On the Connection of Matroids and Greedy Algorithms

Abstract

Matroids are the combinatorial structure and Greedy algorithmic methods always produces optimal solutions for these mathematical models. A greedy method always selects the option that looks best at each step of process of finding optimal solution. In other words, it selects a choice which is optimal choice locally in such a strategy that this locally chosen option may direct to a solution that will be globally optimal. It is true that while selecting locally optimal solution at each stage, Greedy algorithms may not always yield optimal solutions [1-2], but if we can transform an unknown problem into matroid structure, then there must be a greedy algorithm that will always lead optimal solution for that unknown problem. The range of solutions provided by Greedy is large as compared to the applicability of the Matroid structure. In other words the problems that can be translated into Matroid structure is proper subset of set of all problems whether Greedy algorithm produces optimal solution. Matroid structure thus ensures the global optimal solution one can obtain with help of Greedy approach. We study various logarithmic and linear hierarchical based mathematical models from divergence sources to maximize our information for research purposes. We analyze the time complexity and provide constrains over the upper/lower bounds in correspondence with the optimal (maximum/minimum) solution. We try to establish the relationship between the maximization of information divergences, the optimal-likelihood theory, and classified sharing is instituted. We propose integration of unknown rough sets to matroids in this paper. Particularly, we devise methodically the upper and lower tightening bounds on rough matroids which may expand up to the generic combinatorial matroid structure. The relationships are established by the upper and lower tightening bounds approximations of generalized combinatorial rough sets based on different interdependent relation sets, respectively [4]. As we define the generalized lower/upper bounds for rough matroid, we define a new structure for lower/upper greedoid leading to generalization of the greedoid [5]. Additionally, based on the new established relation, the generalized rough set also provides a theory of poset matroid.