On the generality of the greedy algorithm for solving matroid base problems (original) (raw)
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On the Generality of the Greedy Algorithm for
It is well known that the greedy algorithm solves matroid base problems for all linear cost functions and is, in fact, correct if and only if the underlying combinatorial structure of the problem is a matroid. Moreover, the algorithm can be applied to problems with sum, bottleneck, algebraic sum or k-sum objective functions. In this paper, we address matroid base problems with a more general-"universal"-objective function which contains the previous ones as special cases. This universal objective function is of the sum type and associates multiplicative weights with the ordered cost coefficients of the elements of matroid bases such that, by choosing appropriate weights, many different-classical and new-objectives can be modeled. We show that the greedy algorithm is applicable to a larger class of objective functions than commonly known and, as such, it solves universal matroid base problems with non-negative or non-positive weight coefficients. Based on problems with mixed weights and a single (−, +)-sign change in the universal weight vector, we give a characterization of uniform matroids. In case of multiple sign changes, we use partition matroids. For non-uniform matroids, single sign change problems can be reduced to problems in minors obtained by deletion and contraction. Finally, we discuss how special instances of universal bipartite matching and shortest
The 2-quasi-greedy algorithm for cardinality constrained matroid bases
Discrete Applied Mathematics, 1986
The quasi-greedy algorithm, as proposed by Glover and Klingman [8], efficiently solves minimum weight spanning tree problems with a fixed (or bounded) number of edges incident to a specified vertex. As observed in [8], the results carry through to general matroid problems (where a base contains a bounded number of elements from a specified set). We extend this work to provide an efficient 2-quasi-greedy algorithm where a minimum weight base is constrained to have a fixed number of elements from two disjoint sets.
The base-matroid and inverse combinatorial optimization problems
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A new matroid is introduced: this matroid is deÿned starting from any matroid and one of its bases, hence we call it base-matroid. Besides some properties of the base-matroid, a non-trivial algorithm for the solution of the related matroid optimization problem is presented. The new matroid has application in the ÿeld of inverse combinatorial optimization problems. We discuss in detail the LP formulation of the inverse matroid optimization problem and we propose an e cient algorithm for computing its primal and dual solutions. ?
Decomposable multi-parameter matroid optimization problems
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A framework for solving certain multi-dimensional parametric matroid optimization problems in randomized linear time by prune-and-search is presented. The common feature of these problems, which include the multi-parameter minimum spanning tree problem on planar and dense graphs, is that their ÿxed-parameter versions are solvable by tournament-like algorithms whose structure is represented by a balanced decomposition tree.
SIAM Journal on Discrete Mathematics, 2003
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On the Connection of Matroids and Greedy Algorithms
IJEER, 2022
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, 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. 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. Additionally, based on the new established relation, the generalized rough set also provides a theory of poset matroid.
Parametric nonlinear discrete optimization over well-described sets and matroid intersections
Mathematical Programming, 2010
We address optimization of parametric nonlinear functions of the form 1 f (W x), where f : R d → R is a nonlinear function, W is a d × n matrix, and feasible 2 x are in some large finite set F of integer points in R n . Generally, such problems 3 are intractable, so we obtain positive algorithmic results by looking at broad natural 4 classes of f, W and F. One of our main motivations is multi-objective discrete opti-5 mization, where f trades off the linear functions given by the rows of W . Another 6 motivation is that we want to extend as much as possible the known results about 7 polynomial-time linear optimization over trees, assignments, matroids, polymatroids, 8
Lecture Notes in Computer Science, 2014
This article introduces the lazy matroid problem, which captures the goal of saving time or money in certain task selection scenarios. We are given a budget B and a matroid M with weights on its elements. The problem consists in finding an independent set F of minimum weight. In addition, F is feasible if its augmentation with any new element x implies that either F + x exceeds B or F + x is dependent. Our first result is a polynomial time approximation scheme for this NPhard problem which generalizes a recently studied version of the lazy bureaucrat problem. We next study the approximability of a more general setting called lazy staff matroid. In this generalization, every element of M has a multidimensional weight. We show that approximating this generalization is much harder than for the lazy matroid problem since it includes the independent dominating set problem.