, where m elements are examined in each iteration. When m = 1 or 2, the algorithm reduces to the standard greedy algorithm. For m = 3 we provide a complete characterization of the independence system, cal ..."
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We consider a generalization of the well known greedy algorithm, called mstep greedy algorithm, where m elements are examined in each iteration. When m = 1 or 2, the algorithm reduces to the standard greedy algorithm. For m = 3 we provide a complete characterization of the independence system, called trioid, where the mstep greedy algorithm guarantees an optimal solution for all weight functions. We also characterize the trioid polytope and propose a generalization of submodular functions.