Giorgio Ausiello

Structure Preserving Reductions Among Convex Optimization Problems

Abstract In this paper we introduce the concept of convex optimization problem. Convex optimization problems are studied by giving a formalization of the concept of combinatorial structure, in terms of spectra of approximate solutions, and of reduction which preserves such structure. On this basis a classification of convex NP -optimization problems is introduced and is applied to study the combinatorial structure of several optimization problems associated to well-known NP -complete sets. Conditions on the approximability of such optimization problems are also given and it is shown that structurally isomorphic problems have similar approximability properties.

Minimal representation of directed hypergraphs

In this paper the problem of minimal representations for particular classes of directed hypergraphs is analyzed. Various concepts of minimal representations of directed hypergraphs (called minimal equivalent hypergraphs) are introduced as extensions to the concepts of transitive reduction and minimum equivalent graph of directed graphs. In particular, we consider equivalent hypergraphs which are minimal with respect to all parameters which may be adopted to characterize a given hypergraph (number of hyperarcs, number of adjacency lists required for the representation, length of the overall description,etc.). The relationships among the various concepts of minimality are discussed and their computational properties are analyzed. In order to derive such results, a graph representation of hypergraphs is introduced.

Approximate solution of NP optimization problems

Abstract This paper presents the main results obtained in the field of approximation algorithms in a unified framework. Most of these results have been revisited in order to emphasize two basic tools useful for characterizing approximation classes, that is, combinatorial properties of problems and approximation preserving reducibilities. In particular, after reviewing the most important combinatorial characterizations of the classes PTAS and FPTAS, we concentrate on the class APX and, as a concluding result, we show that this class coincides with the class of optimization problems which are reducible to the maximum satisfiability problem with respect to a polynomial-time approximation preserving reducibility.

Graph Algorithms for Functional Dependency Manipulation

Abstract. A graph-theoretic approach for the representation of functional dependenoes in relauonal databases is introduced and applied in the construction of algorithms for manipulating dependencies. This approach allows a homogeneous treatment of several problems (closure, minimization, etc.), which leads to simpler proofs and, m some cases, more efficient algorithms than in the current literature. Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumencal Algorithms and Problems--computations on d, screte structures; G.2.2 [Diserete Mathematics]: Graph Theory--graph algorithms; H.2.1 [Database Management]: Logical Design-normal forms; schema and subschema General Terms: Algorithms, Design, Management, Theory Additional Key Words and Phrases: Closure, computational complexity, functional dependency, FD- graph, minimal coverings, relational database 1. Introduction

Directed Hypergraphs: Problems, Algorithmic Results, and a Novel Decremental Approach

The purpose of this paper is twofold. First, we review several basic combinatorial problems that have been stated in terms of directed hypergraphs and have been studied in the literature in the framework of different application domains. Among them, transitive closure, transitive reduction, flow and cut problems, and minimum weight traversal problems. For such problems we illustrate some of the most important algorithmic results in the context of both static and dynamic applications. Second, we address a specific dynamic problem which finds several interesting applications, especially in the framework of knowledge representation: the maintenance of minimum weight hyperpaths under hyperarc weight increases and hyperarc deletions. For such problem we provide a new efficient algorithm applicable for a wide class of hyperpath weight measures.

Inclusion and equivalence between relational database schemata

Abstract Conceptual relations among relational database schemata are investigated. Two different definitions of inclusion and a definition of equivalence between schemata are given. Several examples of practical situations adequately handled by our definitions are shown. Finally, necessary and sufficient conditions for inclusion and equivalence are proved for two classes of schemata, meaningful in the relational theory.

Toward a unified approach for the classification of NP-complete optimization problems

Abstract Two notions which have been introduced with the aim of classifying NP-complete optimization problems are compared: the notion of strong NP-completeness, due to Garey and Johnson, and that of simple and rigid problems, due to Paz and Moran. In particular, we show under what conditions reductions preserve rigidity, simplicity, strong simplicity and p-simplicity and we show that under reasonable hypothesis, p-simple problems are solved by pseudo-polynomial algorithms and strong NP-complete problems are weakly rigid.

Algorithms for the on-line quota traveling salesman problem

The Quota Traveling Salesman Problem is a generalization of the well-known Traveling Salesman Problem. The goal of the traveling salesman is, in this case, to reach a given quota of sales, minimizing the amount of time. In this paper we address the on-line version of the problem, where requests are given over time. We present algorithms for various metric spaces, and analyze their performance in the usual framework of competitive analysis. In particular we present a 2-competitive algorithm that matches the lower bound for general metric spaces. In the case of the halfline metric space, we show that it is helpful not to move at full speed, and this approach is also used to derive the best on-line polynomial time algorithm known so far for the On-Line TSP (in the homing version).

Dynamic maintenance of directed hypergraphs

Abstract In this paper we are concerned with the on-line maintenance of directed hypergraphs, a generalization of directed graphs previously introduced in the literature. In particular, we show how to maintain efficiently information about hyperpaths while new hyperarcs are inserted. We present a data structure which allows us to check whether there exists a hyperpath between an arbitrarily given pair of nodes in constant time and to return such a hyperpath in a time which is linear in its size. The total time required to maintain the data structure during the insertion of new hyperarcs is O( mn ), where m is the total size of the description of the hyperarcs and n is the number of nodes. This generalizes a previous result known for directed graphs and has applications in several areas of computer science, such as rewriting systems, database schemes, logic programming and problem solving. An extension of these results to hyperpaths between sets of nodes is also presented.

On-line algorithms for polynomially solvable satisfiability problems

Abstract Given a propositional Horn formula, we show how to maintain on-line information about its satisfiability during the insertion of new clauses. A data structure is presented which answers each satisfiability question in O (1) time and inserts a new clause of length q in O ( q ) amortized time. This significantly outperforms previously known solutions of the same problem. This result is extended also to a particular class of non-Horn formulae already considered in the literature, for which the space bound is improved. Other operations are considered, such as testing whether a given hypothesis is consistent with a satisfying interpretation of the given formula and determining a truth assignment which satisfies a given formula. The on-line time and space complexity of these operations is also analyzed.