Abstract
A long-standing problem in algorithm design has been to characterize the class of problems for which greedy algorithms exist. Many greedy problems can be described using algebraic structures called matroids, which were later generalized to greedoids. Once in this form, the original problem can be solved using Edmonds’ Greedy Algorithm. However there are several practical problems with greedy solutions that either do not have a greedoid representation (e.g. Activity Selection) or for which none is known (e.g. Huffman Coding). This paper presents a new characterization of greedy problems that is strictly more general than greedoids, in that it includes all greedoids, as well as problems such as Activity Selection and Huffman Coding. Unlike matroids, our characterization is an axiomatization of a form of Branch and Bound Search, where greediness is associated with the existence of an appropriate dominance relation. Starting from a definition of optimality of the required solution we derive a recurrence relation. This recurrence can then be transformed into a correct-by-construction program that solves problems in our greedy class, analogous to the Greedy Algorithm.
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Nedunuri, S., Smith, D.R., Cook, W.R. (2010). A Class of Greedy Algorithms and Its Relation to Greedoids. In: Cavalcanti, A., Deharbe, D., Gaudel, MC., Woodcock, J. (eds) Theoretical Aspects of Computing – ICTAC 2010. ICTAC 2010. Lecture Notes in Computer Science, vol 6255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14808-8_24
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DOI: https://doi.org/10.1007/978-3-642-14808-8_24
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