Facial structures of lattice path matroid polytopes

A lattice path matroid is a transversal matroid corresponding to a pair of lattice paths on the plane. A matroid base polytope is the polytope whose vertices are the incidence vectors of the bases of the given matroid. In this paper, we study the facial structures of matroid base polytopes corresponding to lattice path matroids. In the case of a border strip, we show that all faces of a lattice path matroid polytope can be described by certain subsets of deletions, contractions, and direct sums. In particular, we express them in terms of the lattice path obtained from the border strip. Subsequently, the facial structures of a lattice path matroid polytope for a general case are explained in terms of certain tilings of skew shapes inside the given region.     

Solving the linear matroid parity problem as a sequence of matroid intersection problems

In this paper, we present an O(r ⁴ n) algorithm for the linear matroid parity problem. Our solution technique is to introduce a modest generalization, the non-simple parity problem, and identify an important subclass of non-simple parity problems called easy parity problems which can be solved as matroid intersection problems. We then show how to solve any linear matroid parity problem parametrically as a sequence of easy parity problems.In contrast to other algorithmic work on this problem, we focus on general structural properties of dual solutions rather than on local primal structures. In a companion paper, we develop these ideas into a duality theory for the parity problem.     

Extended formulations for matroid polytopes through randomized protocols

The hitting number of a polytope P is the smallest size of a subset of vertices of P such that every facet of P has a vertex in the subset. We show that, if P is the base polytope of any matroid, then P admits an extended formulation of linear size on the hitting number of P. Our results generalize those of the spanning tree polytope given by Martin and Wong, and extend to polymatroids.     

A new algorithm for the intersection of a line with the independent set polytope of a matroid

We present a new algorithm for the problem of determining the intersection of a half-line with the independent set polytope of a matroid. We show it can also be used to compute the strength of a graph and the corresponding partition using successive contractions. The algorithm is based on the maximization of successive linear forms on the boundary of the polytope. We prove it is a polynomial algorithm in probability with average number of iterations in O(n⁵). Finally, numerical tests reveal that it should only require O(n²) iterations in practice.     

An algorithm for generalized fractional programs

An algorithm is suggested that finds the constrained minimum of the maximum of finitely many ratios. The method involves a sequence of linear (convex) subproblems if the ratios are linear (convex-concave). Convergence results as well as rate of convergence results are derived. Special consideration is given to the case of (a) compact feasible regions and (b) linear ratios.The research of S. Schaible was supported by Grant Nos. A4534 and A5408 from NSERC. The authors thank two anonymous referees for their helpful remarks.     

Polynomials with the half-plane property and matroid theory

A polynomial f is said to have the half-plane property if there is an open half-plane H⊂C, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions relating multivariate polynomials with the half-plane property to matroid theory.

(1)

We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous.

(2)

We prove that a multivariate multi-affine polynomial f∈R[z₁,…,zn] has the half-plane property (with respect to the upper half-plane) if and only if

     

A new formula for an evaluation of the Tutte polynomial of a matroid

Given a matroid M and its Tutte polynomial T_M(x,y), T_M(0,1) is an invariant of M with various interesting combinatorial and topological interpretations. Being a Tutte–Grothendieck invariant, T_M(0,1) may be computed via deletion–contraction recursions. In this note we derive a new recursion formula for this invariant that involves contractions of M through the circuits containing a fixed element of M.     

Random sampling and greedy sparsification for matroid optimization problems

Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems on matroids: finding an optimum matroid basis and packing disjoint matroid bases. Application of these ideas to the graphic matroid led to fast algorithms for minimum spanning trees and minimum cuts. An optimum matroid basis is typically found by agreedy algorithm that grows an independent set into an optimum basis one element at a time. This continuous change in the independent set can make it hard to perform the independence tests needed by the greedy algorithm. We simplify matters by using sampling to reduce the problem of finding an optimum matroid basis to the problem of verifying that a givenfixed basis is optimum, showing that the two problems can be solved in roughly the same time. Another application of sampling is to packing matroid bases, also known as matroid partitioning. Sampling reduces the number of bases that must be packed. We combine sampling with a greedy packing strategy that reduces the size of the matroid. Together, these techniques give accelerated packing algorithms. We give particular attention to the problem of packing spanning trees in graphs, which has applications in network reliability analysis. Our results can be seen as generalizing certain results from random graph theory. The techniques have also been effective for other packing problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Some of this work done at Stanford University, supported by National Science Foundation and Hertz Foundation Graduate Fellowships, and NSF Young Investigator Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation and Xerox Corporation. Also supported by NSF award 962-4239.     

A constrained independent set problem for matroids

In this note, we study a constrained independent set problem for matroids. The 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. We show how earlier results of Hefner and Kleinschmidt on the so-called MS-matchings fit in our framework.     

Matroid matching with Dilworth truncation

Let H=(V,E) be a hypergraph and let k?1 and l?0 be fixed integers. Let M be the matroid with ground-set E s.t. a set F⊆E is independent if and only if each X⊆V with k|X|-l?0 spans at most k|X|-l hyperedges of F. We prove that if H is dense enough, then M satisfies the double circuit property, thus Lovász’ min-max formula on the maximum matroid matching holds for M. Our result implies the Berge-Tutte formula on the maximum matching of graphs (k=1, l=0), generalizes Lovász’ graphic matroid (cycle matroid) matching formula to hypergraphs (k=l=1) and gives a min-max formula for the maximum matroid matching in the two-dimensional rigidity matroid (k=2, l=3).     

