Valuated matroids: a new look at the greedy algorithm

Abstract"Convex analysis" is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz's exchange property. It includes linear functions on matroids, valuations on matroids (in the sense of Dress and Wenzel), and separable concave functions on the integral base polytope of submodular systems. It is shown that a function ω has the Steinitz exchange property if and only if it can be extended to a concave function ω such that the maximizers of (ω+any linear function) form an integral base polytope. A Fenchel-type min-max theorem and discrete separation theorems are established which imply, as immediate consequences, Frank's discrete separation theorem for submodular functions, Edmonds' intersection theorem, Fujishige's Fencheltype min-max theorem for submodular functions, and also Frank's weight splitting theorem for weighted matroid intersection.

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“…This is a straightforward extension of a similar result of [5], [8] for a matroid valuation. The analogy to concave functions should be obvious.…”

Section: Theorem 23 ([32 Lemma 24]) For X Y ∈ B We Havementioning

confidence: 79%

Convexity and Steinitz's exchange property

Murota

1996

Integer Programming and Combinatorial Optimization

121

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“…Indeed, the explicit algorithms given in [16] are too complex in practice. We generalise the algorithmic approach to tropical projection developed by Ardila [1], and further by Rincón [23], for ordinary matroids, to the case of valuated matroids, defined by Dress and collaborators [9]. Namely, if p is a valuated matroid of rank n on [m], and L p is the tropical linear space [24] attached to it, then we have the following result.…”

Section: Theoremmentioning

confidence: 99%

“…Since i ∈ B 1 \B , we apply axiom (17) Hence B is locally ω-minimum. According to [9], the global minimum of p ω is obtained by choosing the local minimum at each step. Hence B is also a global minimum of p ω .…”

Section: Moreover B Is Also ω-Minimalmentioning

confidence: 99%

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Gérard–Levelt membranes

Corel

2012

J Algebr Comb

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We present an unexpected application of tropical convexity to the determination of invariants for linear systems of differential equations. We show that the classical Gérard-Levelt lattice saturation procedure can be geometrically understood in terms of a projection on the tropical linear space attached to a subset of the local affine Bruhat-Tits building, which we call the Gérard-Levelt membrane. This provides a way to compute the true Poincaré rank, but also the Katz rank of a meromorphic connection without having to perform either gauge transforms or ramifications of the variable. We finally present an efficient algorithm to compute this tropical projection map, generalising Ardila's method for Bergman fans to the case of the tight-span of a valuated matroid.

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“…where χ v ∈ {0, 1} V is the characteristic vector of v ∈ V. This definition of an M-convex function is motivated by valuated matroids introduced by Dress and Wenzel [4], [5]. Mconvex functions have various desirable properties of "discrete convexity" such as extensibility to ordinary convex functions, conjugacy, duality, etc.…”

Section: Introductionmentioning

confidence: 99%

A Steepest Descent Algorithm for M-Convex Functions on Jump Systems

Murota

Tanaka

2006

IEICE Transactions on Fundamentals of Electronics, Communicatio

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SUMMARYThe concept of M-convex functions has recently been generalized for functions defined on constant-parity jump systems. The bmatching problem and its generalization provide canonical examples of Mconvex functions on jump systems. In this paper, we propose a steepest descent algorithm for minimizing an M-convex function on a constant-parity jump system. key words: jump system, discrete convex function, local optimality, steepest descent algorithm

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Valuated matroids: a new look at the greedy algorithm

Cited by 98 publications

References 4 publications

Convexity and Steinitz's exchange property

Convexity and Steinitz's exchange property

Gérard–Levelt membranes

A Steepest Descent Algorithm for M-Convex Functions on Jump Systems

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