Showing papers on "Greedy algorithm published in 1981"

On a greedy heuristic for complete matching

Abstract: Finding a minimum weighted complete matching on a set of vertices in which the distances satisfy the triangle inequality is of general interest and of particular importance when drawing graphs on a mechanical plotter. The “greedy” heuristic of repeatedly matching the two closest unmatched points can be implemented in worst-case time $O(n^2 \log n)$, a reasonable savings compared to the general minimum weighted matching algorithm which requires time proportional to $n^3 $ to find the minimum cost matching in a weighted graph. We show that, for an even number n of vertices whose distances satisfy the triangle inequality, the ratio of the cost of the matching produced by this greedy heuristic to the cost of the minimal matching is at most ${}_3^4 n^{\lg _2^3 } - 1$, $\lg _2^3 \approx 0.58496$, and there are examples that achieve this bound. We conclude that this greedy heuristic, although desirable because of its simplicity, would be a poor choice for this problem.

Analysis of Greedy Solutions for a Replacement Part Sequencing Problem

Abstract: This paper considers the problem of sequencing the installation of replacement parts for the repeated repair of a machine consisting of two working parts. A finite inventory of spares is initially available for each type of part. Each spare has a known deterministic field life, and spares of each type may have different field lives. The inventories are eventually depleted since no replacement or repairs of parts occur. The primary objective is to obtain a sequencing policy which minimizes the total number of installations replacements. This problem is a special case of many opportunistic replacement problems and is called a “build” problem. The motivation for this work is based on the maintenance of modular gas turbine aircraft engines used by domestic airlines and the U.S. Air Force. We show this build problem is NP-hard which indicates the necessity of efficient heuristic solution procedures. This paper considers several “greedy” type algorithms based on grouping parts according to exact matching of field lives. Then, a generalization is presented which allows approximate matches. The focus of the analysis is to obtain tight worst case bounds on the number of replacements obtained by each of the greedy algorithms.

Analysis of a heuristic for full trie minimization

Abstract: A trie is a distributed-key search tree in which records from a file correspond to leaves in the tree. Retrieval consists of following a path from one root to a leaf, where the choice of edge at each node is determined by attribute values of the key. For full tries, those in which all leaves lie at the same depth, the problem of finding an ordering of attributes which yields a minimum size trie is NP-complete.This paper considers a “greedy” heuristic for constructing low-cost tries. It presents simulation experiments which show that the greedy method tends to produce tries with small size, and analysis leading to a worst case bound on approximations produced by the heuristic. It also shows a class of files for which the greedy method may perform badly, producing tries of high cost.

Optimization problems on graphs with independent random edge weights

Abstract: Optimization problems on complete graphs with edge weights drawn independently, from a fixed distribution, are considered. Several methods for analyzing these problems are discussed, including greedy methods, applications of Boole’s inequality, and exploitation of relationships with results about random unweighted graphs. These techniques are illustrated in the case in which the edge weights are drawn from a normal distribution; in particular, we investigate the expected behavior of the minimum weight clique on k vertices. We describe the asymptotic behavior (in probability and/or almost surely) of the random variable which describes the optimum; we also discuss the asymptotic behavior of its mean. Finally techniques are demonstrated by which we may determine an asymptotic description of the behavior of a greedy algorithm for this problem.

Technical Note—Analysis of a Preference Order Traveling Salesman Problem

Abstract: Application is made to the preference order dynamic programming solution procedure proposed by Kao for a stochastic traveling salesman problem. Although the procedure is flawed from the myopic interpretation of the monotonicity condition, it may be used as a convenient heuristic tool for solving stochastic problems.

