Showing papers on "Greedy algorithm published in 1984"

Heuristic Algorithms for Task Assignment in Distributed Systems.

Abstract: Investigate the problem of static task assignment in distributed computing systems, i.e. given a set of k communicating tasks to be executed on a distributed system of n processors, to which processor should each task be assigned? The author proposes a family of heuristic algorithms for Stone's classic model of communicating tasks whose goal is the minimization of the total execution and communication costs incurred by an assignment. In addition, she augments this model to include interference costs which reflect the degree of incompatibility between two tasks. Whereas high communication costs serve as a force of attraction between tasks, causing them to be assigned to the same processor, interference costs serve as a force of repulsion between tasks, causing them to be distributed over many processors. The inclusion of interference costs in the model yields assignments with greater concurrency, thus overcoming the tendency of Stone's model to assign all tasks to one or a few processors. Simulation results show that the algorithms perform well and in particular, that the highly efficient Simple Greedy Algorithm performs almost as well as more complex heuristic algorithms. >

Greedoids - A Structural Framework for the Greedy Algorithm

Abstract: In this paper we introduce combinatorial structures called “greedoids” which are more general than matroids and which can be considered as relaxations of them. These structures are characterized by the optimality of the greedy solution for a broad class of objective functions of which breadth first search, shortest path, scheduling under precedence constraints are special cases. Besides some basic structural and algorithmic facts about greedoids we mainly discuss many different examples of greedoids. A subsequent paper will deal with more structural properties of greedoids.

Parallelism and greedy algorithms

Abstract: A number of greedy algorithms are examined and are shown to be probably inherently sequential. Greedy algorithms are presented for finding a maximal path, for finding a maximal set of disjoint paths in a layered dag, and for finding the largest induced subgraph of a graph that has all vertices of degree at least k. It is shown that for all of these algorithms, the problem of determining if a given node is in the solution set of the algorithm is P-complete. This means that it is unlikely that these sequential algorithms can be sped up significantly using parallelism.

Sparse null bases and marriage theorems

Abstract: This dissertation considers the problem of constructing the sparsest basis for the null space of a constraint matrix. This problem arises in the design of practical algorithms for large-scale numerical optimization problems. Suprisingly, this problem can be formulated as a combinatorial optimization problem under a non-degeneracy assumption on the constraint matrix. The theory of matchings in bipartite graphs--marriage theorems--can then be used to obtain the nonzero positions in a null basis. Numerically stable matrix factorizations are used in the next stage to compute the null basis. We use conformal decompositions to characterize the columns of a sparsest null basis. Matroid theory is used to prove that a greedy algorithm constructs a sparsest null basis. We prove that finding a sparsest null basis is NP-hard by showing that associated matroidal and graph-theoretic problems are NP-complete. We propose two approximation algorithms to construct sparse null bases. Both of them make use of the Dulmage-Mendelsohn decomposition of rectangular matrices. One algorithm is a sparsity exploiting variant of the variable-reduction technique. The second is a locally greedy algorithm that constructs a null basis with an upper triangular submatrix. These results are extended to computing sparse orthogonal null bases. We show that the sparsest null basis for an n-vector computed as a product of Givens rotations has n log(,2) n nonzeros. A generalization for dense t x n matrices constructs an orthogonal null basis with nt log(,2)n/t nonzeros. We also classify all known methods for constructing null bases, and show some unexpected equivalences between some of them.

Worst-case analysis of greedy algorithms for the subset-sum problem

Abstract: Given a set ofn positive integers and another positive integerW, the Subset-Sum Problem is to find that subset whose sum is closest to, without exceeding,W. We present a polynomial approximation scheme for this problem and prove that its worst-case performance dominates that of Johnson's well-known scheme.

An Efficient Single-Row Routing Algorithm

Abstract: In this paper, we present a heuristic algorithm for single-row routing. Our approach is based on the interval graphical representation of the given net list. The objective function for minimization is the street congestion. The problem is known to be intractable in the sense of NP-completeness, thus a polynomial-time heuristic algorithm is proposed. It has been implemented and tested with various examples. So far it has always produced optimal solutions.

