English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The Max-Flow Min-Cut theorem is the classical duality result for the Max-Flowproblem, which considers flow of a single commodity. We study a multiplecommodity generalization of Max-Flow in which flows are composed of real-valuedk-vectors through networks with arc capacities formed by regions in \R^k. Giventhe absence of a clear notion of ordering in the multicommodity case, we definethe generalized max flow as the feasible region of all flow values.  We define a collection of concepts and operations on flows and cuts in themulticommodity setting. We study the mutual capacity of a set of cuts, definedas the set of flows that can pass through all cuts in the set. We present amethod to calculate the mutual capacity of pairs of cuts, and then generalizethe same to a method of calculation for arbitrary sets of cuts. We show thatthe mutual capacity is exactly the set of feasible flows in the network, andhence is equal to the max flow. Furthermore, we present a simple class of themulticommodity max flow problem where computations using this tight dualityresult could run significantly faster than default brute force computations.  We also study more tractable special cases of the multicommodity max flowproblem where the objective is to transport a maximum real or integer multipleof a given vector through the network. We devise an augmenting cycle searchalgorithm that reduces the optimization problem to one with m constraints in atmost \R^{(m-n+1)k} space from one that requires mn constraints in \R^{mk} spacefor a network with n nodes and m edges. We present efficient algorithms thatcompute eps-approximations to both the ratio and the integer ratio maximum flowproblems.

A Tight Max-Flow Min-Cut Duality Theorem for Non-Linear Multicommodity Flows