Matroids And Greedy Algorithms. A Deeper Justification of Using Greedy Approach To Find A Maximal set of a Matroid

Syed Aqib Haider

doi:10.17951/ai.2016.16.2.48

Abstract

<p>Greedy algorithms are used in solving a diverse set of problems in small computation time. However, for solving problems using greedy approach, it must be proved that the greedy strategy applies. The greedy approach relies on selection of optimal choice at a local level reducing the problem to a single sub problem, which actually leads to a globally optimal solution. Finding a maximal set from the independent set of a matroid M(S, I) also uses greedy approach and justification is also provided in standard literature (e.g. Introduction to Algorithms by Cormen et .al.). However, the justification does not clearly explain the equivalence of using greedy algorithm and contraction of M by the selected element. This paper thus attempts to give a lucid explanation of the fact that the greedy algorithm is equivalent to reducing the Matroid into its contraction by selected element. This approach also provides motivation for research on the selection of the test used in algorithm which might lead to smaller computation time of the algorithm.</p>

Highlights

THE MATROID THEORY AND GREEDY APPROACHThis section will given formal definition of matroid theory, the problem of finding a maximal set and the greedy algorithm that is used for solving it
Introduction to AlgorithmsThird Edition.” Cormen,Leiserson,Rivest and Stein.[2] B
Let the loop run N time and let xk denote element selected at the kth iteration such that optimal set formed is A = {x1 x2.. xN} xn is selected if AU {xn} ε I, where A = {x1, x2..xn-1} Assuming that till (k-1)th iteration, all elements selected were partt of corresponding contraction, i.e. {xi-1,xi} ε Ii-1 for all i = 2. k-1 for i =2 A = {x1} ⇒A U {x1} ε I if and only if {x1,x2} ∈ I

Summary

THE MATROID THEORY AND GREEDY APPROACH

This section will given formal definition of matroid theory, the problem of finding a maximal set and the greedy algorithm that is used for solving it. Matroid: A matroid M (S, I) is an ordered pair of two sets S (which must be finite) and I if and only if I ≠ ɸ and I is a nonempty set of some subsets of S such that if. 3.For each element of x ∈ S if A ∪ {x} ∈ I A = A ∪ {x} 4.return A The above algorithm return an optimal solution. Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.pl Data: 08/11/2021 13:23:02

VALIDITY OF MAXWEIGHT

EQUIVALENCE OF MAXWEIGHT AND CONTRACTION OF MATROID

CONCLUSION