Greedoids and Linear Objective Functions

Abstract: Greedoids were introduced by the authors as generalizations of matroids providing a framework for the greedy algorithm. They can be characterized algorithmically via the optimality of the greedy algorithm for a class of objective functions, which are in general not linear and do not include all linear functions. It is therefore natural to ask the following questions: (1) What are those linear objective functions which can be optimized over any greedoid by the greedy algorithm; (2) what are those greedoids over which the linear objective function can be optimized by the greedy algorithm. This paper gives an answer to both questions. Moreover, it gives slimming procedures for obtaining such greedoids from matroids and it gives briefly some (negative) oracle results about greedoid optimization and greedoid recognition.

A probabilistic analysis of some greedy cardinality matching algorithms

Abstract: This paper deals with the expected cardinality of greedy matchings in random graphs. Different versions of the greedy heuristic for the cardinality matching problem are considered. Experimental data and some theoretical results are reported.

Matroids on Ordered Sets and the Greedy Algorithm

Abstract: Generalized independence systems and classes of objective functions are investigated for which the greedy algorithm works well Those systems may be viewed as matroids on ordered ground sets and include, in particular, systems of integral vectors of integral polymatroids The greedy algorithm can be understood as being performed in an associated matroid on an unordered set, the ‘Dilworth completion’ This allows to derive worst case bounds for the greedy heuristic for certain ordered systems of integral vectors

The Sparse Null Space Basis Problem

Abstract: The sparse null space basis problem is the following: $A t \times n$ matrix $A (t > n)$ is given. Find a matrix $N$, with the fewest nonzero entries in it, whose columns span the null space of $A$. This problem arises in the design of practical algorithms for large-scale numerical optimization problems. Surprisingly, this problem can be formulated as a combinatorial optimization problem under a non-degeneracy assumption on $A$. THe theory of matchings in bipartite graphs marriage theorems can then be used to obtain the nonzero positions in $N$. Numerically stable matrix factorizations are used in the next phase to compute $N$. We use conformal decompositions to characterize the columns of a sparsest null basis. Matroid theory is used to prove that a greedy algorithm constructs a sparsest null basis. We prove that finding a sparsest null basis is NP-hard by showing that associated matroidal and graph-theoretic problems are NP-complete. We propose two approximation algorithms to construct sparse null bases. Both of them make use of the Dulmage-Mendelsohn decomposition of rectangular matrices. One algorithm is a sparsity exploiting variant of the variable-reduction technique. The second is a locally greedy algorithm that constructs a null basis with an upper triangular submatrix. These results are extended to computing sparse orthogonal null bases. We discuss how this ``two-phase'''' approach can construct sparser null bases than a purely numerical approach; it is also potentially faster than the latter. Finally, we classify all known methods for constructing null bases, and show some unexpected equivalences between some of them.

A Greedy Algorithm for Solving Network Flow Problems in Trees

Abstract: Let (H,*,⩽) be a d-monoid and T an intree with root r. For each vertex j let L(j) be the set of leaves of the subtree rooted in j. Associated with the leaves of T are monotone functions from Z + into H satisfying some type of convexity properties. A Greedy algorithm is given which solves the following integer algebraic network flow problem: Minimize for all j∈T.

A Greedy Algorithm for the Transhipment along a Single Road Problem.

Abstract: : This paper presents a specialized algorithm for the transhipment along a single line problem. The problem is a specially structured network flow problem for which basis structure is such that a greedy algorithm can be employed for solution. The specialized algorithm is on the order of a hundred times faster than the primal simplex method on a graph. Additional keywords: Fortran, Cost Effectiveness, Naval Research. (Author)

Greedy Solutions for General Set Covering Problems

Abstract: Set covering problems are well known as hard [4]. Therefore only small problems can be solved effectively by standard methods of integer programming. The solution of larger problems requires the use of methods with polynomial time behaviour. Thus the choice of methods is restricted to heuristics, which can eventually be combined in the framework of branch-and-bound to give ‘hybrid’ algorithms. In [1] Balas and Ho report on an algorithm of this type, based on several heuristics, subgradient optimization, cutting planes, and implicit enumeration.