Simplicial and Cellular Trees

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed first by Bolker, Kalai and Adin, and more recently by numerous authors, the fundamental topological properties of a tree --- namely acyclicity and connectedness --- can be generalized to arbitrary dimension as the vanishing of certain cellular homology groups. This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley's formula and the matrix-tree theorem. A subtlety of the higher-dimensional case is that enumeration must account for the possibility of torsion homology in trees, which is always trivial for graphs. Cellular trees are the starting point for further high-dimensional extensions of concepts from algebraic graph theory including the critical group, cut and flow spaces, and discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for forthcoming IMA volume "Recent Trends in Combinatorics

The algebra of flows in graphs

We define a contravariant functor K from the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graph X, an abelian group B, and a nonnegative integer j, an element of Hom(K^j(X),B) is a coherent family of B-valued flows on the set of all graphs obtained by contracting some (j-1)-set of edges of X; in particular, Hom(K^1(X),R) is the familiar (real) ``cycle-space'' of X. We show that K(X) is torsion-free and that its Poincare polynomial is the specialization t^{n-k}T_X(1/t,1+t) of the Tutte polynomial of X (here X has n vertices and k components). Functoriality of K induces a functorial coalgebra structure on K(X); dualizing, for any ring B we obtain a functorial B-algebra structure on Hom(K(X),B). When B is commutative we present this algebra as a quotient of a divided power algebra, leading to some interesting inequalities on the coefficients of the above Poincare polynomial. We also provide a formula for the theta function of the lattice of integer-valued flows in X, and conclude with ten open problems.Comment: 31 pages, 1 figur

Some recent results in the analysis of greedy algorithms for assignment problems

We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems

Cuts and flows of cell complexes

We study the vector spaces and integer lattices of cuts and flows associated with an arbitrary finite CW complex, and their relationships to group invariants including the critical group of a complex. Our results extend to higher dimension the theory of cuts and flows in graphs, most notably the work of Bacher, de la Harpe and Nagnibeda. We construct explicit bases for the cut and flow spaces, interpret their coefficients topologically, and give sufficient conditions for them to be integral bases of the cut and flow lattices. Second, we determine the precise relationships between the discriminant groups of the cut and flow lattices and the higher critical and cocritical groups with error terms corresponding to torsion (co)homology. As an application, we generalize a result of Kotani and Sunada to give bounds for the complexity, girth, and connectivity of a complex in terms of Hermite's constant.Comment: 30 pages. Final version, to appear in Journal of Algebraic Combinatoric

On Correcting Inputs: Inverse Optimization for Online Structured Prediction

Algorithm designers typically assume that the input data is correct, and then proceed to find "optimal" or "sub-optimal" solutions using this input data. However this assumption of correct data does not always hold in practice, especially in the context of online learning systems where the objective is to learn appropriate feature weights given some training samples. Such scenarios necessitate the study of inverse optimization problems where one is given an input instance as well as a desired output and the task is to adjust the input data so that the given output is indeed optimal. Motivated by learning structured prediction models, in this paper we consider inverse optimization with a margin, i.e., we require the given output to be better than all other feasible outputs by a desired margin. We consider such inverse optimization problems for maximum weight matroid basis, matroid intersection, perfect matchings, minimum cost maximum flows, and shortest paths and derive the first known results for such problems with a non-zero margin. The effectiveness of these algorithmic approaches to online learning for structured prediction is also discussed.Comment: Conference version to appear in FSTTCS, 201