On Extremal Multiflows

Min-cost multiflows in node-capacitated undirected networks

We consider an undirected graph G = (VG,EG) with a set T ⊆ VG of terminals, and with nonnegative integer capacities c(v) and costs a(v) of nodes v ∈ VG. A path in G is a T -path if its ends are distinct terminals. By a multiflow we mean a function F assigning to each T -path P a nonnegative rational weight F(P ), and a multiflow is called feasible if the sum of weights of T -paths through each ...

Half-integrality of node-capacitated multiflows and tree-shaped facility locations on trees

In this paper, we establish a novel duality relationship between node-capacitated multiflows and tree-shaped facility locations. We prove that the maximum value of a tree-distance-weighted maximum node-capacitated multiflow problem is equal to the minimum value of the problem of locating subtrees in a tree, and the maximum is attained by a half-integral multiflow. Utilizing this duality, we sho...

On Combinatorial Properties of Binary Spaces

Abs t rac t . A binary clutter is the family of inclusionwise minimal supports of vectors of affine spaces over GF(2). Binary clutters generalize various objects studied in Combinatorial Optimization, such as paths, Chinese Postman Tours, multiflows and one-sided circuits on surfaces. The present work establishes connections among three matroids associated with binary clutters, and between any ...

Minimum cost multiflows in undirected networks

Hyperinvariant subspaces and quasinilpotent operators

For a bounded linear operator on Hilbert space we define a sequence of the so-called weakly extremal vectors‎. ‎We study the properties of weakly extremal vectors and show that the orthogonality equation is valid for weakly extremal vectors‎. ‎Also we show that any quasinilpotent operator $T$ has an hypernoncyclic vector‎, ‎and so $T$ has a nontrivial hyperinvariant subspace‎.