Greed and Majorization

We present a straightforward linear algebraic model of greed, based only on extensions of classical majorization and convexity theory. This gives an alternative to other models of greedy-solvable problems such as matroids, greedoids, submodular functions, etc., and it is able to express established examples of greedy-solvable optimization problems that they cannot. The linear algebraic approach is also much closer in spirit to established practice in operations research and numerical optimization. The essence of the approach is to modeìexchanges' with certain linear transformations. Modeling solutions as vectors also, we then exploit the fact that these exchanges deene an ordering on solutions. When the exchanges are doubly-stochastic matrices, this ordering is the majorization ordering developed by Hardy, Littlewood, and PP olya in their pioneering work on inequality theory. We generalize majoriza-tion to permit any matrix semigroup of exchanges, but nd that several kinds of stochastic matrices make particularly useful families of exchanges. We also show that greedy-solvable problems can be formalized as optimization problems in which the objective preserves an exchange ordering (i.e., is monotone with respect to it). We outline greedy algorithms for such problems, including those that exploit additional properties of the objective, such as convexity or submodularity. Examples from the literature illustrate how several well-known applications of greed can be expressed.