Balanced matroids

Feder, Tomás; Mihail, Milena

doi:10.1145/129712.129716

Cited by 141 publications

(174 citation statements)

References 17 publications

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“…There are many important examples of negatively dependent "repelling" random variables in probability theory, combinatorics, stochastic processes and statistical mechanics: uniform random spanning tree measures [16], symmetric exclusion processes [52,55], random cluster models (with q < 1) [33,42,65], balanced and Rayleigh matroids [20,29,68,70,72], competing urns models [26], etc; see, e.g., [45,65] and the references therein for a discussion of some of these examples and several others. To add to this list, in §3 we show that both the inequalities characterizing multi-affine real stable polynomials [8,9,15] and Hadamard-FischerKotelyansky type inequalities in matrix theory [28,40,39] may in fact be viewed as natural manifestations of negative dependence properties.…”

Section: F Dµ Gdµ ≤ F Gdµmentioning

confidence: 99%

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Negative dependence and the geometry of polynomials

Borcea

Brändén

Liggett

2008

J. Amer. Math. Soc.

205

312

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We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures and uniform random spanning tree measures) and distributions for symmetric exclusion processes. We show that strongly Rayleigh measures enjoy all virtues of negative dependence, and we also prove a series of conjectures due to Liggett, Pemantle, and Wagner, respectively. Moreover, we extend Lyons’ recent results on determinantal measures, and we construct counterexamples to several conjectures of Pemantle and Wagner on negative dependence and ultra log-concave rank sequences.

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Section: F Dµ Gdµ ≤ F Gdµmentioning

confidence: 99%

“…It was first considered for uniform measures on the set of bases of a matroid [20] as a strengthening of weaker notions studied in [29,69]. The following properties of Rayleigh polynomials are consequences of Definition 2.5.…”

Section: Definition 23mentioning

confidence: 99%

Negative dependence and the geometry of polynomials

Borcea

Brändén

Liggett

2008

J. Amer. Math. Soc.

205

312

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“…We use that to prove Corollaries 5.17 and 5. 19. But before that, we state the following technical lemma whose proof is deferred to the appendix.…”

Section: Cut Classes With Inside Atomsmentioning

confidence: 99%

“…Feder and Mihail [19] proved that uniform measures on balanced matroids (and in particular on spanning trees) have negative association. Borcea et al in [8] proved that the strongly Rayleigh measure satisfy the strongest form of negative association known as CNA+ which we do not describe here.…”

Section: Random Spanning Trees and Strongly Rayleigh Measuresmentioning

confidence: 99%

A Randomized Rounding Approach to the Traveling Salesman Problem

Gharan

Saberi

Singh

2011

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

115

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For some positive constant 0 , we give a ( 3 2 − 0 )-approximation algorithm for the following problem: given a graph G 0 = (V, E 0 ), find the shortest tour that visits every vertex at least once. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in G 0 . The result improves on the 3 2 -approximation algorithm due to Christofides [13] for this special case.Similar to Christofides, our algorithm finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or T-join) of that tree. The main difference is in the selection of the spanning tree. Except in certain cases where the solution of LP is nearly integral, we select the spanning tree randomly by sampling from a maximum entropy distribution defined by the linear programming relaxation.Despite the simplicity of the algorithm, the analysis builds on a variety of ideas such as properties of strongly Rayleigh measures from probability theory, graph theoretical results on the structure of near minimum cuts, and the integrality of the T-join polytope from polyhedral theory. Also, as a byproduct of our result, we show new properties of the near minimum cuts of any graph, which may be of independent interest.

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“…Additionally, improvements can be achieved in the following static problems (see [4,3]): randomly sampling spanning forests of a given graph [6] and finding a color-constrained minimum spanning tree [8].…”

Section: Introductionmentioning

confidence: 99%

Maintaining Minimum Spanning Forests in Dynamic Graphs

Henzinger

King

2001

SIAM J. Comput.

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Abstract. We present the first fully dynamic algorithm for maintaining a minimum spanning forest in time o( √ n) per operation. To be precise, the algorithm uses O(n 1/3 log n) amortized time per update operation. The algorithm is fairly simple and deterministic. An immediate consequence is the first fully dynamic deterministic algorithm for maintaining connectivity and bipartiteness in amortized time O(n 1/3 log n) per update, with O(1) worst case time per query.

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Balanced matroids

Abstract: DalanceaCenter CA 95120We introduce the notion of 'balance" , and say that a rnatroid is balanced if the matroid and all its minors satisfy the property that, for a randomly chosen baais, the presence of an element can only make any other element less likely.

Cited by 141 publications

References 17 publications

Negative dependence and the geometry of polynomials

Negative dependence and the geometry of polynomials

A Randomized Rounding Approach to the Traveling Salesman Problem

Maintaining Minimum Spanning Forests in Dynamic Graphs

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