New results on knapsack problems
Abstract
This dissertation is composed of three self-contained essays on unbounded knapsack problems. Chapter 1 derives sharp bounds on turnpike theorems for the unbounded knapsack problem. Turnpike theorems specify when it is optimal to load at least one unit of the best item (i.e., the one with the highest "bang-for-buck" ratio) and, thus can be used for problem preprocessing. The successive application of the turnpike theorems can drastically reduce the size of the knapsack problems to be solved. Two theorems subsume known results as special cases. The third one is an entirely different result. It is proved that all three theorems specify sharp bounds in the sense that no smaller bounds can be found under the assumed conditions. It is also shown that two of the bounds can be obtained in constant time. Computational results on randomly generated problems demonstrate the effectiveness of the turnpike theorems both in terms of how often they can be applied and the resulting reduction in the size of the knapsack problems. Chapter 2 considers loading not only the best but also the second best item to achieve even smaller bounds. The new theorems are based upon a classical result on the Frobenius number. Chapter 3 introduces a "greedy bound" that exists in all coin systems. It then proposes a partial greedy method to solve the generalized changing-making problem which minimizes the total cost of using a set of coins to make a given amount of change. The resulting time complexity is bounded only by the number of denominations and the largest denomination of the coin system, not the change amount. It is further demonstrated that the greedy bound can be used in testing whether a coin system is canonical (i.e., the greedy algorithm always produces an optimal representation for any change amount). It is also shown that the greedy bound may outperform existing bounds reported in the literature. Finally, an algebraic structure of compositions/decompositions is presented to provide a novel and efficient method to analyze large coin systems.
Degree
Ph.D.
Advisors
Morin, Purdue University.
Subject Area
Management|Operations research
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