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.     

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.     

Equivalence-free exhaustive generation of matroid representations

In this paper we present an algorithm for the problem of exhaustive equivalence-free generation of 3-connected matroids which are represented by a matrix over some finite (partial) field, and which contain a given minor. The nature of this problem is exponential, and it appears to be much harder than, say, isomorph-free generation of graphs. Still, our algorithm is very suitable for practical use, and it has been successfully implemented in our matroid computing package MACEK [http://www.mcs.vuw.ac.nz/research/macek, 2001-05].     

On the (co)girth of a connected matroid

This article studies the girth and cogirth problems for a connected matroid. The problem of finding the cogirth of a graphic matroid has been intensively studied, but studies on the equivalent problem for a vector matroid or a general matroid have been rarely reported. Based on the duality and connectivity of a matroid, we prove properties associated with the girth and cogirth of a matroid whose contraction or restriction is disconnected. Then, we devise algorithms that find the cogirth of a matroid M from the matroids associated with the direct sum components of the restriction of M. As a result, the problem of finding the (co)girth of a matroid can be decomposed into a set of smaller sub-problems, which helps alleviate the computation. Finally, we implement and demonstrate the application of our algorithms to vector matroids.     

Complementary bases of a matroid

Let e₁, e′₁, e₂, e′₂, …, e_n, e′_n be the elements of matroid M. Suppose that {e₁, e₂, …;, e_n} is a base of M and that every circuit of M contains at least m + 1 elements. We prove that there exist at least 2^m bases, called complementary bases, of M with the property that only one of each complementary pair e_j, e′_j is contained in any base.We also prove an analogous result for the case where E is partitioned into E₁, E₂, …, E_n and the initial base contains |E_j| ? 1 elements from partition E_j.     

Persistency and matroid intersection

In this paper, we show that for any independence system, the problem of finding a persistency partition of the ground set and that of finding a maximum weight independent set are polynomially equivalent. This research has been partially funded by the Greek Ministry of Education under the program Pythagoras.     

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.     

Envy-free matchings with one-sided preferences and matroid constraints

In this paper, we consider the problem of checking the existence of an envy-free matching in a many-to-one matching model with one-sided preferences and matroid constraints. For this problem, we propose a polynomial-time algorithm which is a generalization of the algorithm proposed by Gan, Suksompong, and Voudouris for the one-to-one setting. Furthermore, we consider a stronger variant of envy-freeness.     

A matroid operation that commutes with duality

In this note, we propose an operation for matroids that commutes with duality having deletions and contractions as extremal cases. Crapo and Schmitt's free product of matroids is one of its consequences. A special case of this operation can be used as an inductive tool because it reduces the number of elements of a matroid by two and it is invariant by duality.     

k-Best constrained bases of a matroid

We propose a method for finding a set ofk-best bases of an arbitrary matroid where the bases are required to satisfy additional partitionlike constraints. An application of this problem is discussed.Research partly supported by Sonderforschungsbereich 303 der Deutschen Forschungsgemeinschaft and by the Austrian Science Foundation, Project P6004.     

Bounding the number of bases of a matroid

Letb(M) andc(M), respectively, be the number of bases and circuits of a matroidM. For any given minor closed class? of matroids, the following two questions, are investigated in this paper. (1) When is there a polynomial functionp(x) such thatb(M)≤p(c(m)|E(M)|) for every matroidM in?? (2) When is there a polynomial functionp(x) such thatb(M)≤p(|E(M)|) for every matroidM in?? Let us denote byM _Mn the direct sum ofn copies ofU _1,2. We prove that the answer to the first question is affirmative if and only if someM _Mn is not in?. Furthermore, if all the members of? are representable over a fixed finite field, then we prove that the answer to the second question is affirmative if and only if, also, someM _Mn is not in?.     

Minors of a random binary matroid

Let A be an n × m matrix over GF ₂ where each column consists of k ones, and let M be an arbitrary fixed binary matroid. The matroid growth rate theorem implies that there is a constant C_M such that m ≥ C_Mn² implies that the binary matroid induced by A contains M as a minor. We prove that if the columns of A = A _n,m,k are chosen randomly, then there are constants k_M,L_M such that k ≥ k_M and m ≥ L_Mn implies that A contains M as a minor with high probability .     

Combinatorial Laplacians of matroid complexes

We combinatorially interpret the spectra of discrete Laplace operators from the boundary maps in the simplicial complex of independent sets of a matroid. The interpretation follows from a surprising orthogonal decomposition of the simplicial chain groups. This decomposition is in general finer than the spectral decomposition. As a consequence, the spectra are integral. One corollary to our combinatorial interpretation may be paraphrased as stating that one can ``hear" the characteristic polynomial of a matroid.