On the Kth Best Base of a Matroid (original) (raw)

On the generality of the greedy algorithm for solving matroid base problems

Discrete Applied Mathematics, 2014

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.

On the number of common bases of two matroids

Discrete Mathematics, 1983

Soient M, = (E, 9,) et M2 = (E, S2) deux matro,iiles simples d&finies S'X le mtme ensemble E. Supposant que M, et Mz ont une base commue B,, il s'agit de trouver une borne inferieure sur le nombre de bases communes B M, et M2. Ce probli5me est approchC par I'htude du polytope intersection de K(M,) et K(M,) oii K(Mil est le polytope associ6 des bases du matroide de Mi ; i = 1,2. Nous donnons quelques applications B certains probli3mes d'optimisation combinatoire. M, = (E, $Is,) and M2 = (E, sz) are two simple matroids on the same set E. We assume that M, and M2 have a common basis and we want to know a least bound for the number of common bases of M, and bf2. To solve this problem, we study the dimension of the intersection of the polytopes R(M,) and K(M,); where K(Mi) is the polytope having as extreme points the representative vectors of the bases of Mi, i = 1,2. Applications of the least number of common bases of two matroids are given for some combinatorial optimisation problems.

Matroid Intersections, Polymatroid Inequalities, and Related Problems

Lecture Notes in Computer Science, 2002

Given m matroids M1, . . . , Mm on the common ground set V , it is shown that all maximal subsets of V , independent in the m matroids, can be generated in quasi-polynomial time. More generally, given a system of polymatroid inequalities f1(X) ≥ t1, . . . , fm(X) ≥ tm with quasi-polynomially bounded right hand sides t1, . . . , tm, all minimal feasible solutions X ⊆ V to the system can be generated in incremental quasi-polynomial time. Our proof of these results is based on a combinatorial inequality for polymatroid functions which may be of independent interest. Precisely, for a polymatroid function f and an integer threshold t ≥ 1, let α = α(f, t) denote the number of maximal sets X ⊆ V satisfying f (X) < t, let β = β(f, t) be the number of minimal sets X ⊆ V for which f (X) ≥ t, and let n = |V |. We show that α ≤ max{n, β (log t)/c }, where c = c(n, β) is the unique positive root of the equation 2 c (n c/ log β − 1) = 1. In particular, our bound implies that α ≤ (nβ) log t . We also give examples of polymatroid functions with arbitrarily large t, n, α and β for which α = β (1−o(1)) log t/c .

Matroid Bases with Cardinality Constraints on the Intersection

ArXiv, 2019

Given two matroids mathcalM1=(E,mathcalB1)\mathcal{M}_{1} = (E, \mathcal{B}_{1})mathcalM1=(E,mathcalB1) and mathcalM2=(E,mathcalB2)\mathcal{M}_{2} = (E, \mathcal{B}_{2})mathcalM2=(E,mathcalB2) on a common ground set EEE with base sets mathcalB1\mathcal{B}_{1}mathcalB1 and mathcalB2\mathcal{B}_{2}mathcalB2, some integer kinmathbbNk \in \mathbb{N}kinmathbbN, and two cost functions c1,c2colonErightarrowmathbbRc_{1}, c_{2} \colon E \rightarrow \mathbb{R}c1,c2colonErightarrowmathbbR, we consider the optimization problem to find a basis XinmathcalB1X \in \mathcal{B}_{1}XinmathcalB1 and a basis YinmathcalB2Y \in \mathcal{B}_{2}YinmathcalB2 minimizing cost sumeinXc1(e)+sumeinYc2(e)\sum_{e\in X} c_1(e)+\sum_{e\in Y} c_2(e)sum_einXc1(e)+sumeinYc_2(e) subject to either a lower bound constraint ∣XcapY∣lek|X \cap Y| \le k∣XcapY∣lek, an upper bound constraint ∣XcapY∣gek|X \cap Y| \ge k∣XcapY∣gek, or an equality constraint ∣XcapY∣=k|X \cap Y| = k∣XcapY∣=k on the size of the intersection of the two bases XXX and YYY. The problem with lower bound constraint turns out to be a generalization of the Recoverable Robust Matroid problem under interval uncertainty representation for which the question for a strongly polynomial-time algorithm was left as an open question by Hradovich et al. We show that the two problems with lower and upp...

A constrained independent set problem for matroids

Operations Research Letters, 2004

In this note, we study a constrained independent set problem for matroids and certain generalizations. The basic problem can be regarded as an ordered version of the matroid parity problem. By a reduction of this problem to matroid intersection, we prove a min-max formula. Studying the weighted case and a delta-matroid generalization, we prove that some of them are not more complex than matroid intersection, but others are as hard as matroid parity. We show how earlier results of Hefner and Kleinschmidt on so called MS-matchings fit in our framework. We also point out another connection to electric networks.

The Maximum Matroid of a Graph

Cornell University - arXiv, 2019

The ground set for all matroids in this paper is the set of all edges of a complete graph. The notion of a maximum matroid for a graph is introduced. The maximum matroid for K 3 (or 3-cycle) is shown to be the cycle (or graphic) matroid. This result is persued in two directions-to determine the maximum matroid for the m-cycle and to determine the maximum matroid for the complete graph K m. While the maximum matroid for an m-cycle is determined for all m ≥ 4, the determination of the maximum matroids for the complete graphs is more complex. The maximum matroid for K 4 is the matroid whose bases are the Laman graphs, related to structural rigidity of frameworks in the plane. The maximum matroid for K 5 is related to a famous 153 year old open problem of J. C. Maxwell. An Algorithm A is provided, whose input is a given graph G and whose output is a matroid M A (G), defined in terms of its closure operator. The matroid M A (G) is proved to be the maximum matroid for G, implying that every graph has a unique maximum matroid.

Minimum partition of a matroid into independent subsets

Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics, 1965

A matroid M is a finit e se t M of e le me nts with a famil y of subsets, calle d inde pe nde nt, s uc h th a t (I) every su bset of an ind e pe nd e nt se t is indepe nde nt, and (2) for e ve ry s ubset A of M , all maximal inde pe nd e nt s ub sets of A have th e sa me ca rdinality , calle d th e rank r\A) o f A. It is proved that a matroid ca n be partitione d into as few as k sets, eac h ind e pe nd e nt , if and o nly if e ve ry s ub se t A has cardinality at mos t k . r(A ).

A note on packing spanning trees in graphs and bases in matroids

Cornell University - arXiv, 2012

We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We provide descriptions of such graphs and matroids, showing that such a graph (or matroid) has a unique decomposition. In the case of graphs, our results are relevant for certain communication protocols.