Fenchel-type duality for matroid valuations

The weighted matroid intersection problem has recently been extended to the valuated matroid intersection problem: Given a pair of valuated matroidsM _i = (V, _i , _i ) (i = 1,2) defined on a common ground setV, find a common baseB _{1 2} that maximizes ₁ (B) + ₂ (B). This paper develops a Fenchel-type duality theory related to this problem with a view to establishing a convexity framework for nonlinear integer programming. A Fenchel-type min max theorem and a discrete separation theorem are given. Furthermore, the subdifferentials of matroid valuations are investigated. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Part of this paper has been presented at fifth SIAM Conference on Optimization, Victoria, May 1996.This work was done while the author was at Forschungsinstitut für Diskrete Mathematik, Universität Bonn, 1994–1995.     

The perfect matching polytope and solid bricks

The perfect matching polytope of a graph G is the convex hull of the set of incidence vectors of perfect matchings of G. Edmonds (J. Res. Nat. Bur. Standards Sect. B 69B 1965 125) showed that a vector x in Q^E belongs to the perfect matching polytope of G if and only if it satisfies the inequalities: (i) x0 (non-negativity), (ii) x(∂(v))=1, for all vV (degree constraints) and (iii) x(∂(S))1, for all odd subsets S of V (odd set constraints). In this paper, we characterize graphs whose perfect matching polytopes are determined by non-negativity and the degree constraints. We also present a proof of a recent theorem of Reed and Wakabayashi.     

An efficient heuristic algorithm for minimum matching

We give a new heuristic algorithm for minimum matching problems and apply it to the Euclidean problem with random vertices in 2 dimensions. The algorithm is based on simulated annealing and performs in practice faster than previous heuristic algorithms yielding suboptimal solutions of the same good quality. From configurations with up toN=20.000 vertices in the unit square we estimate that the length of a minimum matching scales asymptotically asLN with (=0.3123±0.0016.

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Zusammenfassung Wir stellen einen neuen heuristischen Algorithmus für minimale Matching-Probleme vor und wenden diesen auf das euklidische Problem mit zufÄlliger Punkteverteilung in 2 Dimensionen an. Auf Simulated Annealing basierend lÄuft der Algorithmus schneller als frühere heuristische Algorithmen und erreicht dabei suboptimale Lösungen gleich guter QualitÄt. Aus Konfigurationen mit bis zuN=20.000 Punkten im Einheitsquadrat schÄtzen wir, da\ für die LÄnge des minimalen Matchings asymptotischLN mit=0.3123±0.0016 gilt.

     

A matroid-friendly basis for the quasisymmetric functions

A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the quasisymmetric functions associated to matroids by the Hopf algebra morphism F of Billera, Jia, and Reiner [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646]. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F distinguishes isomorphism classes of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer to the Hilbert basis question raised in [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646] is given.     

Sharp weighted bounds for fractional integral operators

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.     

An algorithm for finding extremal polytope norms of matrix families

In this paper the problem of the computation of the joint spectral radius of a finite set of matrices is considered. We present an algorithm which, under some suitable assumptions, is able to check if a certain product in the multiplicative semigroup is spectrum maximizing. The algorithm proceeds by attempting to construct a suitable extremal norm for the family, namely a complex polytope norm. As examples for testing our technique, we first consider the set of two 2-dimensional matrices recently analyzed by Blondel, Nesterov and Theys to disprove the finiteness conjecture, and then a set of 3-dimensional matrices arising in the zero-stability analysis of the 4-step BDF formula for ordinary differential equations.     

Two algorithms for weighted matroid intersection

Consider a finite setE, a weight functionw:E→R, and two matroidsM ₁ andM ₂ defined onE. The weighted matroid intersection problem consists of finding a setI⊆E, independent in both matroids, that maximizes Σ{w(e):e inI}. We present an algorithm of complexity O(nr(r+c+logn)) for this problem, wheren=|E|,r=min(rank(M ₁), rank (M ₂)),c=max (c ₁,c ₂) and, fori=1,2,c _i is the complexity of finding the circuit ofI∪{e} inM _i (or show that none exists) wheree is inE andI⊆E is independent inM ₁ andM ₂. A related problem is to find a maximum weight set, independent in both matroids, and of given cardinalityk (if one exists). Our algorithm also solves this problem. In addition, we present a second algorithm that, given a feasible solution of cardinalityk, finds an optimal one of the same cardinality. A sensitivity analysis on the weights is easy to perform using this approach. Our two algorithms are related to existing algorithms. In fact, our framework provides new simple proofs of their validity. Other contributions of this paper are the existence of nonnegative reduced weights (Theorem 6), allowing the improved complexity bound, and the introduction of artificial elements, allowing an improved start and flexibility in the implementation of the algorithms. This research was supported in part by NSF grant ECS 8503192 to Carnegie-Mellon University.