New algorithms for deterministic and stochastic vehicle routing problems

Abstract: In this dissertation, a comprehensive approach to stochastic vehicle routing is developed. Effective solution procedures for various forms of this complex problem are proposed and computationally tested. These solution procedures are adapted from solution procedures that are effective for related problems such as the Traveling Salesman Problem and the Vehicle Routing Problem. In general, given a set of points on a graph (locations on a map) the problem is to route one or more vehicles in such a way that each point and the total distance traveled is minimized. When there is one vehicle which must visit every location and return to its start location, the problem is called the Traveling Salesman Problem. This problem has been widely studied and has many applications in the transportation and scheduling fields. When there are multiple vehicles, each with a specific capacity, and a portion of this capacity is demanded at each point visited (e.g., delivery trucks), the problem is called the Vehicle Routing Problem. Such diverse operations as garbage collection, school busing, small package air mail service, along with a host of commercial distribution systems (e.g., milk, petroleum, etc.) fit this description and have benefited from algorithms designed to efficiently route the vehicles involved. When the demands at the various points are random variables whose values only become known when the vehicle serving that point arrives, the problem is a Stochastic Vehicle Routing Problem and must be treated differently than a deterministic problem above. In stochastic vehicle routing, the problem is to design a route system which has a short overall distance, but at the same time meets the demands on each route with regularity. If too many demand points are placed on one route, the vehicle assigned to that route will often be too small to meet all the demands. When this occurs, either some of the customers will not be served, or a special expense will be incurred in finishing the failed route. Both of these alternatives involve a cost which must be considered when designing the route system. Before specifically addressing the Stochastic Vehicle Routing Problem, this dissertation presents two efficient algorithms for the Traveling Salesman Problem. These algorithms are described in detail and then tested on a set of problems from the literature. The new algorithms produce better solutions to these test problems than several similar heuristic algorithms that have been proposed by other authors. After discussing these algorithms for the Traveling Salesman Problem, a new heuristic algorithm for deterministic vehicle routing is presented. It is based on an effective algorithm for the Traveling Salesman Problem, and has generated the best known solutions to several test problems in the literature. Having presented these new algorithms, the Stochastic Vehicle Routing Problem is formulated in two different forms, a chance constrained model and a penalty function model. These models are developed and tested computationally. The new vehicle routing algorithm presented previously is adapted to handle stochastic demands and is used to generate solutions to some test problems. The final phase of this dissertation involves adapting the stochastic vehicle routing models to the Subscriber Bus Routing Problem. This problem arises when a subscriber bus system in which customers sign up in advance for bus service to and from some large employment center is used as an alternative to the personal car. Such systems have arisen as a result of the recent escalation of automobile and gasoline costs.

Worst case bounds for the Euclidean matching problem

Abstract: It is shown how the classical mathematical theory of sphere packing can be used to obtain bounds for a greedy heuristic for the bounded euclidean matching problem. In the case of 2 dimensions, bounds are obtained directly. For higher dimensions, an appeal is made to known bounds for the sphere packing problem that have appeared in the mathematical literature.

A Greedy Heuristic for Single Machine Sequencing with Precedence Constraints

Abstract: We present a greedy heuristic for the n job/1 machine scheduling problem with precedence constraints. This method is useful whenever the manager's optimization criteria is the sum of weighted or unweighted completion times, the sum of weighted or unweighted flow times, with or without release dates, the sum of weighted or unweighted working times, the sum of weighted or unweighted lateness, average completion time, average flow time, average waiting time or average lateness. The greedy heuristic found the optimal solution for 58 of 68 test problems for which a branch and bound method was used to find the optimal solution. The heuristic is, of course, much easier to implement and executes in less time. The greedy heuristic fared well in comparison with a simple myopic heuristic presented by Morton and Dharan Morton. T. E., B. G. Dharan. 1978. Algoristics for single machine sequencing with precedence constraints. Management Sci.24 10 1011-1020..

Exponential Lower Bounds on a Class of Knapsack Algorithms

Abstract: The 0-1 knapsack problem is a well known NP-hard optimization problem. The proof of the generally believed hypothesis that there is no polynomial optimization algorithm for this problem would settle the famous NP ≠ P question. We show here the nonexistence of a polynomial knapsack algorithm within the class of algorithms having restricted access to the input. Algorithms for the problem max{cS: aS ≤ b, S ⊆ En} that can only check the sign of aS-b cannot have a better performance guarantee than the greedy algorithm; furthermore even if they can exploit the order of the densities ci/ai, 1 ≤ i ≤ n, or of the weights aS, S ⊆ En, they cannot be guaranteed to find the optimum.

The greedy algorithm as a combinatorial principle.

Abstract: One of the best known results of combinatorial matching theory is Hall’s “marriage theorem”. In fact, matching theory may be based on this theorem (cf. [5]). It can either be proved directly or derived from stronger theories, e.g., the theory of flows in networks [4] or the theory of polyhedral matroids [2]. The latter theories are both usually seen as manifestations of the duality principle in linear programming — an “explanation” which is not very satisfactory from a purely combinatorial point of view. In this note, we want to give an outline how a combinatorial theory including, in particular, matching theory may be based on a very simple combinatorial principle. This principle states that, under certain restrictions, an optimal combinatorial object can be constructed in a straight-forward manner, namely by the “greedy algorithm”. It seems to be an open problem to give a definition of “combinatorics” which everyone agrees upon. For our purposes, the following standpoint is appropriate: combinatorics is the study of the processes involved in building up a combinatorial object step by step so that certain requirements are met (cf. [8]). What we study here are the implications if we know that an optimal combinatorial object can be obtained by taking the greedy algorithm as our rule of construction.