2 research outputs found
A dual descent algorithm for node-capacitated multiflow problems and its applications
In this paper, we develop an -time algorithm
to find a half-integral node-capacitated multiflow of the maximum total
flow-value in a network with nodes, edges, and terminals, where
denotes the time complexity of solving the maximum
submodular flow problem in a network with nodes, edges, and the
complexity of computing the exchange capacity of the submodular
function describing the problem. By using Fujishige-Zhang algorithm for
submodular flow, we can find a maximum half-integral multiflow in time. This is the first combinatorial strongly polynomial time algorithm
for this problem. Our algorithm is built on a developing theory of discrete
convex functions on certain graph structures. Applications include
"ellipsoid-free" combinatorial implementations of a 2-approximation algorithm
for the minimum node-multiway cut problem by Garg, Vazirani, and Yannakakis.Comment: To appear in ACM Transactions on Algorithm
L-extendable functions and a proximity scaling algorithm for minimum cost multiflow problem
In this paper, we develop a theory of new classes of discrete convex
functions, called L-extendable functions and alternating L-convex functions,
defined on the product of trees. We establish basic properties for
optimization: a local-to-global optimality criterion, the steepest descend
algorithm by successive -submodular function minimizations, the persistency
property, and the proximity theorem. Our theory is motivated by minimum cost
free multiflow problem. To this problem, Goldberg and Karzanov gave two
combinatorial weakly polynomial time algorithms based on capacity and cost
scalings, without explicit running time. As an application of our theory, we
present a new simple polynomial proximity scaling algorithm to solve minimum
cost free multiflow problem in time, where
is the number of nodes, is the number of edges, is the number of
terminals, is the maximum of edge-costs, is the total sum of
edge-capacities, and denotes the time complexity to find a
maximum flow in a network of nodes and edges. Our algorithm is
designed to solve, in the same time complexity, a more general class of
multiflow problems, minimum cost node-demand multiflow problem, and is the
first combinatorial polynomial time algorithm to this class of problems. We
also give an application to network design problem.Comment: 39 pages; Discrete Optimization, to appea