Diierence matching is a generalization of rst-order matching where terms are made identical both by variable instantiation and by structure hiding. After matching, the hidden structure may be removed by a type of controlled rewriting, called rippling, that leaves the rest of the term unaltered. Rippling has proved highly successful in inductive theorem proving. Diierence matching allows us to u...

We consider the well-studied problem of finding a perfect matching in d-regular bipartite graphs with 2n vertices and m = nd edges. While the best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes O(m √ n) time, in regular bipartite graphs, a perfect matching is known to be computable in O(m) time. Very recently, the O(m) bound was improved to O(min{m, n 2.5 ln n d }...

We prove that Menger's theorem is valid for infinite graphs, in the following strong form: let A and B be two sets of vertices in a possibly infinite digraph. Then there exist a set P of disjoint A-B paths, and a set S of vertices separating A from B, such that S consists of a choice of precisely one vertex from each path in P. This settles an old conjecture of Erd˝ os. 1. History of the proble...

Given n uniformly and independently distributed points in a ball of unit volume in dimension d, it is well established that the length of several combinatorial optimization problems (including the minimum spanning tree (MST), the minimum matching (M), the traveling salesman problem (TSP), etc.) on these n points obeys laws of large numbers. It has been a long open conjecture that a central limi...

We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N . Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the alge...

For a graph G, we show a theorem that establishes a correspondence between the fine Hilbert series of the Stanley-Reisner ring of the clique complex for the complementary graph of G and the fine subgraph polynomial of G. We obtain from this theorem some corollaries regarding the independent set complex and the matching complex.

We propose and study a new constrained independence system. We obtain a sequence of results, including a matching theorem for bases of the system and introducing a set of light elements which give a lower bound for the objective function of a minimization problem in the system. We then demonstrate that the set of triangulations of a planar point set can be modeled as constrained independence sy...

We consider one-to-one, one-sided matching (roommate) problems in which agents can either be matched as pairs or remain single. We introduce a so-called bi-choice graph for each pair of stable matchings and characterize its structure. Exploiting this structure we obtain as a corollary the “lone wolf” theorem and a decomposability result. The latter result together with transitivity of blocking ...

A characterization theorem of the core of two-sided matching problems is proved. The theorem characterizes the core by means of four axioms, the Pareto optimality, consistency, converse consistency, and anonymity. It is shown that all of them are logically independent.

This paper shows that if interaction is not anonymous then costly messages can establish a folk theorem in repeated matching games. This result holds for all population sizes and a broad class of matching rules— including many choice based rules. Cooperation is achieved through a formalization of a “reference”strategy.