Michael D. Grigoriadis

A fast parametric maximum flow algorithm and applications

The classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter, and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition. Finding the desired parameter value requires solving a sequence of related maximum flow problems. In this paper it is shown that the recent maximum flow algorithm of Goldberg and Tarjan can be extended to solve an important class of such parametric maximum flow problems, at the cost of only a constant factor in its worst-case time bound. Faster algorithms for a variety of combinatorial optimization problems follow from the result.

Fast Approximation Schemes for Convex Programs with Many Blocks and Coupling Constraints

This paper presents block-coordinate descent algorithms for the approximate solution of large structured convex programming problems. The constraints of such problems consist of K disjoint convex compact sets

Coordination complexity of parallel price-directive decomposition

B^k

Approximate Max-Min Resource Sharing for Structured Concave Optimization

called blocks, and M nonnegative-valued convex block-separable inequalities called coupling or resource constraints. The algorithms are based on an exponential potential function reduction technique. It is shown that feasibility as well as min-mix resource-sharing problems for such constraints can be solved to a relative accuracy

Use of dynamic trees in a network simplex algorithm for the maximum flow problem

\varepsilon

Approximate minimum-cost multicommodity flows in O˜(e -2 KNM ) time

in

An exponential‐function reduction method for block‐angular convex programs

O( K\ln M ( \varepsilon^{ - 2} + \ln K ) )

Approximate Lagrangian Decomposition with a Modified Karmarkar Logarithmic Potential

iterations, each of which solves K block problems to a comparable accuracy, either sequentially or in parallel. The same bound holds for the expected number of iterations of a randomized variant of the algorithm which uniformly selects a random block to process at each iteration. An extension to objective and constraint functions of arbitrary sign is also presented. The above results yield fast approximatio...

An Interior Point Method for Bordered Block-Diagonal Linear Programs

The general block-angular convex resource sharing problem in K blocks and M nonnegative block-separable coupling constraints is considered. Applications of this model are in combinatorial optimization, network flows, scheduling, communication networks, engineering design, and finance. This paper studies the coordination complexity of approximate price-directive decomposition PDD for this problem, i.e., the number of iterations required to solve the problem to a fixed relative accuracy as a function of K and M. First a simple PDD method based on the classical logarithmic potential is shown to be optimal up to a logarithmic factor in M in the class of all PDD methods that work with the original unrestricted blocks. It is then shown that logarithmic and exponential potentials generate a polylogarithmically-optimal algorithm for a wider class of PDD methods which can restrict the blocks by the coupling constraints. As an application, the fastest currently-known deterministic approximation algorithm for minimum-cost multicommodity flows is obtained.

Approximate Structured Optimization by Cyclic Block-Coordinate Descent

We present a Lagrangian decomposition algorithm which uses logarithmic potential reduction to compute an