Perhaps the two most fundamental well-solved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. We review the basic results for both, and describe some more recent advances. Then we discuss extensions of these models, in particular, two recent ones—jump systems and path-matchings.

The notions of relaxed submonotone and relaxed monotone mappings in Banach spaces are introduced and many of their properties are investigated. For example, the Clarke subdifferential of a locally Lipschitz function in a separable Banach space is relaxed submonotone on a residual subset. For example, it is shown that this property need not be valid on the whole space. We prove, under certain hy...

Let two linear matroids have the same rank in matroid intersection. A maximum linear matroid intersection (maximum linear matroid parity set) is called a basic matroid intersection (basic matroid parity set), if its size is the rank of the matroid. We present that enumerating all basic matroid intersections (basic matroid parity sets) is in NC, provided that there are polynomial bounded basic m...

This paper addresses the problem of estimation when the cross-correlation in the errors between diierent random variables are unknown. A new data fusion algorithm, the Covariance Intersection Algorithm(CI), is presented. It is proved that this algorithm yields consistent estimates irrespective of the actual correlations. This property is illustrated in an application of decentralised estimation...

We consider the problem of determining when two delta-matroids on the same ground-set have a common base. Our approach is to adapt the theory of matchings in 2-polymatroids developed by Lovász to a new abstract system, which we call a parity system. Examples of parity systems may be obtained by combining either, two deltamatroids, or two orthogonal 2-polymatroids, on the same ground-sets. We sh...

Consider two families of closed oriented curves in a manifold M. At each point of intersection of a curve of one family with a curve of the other family, form a new closed curve by going around the first curve and then going around the second. Typically, an i-dimensional family and a j-dimensional family will produce an i + j − d + 2-dimensional family. Our purpose is to describe a mathematical...

Tangle machines are a topologically inspired diagrammatic formalism to describe information flow in networks. This paper begins with an expository account of tangle machines motivated by the problem of describing ‘covariance intersection’ fusion of Gaussian estimators in networks. It then discusses learning tangle machines from data. It concludes with two examples in which tangle machines tell ...

The intersection between 3D objects plays a prominent role in spatial reasoning, and computer vision. Detection of intersection between objects can be based on the triangulated boundaries of the objects, leading to computing triangle-triangle intersection. Traditionally there are separate algorithms for cross and coplanar intersection. For qualitative reasoning, intersection detection is suffic...