Gian-Carlo Rota said that “Anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day.” [20] Hassler Whitney introduced the theory of matroids in 1935 and developed a striking number of their basic properties as well as different ways to formulate the notion of a matroid. As more and more connections between matroi...

Cunningham and Geelen (1997) introduced the notion of independent path-matchings, which generalize both matchings and matroid intersection. Path-matchings are yet generalized to even factors in digraphs by Cunningham and Geelen (2001). Pap (2005) gave a combinatorial algorithm to find a maximum even factor in odd-cycle-symmetric digraphs, in which each arc in any odd dicycle has the reverse arc...

A Mader matroid is obtained as follows. Let G = (V,E) be an undirected graph and let S1, . . . , Sk be disjoint subsets of V . Define S := S1 ∪ · · · ∪ Sk. Let I be the collection of all subsets I of S with the property that there exist disjoint paths P1, . . . , Pt such that each Pj has its ends in different sets Si and has no internal vertices in S and such that I is a subset of the set of en...

Index 74 Preface The three subjects of the title (codes, matroids, and permutation groups) have many interconnections. In particular, in each case, there is a polynomial which captures a lot of information about the structure: we have the weight enumerator of a code, the Tutte polynomial (or rank polynomial) of a matroid, and the cycle index of a permutation group. With any code is associated a...

A central problem in matroid theory—and by extension coding theory and finite geometry— is the problem of constructing matrices with a maximal number of columns such that every square submatrix formed by a subset of the columns is nonsingular. In this project, we apply the tools of analysis to this problem and demonstrate the value of such an approach. First, we introduce various quantities tha...

A matroid M is sequential or has path width 3 if M is 3-connected and its ground set has a sequential ordering, that is, an ordering (e1, e2, . . . , en) such that ({e1, e2, . . . , ek}, {ek+1, ek+2, . . . , en}) is a 3-separation for all k in {3, 4, . . . , n − 3}. This paper proves that every sequential matroid is easily constructible from a uniform matroid of rank or corank two by a sequence...

We present exact characterizations of structures on which the greedy algorithm produces optimal solutions. Our characterization, which we call matroid embeddings, complete the partial characterizations of Rado, Gale, and Edmonds (matroids), and of Korte and Lovasz (greedoids). We show that the greedy algorithm optimizes all linear objective functions if and only if the problem structure (phrase...

Submodular functions arewell-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call weakly submodular functions. Our extension is somewhat unusual in that it includes some (mildly) supermodular functions. We show that several natura...

Matroid theory gives us powerful techniques for understanding combinatorial optimization problems and for designing polynomialtime algorithms. However, several natural matroid problems, such as 3-matroid intersection, are NP-hard. Here we investigate these problems from the parameterized complexity point of view: instead of the trivial O(n) time brute force algorithm for finding a k-element sol...

We prove that the linear matroid that defines the generic rigidity of d-dimensional bodyrod-bar frameworks (i.e., structures consisting of disjoint bodies and rods mutually linked by bars) can be obtained from the union of ( d+1 2 ) copies of a graphic matroid by applying variants of Dilworth truncation operations nr times, where nr denotes the number of rods. This result leads to an alternativ...