Showing papers on "Greedy algorithm published in 1982"

Worst-Case Analysis of Greedy Heuristics for Integer Programming with Nonnegative Data

Abstract: We give a worst-case analysis for two greedy heuristics for the integer programming problem minimize cx, Ax ≥ b, 0 ≤ x ≤ u, x integer, where the entries in A, b, and c are all nonnegative. The first heuristic is for the case where the entries in A and b are integral, the second only assumes the rows are scaled so that the smallest nonzero entry is at least 1. In both cases we compare the ratio of the value of the greedy solution to that of the integer optimal. The error bound grows logarithmically in the maximum column sum of A for both heuristics.

Maximising Real-Valued Submodular Functions: Primal and Dual Heuristics for Location Problems

Abstract: Recently it has been shown how various results on the behaviour of exact and approximate primal algorithms for certain discrete location problems can be obtained by studying the more general problem of maximising a submodular set function Unfortunately, this more general model tells us little about an important aspect of the discrete location problems, their linear programming relaxations and the possible use of dual solutions in obtaining bounds Here we examine a slightly more specialised model that again generalises the location problems of interest, but now also includes the continuous aspects of the problems missing from the earlier model, namely the problem: max{wy: Σj=1najyj ≤ b, 0 ≤ yj < 1} of maximising a real-valued nondecreasing piecewise linear, concave submodular function subject to a knapsack constraint We show that a continuous greedy heuristic always attains at least 1-e-1 × 100% of the optimal value, and that for the discrete problem an adapted greedy heuristic always attains 35% of the optimal value Specialised to the capacitated and uncapacitated location problems these results permit us to strengthen earlier results, and to obtain new results on dual greedy heuristics

On the Greedy Heuristic for Continuous Covering and Packing Problems

Abstract: Worst-case bounds are given on the performance of the greedy heuristic for a continuous version of the set covering problem. This generalizes results of Chvatal, Johnson and Lovasz for the 0-1 covering problem. The results for the greedy heuristic and for other heuristics are obtained by treating the covering problem as a limiting case of a generalized location problem for which worst-case results are known. An alternative approach involving dual greedy heuristics leads also to worst-case bounds for continuous packing problems.

Worst case analysis of a class of set covering heuristics

Abstract: In [2], Chvatal provided the tight worst case bound of the set covering greedy heuristic. We considered a general class of greedy type set covering heuristics. Their worst case bounds are dominated by that of the greedy heuristic.

A Class of Train-Scheduling Problems

Abstract: We relate the question of determining stop-schedules for trains that deliver traffic on a line of stations to the maximization of submodular set functions. We prove that a greedy algorithm is optimal under certain assumptions and report computational experience on the performance of the greedy heuristic as compared to the exact algorithm when the optimality of the greedy heuristic cannot be guaranteed.

A Minimum-Impact Routing Algorithm

Abstract: A routing heuristic is presented that routes two-terminal nets one at a time, for each net choosing the path so as to avoid adversely impacting the nets not yet routed. An algorithm is presented and proved to correctly implement this heuristic; the computational complexity of that algorithm is shown to be polynomially bounded, but perhaps still too great to be of practical use. Another, speedier algorithm is presented that seems to approximate the heuristic rather closely. Strong evidence is given that the Lee routing algorithm is in some sense inadequate to implement this heuristic. The heuristic has been applied, with very encouraging results, to a specific routing problem: the routing of a channel in which all four sides of the channel may contain terminals. This problem arises in the layout of custom VLSI.

Probabilistic analysis of the solution of the knapsack problem

Abstract: In this paper the value of the optimal solution of a random instance of the Knapsack problem is analyzed. With respect to this value, the performance of a simple greedy heuristic for the solution of this problem is evaluated. The results are compared with the performance of other greedy heuristics.

A new extension principle algorithm for the traveling salesman problem

Abstract: In this paper, an algorithm for solving the asymmetric traveling salesman problem is developed and tested by computation. This algorithm is based on the extension principle by Schoch and uses the assignment problem relaxation of the traveling salesman problem for computing lower bounds. Computational experience with randomly generated test problems indicate that the present algorithm yields good results in runtime which are comparable with the results of Smith/Srinivasan/Thompson. Computational experience are reported for up to 120-node problems with uniformly distributed and approximately normally distributed cost.

On the greedy heuristic for continuous covering and packing problems

Abstract: Worst-case bounds are given on the performance of the greedy heuristic for a continuous version of the set covering problem. This generalizes results of Chvatal, Johnson and Lovasz for the 0-1 covering problem. The results for the greedy heuristic and for other heuristics are obtained by treating the covering problem as a limiting case of a generalized location problem for which worst-case results are known. An alternative approach involving dual greedy heuristics leads also to worst-case bounds for continuous packing problems.

An accelerated experimental design algorithm

Abstract: An efficient algorithm is presented for generating D-optimal designs. The usual sequential D-optimal design algorithm embodies the principle of the greedy algorithm of combinatorial optimization. It is shown that a sufficient condition for applying the accelerated greedy algorithm of M. Minoux to the design problem is satisfied. The actual implementation of the accelerated sequential design algorithm is based on a more general sufficient condition. This allows the evaluation of quadratic forms to replace determinant evaluations. A heap type data structure provides additional efficiency. While the standard sequential design algorithm requires a number of basis function evaluations proportional to the number of iterations, the accelerated design algorithm computation is proportional to a much smaller sum of coefficients.

A characterization of the simplex

Abstract: In the vein of Rado's beautiful characterization of matroids by the functioning of the Greedy Algorithm we prove that our persistent frustrations in obtaining analogues and generalizations of Shapiro's Lemma had a good reason

Generalized augmenting paths for the solution of combinatorial optimization problems

Abstract: Alternating chain procedures can be thought of as generalizations of the greedy algorithm in that instead of accepting the best remaining element, they seek to obtain a better augmentation by examining a wider range of alternatives. It is possible to generalize the notion of an augmenting sequence to include augmentations which are in effect trees as opposed to simply paths such that these augmentations are sufficient to guarantee optimality. Unfortunately, in the worst case, these trees are of exponential size. We examine the application of such generalized augmenting sequences to the solution of NP-complete problems and examine their effectiveness and efficiency.