We derive new cut-set bounds on the achievable rates in a general multi-terminal network with finite number of states. Multiple states are common in communication networks in the form of multiple channel and nodes’ states. Our results are broadly applicable and provide much tighter upper bounds than the known single state min-cut max-flow theorem, and hence form an important new tool to bound t...

Let [n] = {1, 2, . . . , n} and let 2 be the collection of all subsets of [n] ordered by inclusion. C ⊆ 2 is a cutset if it meets every maximal chain in 2, and the width of C ⊆ 2 is the minimum number of chains in a chain decomposition of C. Fix 0 ≤ m ≤ l ≤ n. What is the smallest value of k such that there exists a cutset that consists only of subsets of sizes between m and l, and such that it...

M. Stoer and F. Wagner and independently A. Frank have found a simple proof of the validity of Nagamochi and Ibaraki7s min-cut algorithm. This note points out some nice property of the behavior of Nagamochi and Ibaraki's rnin-cut algorithm, which also gives another simple proof of the validity of their algorithm. The proof relies only on the symmetric submodularity of the cut function. Hence, i...

A tree series over a semiring with partially ordered carrier set can be considered as a fuzzy set. We investigate conditions under which it can also be understood as a fuzzi ed recognizable tree language. In this sense, su cient conditions are presented which, when imposed, ensure that every cut set, i.e., the pre-image of a prime lter of the carrier set, is a recognizable tree language. Moreov...