Showing papers on "Greedy algorithm published in 1975"

A new algorithm for the solution of multi-state dynamic programming problems

Abstract: This paper presents a new algorithm for the solution of multi-state dynamic programming problems, referred to as the Progressive Optimality Algorithm. It is a method of successive approximation using a general two-stage solution. The algorithm is computationally efficient and has minimal storage requirements. A description of the algorithm is given including a proof of convergence. Performance characteristics for a trial problem are summarized.

When the Greedy Solution Solves a Class of Knapsack Problems

Abstract: This paper analyzes a heuristic for the knapsack problem that recursively determines a solution by making a variable with smallest marginal unit cost as large as possible. Recursive necessary and sufficient conditions for the optimality of such “greedy” solutions and a “good” algorithm for verifying these conditions are given. Maximum absolute error for nonoptimal “greedy” solutions is also examined.

Introduction to the Theory of Matroids

Abstract: I. Equivalent Axiomatic Definitions and Elementary Properties of Matroids.- 1.1. The first rank-axiomatic definition of a matroid.- 1.2. The independence-axiomatic definition of a matroid.- 1.3. The second rank-axiomatic definition of a matroid.- 1.4. The circuit-axiomatic definition of a matroid.- 1.5. The basis-axiomatic definition of a matroid.- II. Further Properties of Matroids.- 2.1. The span mapping.- 2.2. The span-axiomatic definition of a matroid.- 2.3. Hyperplanes and cocircuits.- 2.4. The dual matroid.- III. Examples.- 3.1. Linear algebraic examples.- 3.2. Binary matroids.- 3.3. Elementary definitions and results from graph theory.- 3.4. Graph-theoretic examples.- 3.5. Combinatorial examples.- IV. Matroids and the Greedy Algorithm.- 4.1. Matroids and the greedy algorithm.- V. Exchange Properties for Bases of Matroids.- 5.1. Symmetric point exchange.- 5.2. Bijective point replacement.- 5.3. More on minors of a matroid.- 5.4. Symmetric set exchange.- 5.5. Bijective set replacement.- 5.6. A further symmetric set exchange property.

Subgradient Optimization, Matroid Problems and Heuristic Evaluation

Abstract: Many polynomial complete problems can be reduced efficiently to three matroids intersection problems. Subgradient methods are shown to yield very good algorithms for computing tight lower bounds to the solution of these problems. The bounds may be used either to construct heuristically guided (branch-and-bound) methods for solving the problems, or to obtain an upper bound to the difference between exact and approx imate solutions by heuristic methods. The existing experience tend to indicate that such bounds would be quite precise.

Matroids and the Greedy Algorithm

Abstract: Let E be a finite set. (1) Let w: E→IR be a function with w(e) ≥ 0 for all e ∈ E. We extend w to a function w: P (E) → IR by setting w(S) := Σw(e) , S⊂E. The function w is called a weighting of E. (2) A family F(E) of subsets of E with the property [S′⊂:S∈F(E) → S′∈F(E)] is called an independence system on E. (3) If P is a family of subsets of E, then the family {S⊂E : ∃ S′∈ P such that S⊂S′} of subsets of E is the independence system F(P) on E generated by P.