Some Recent Advances in Network Flows

The literature on network flow problems is extensive, and over the past 40 years researchers have made continuous improvements to algorithms for solving several classes of problems. However, the surge of activity on the algorithmic aspects of network flow problems over the past few years has been particularly striking. Several techniques have proven to be very successful in permitting researchers to make these recent contributions: (i) scaling of the problem data; (ii) improved analysis of algorithms, especially amortized average case performance and the use of potential functions; and (iii) enhanced data structures. In this survey, we illustrate some of these techniques and their usefulness in developing faster network flow algorithms. Our discussion focuses on the design of faster algorithms from the worst case perspective and we limit our discussion to the following fundamental problems: the shortest path problem, the maximum flow problem, and the minimum cost flow problem. We consider several representative algorithms from each problem class including the radix heap algorithm for the shortest path problem, preflow push algorithms for the maximum flow problem, and the pseudoflow push algorithms for the minimum cost flow problem

A branch-and-bound algorithm for stable scheduling in single-machine production systems.

Robust scheduling aims at the construction of a schedule that is protected against uncertain events. A stable schedule is a robust schedule that will change little when variations in the input parameters arise. This paper proposes a branch-and-bound algorithm for optimally solving a single-machine scheduling problem with stability objective, when a single job is anticipated to be disrupted.Branch-and-bound; Construction; Event; Job; Robust scheduling; Robustness; Scheduling; Single-machine scheduling; Stability; Systems; Uncertainty;

Lecture Notes on Network Flow Spring 2004

Lecture Notes on Network Flow Spring 200

A strongly polynomial algorithm for generalized flow maximization

A strongly polynomial algorithm is given for the generalized flow maximization problem. It uses a new variant of the scaling technique, called continuous scaling. The main measure of progress is that within a strongly polynomial number of steps, an arc can be identified that must be tight in every dual optimal solution, and thus can be contracted. As a consequence of the result, we also obtain a strongly polynomial algorithm for the linear feasibility problem with at most two nonzero entries per column in the constraint matrix.Comment: minor correction

On Vegh's Strongly Polynomial Algorithm for Generalized Flows

This thesis contains an exposition of the new strongly polynomial algorithm for the generalized flow problem by Laszlo Vegh (2013). It has been a long-standing open question whether such an algorithm exists, until it was resolved by Vegh in 2013. Generalized flows have many applications in economic problems, such as transportation of goods and foreign currency exchange. The main presentation follows Vegh's paper, but this exposition contains some simplifications and differences in the algorithm and its analysis. The main difference is that we consider the running time of the strongly polynomial algorithm up to one arc contraction before starting fresh on a smaller network. This increases the running time of the algorithm slightly, but the analysis becomes easier

An accelerated Newton-dinkelbach method and its application to two variables per inequality systems

We present an accelerated, or “look-ahead” version of the Newton-Dinkelbach method, a well-known technique for solving fractional and parametric optimization problems. This acceleration halves the Bregman divergence between the current iterate and the optimal solution within every two iterations. Using the Bregman divergence as a potential in conjunction with combinatorial arguments, we obtain strongly polynomial algorithms in three applications domains: (i) For linear fractional combinatorial optimization, we show a convergence bound of O(m log m) iterations; the previous best bound was O(m2 log m) by Wang et al. (2006). (ii) We obtain a strongly polynomial label-correcting algorithm for solving linear feasibility systems with two variables per inequality (2VPI). For a 2VPI system with n variables and m constraints, our algorithm runs in O(mn) iterations. Every iteration takes O(mn) time for general 2VPI systems, and O(m + n log n) time for the special case of deterministic Markov Decision Processes (DMDPs). This extends and strengthens a previous result by Madani (2002) that showed a weakly polynomial bound for a variant of the Newton-Dinkelbach method for solving DMDPs. (iii) We give a simplified variant of the parametric submodular function minimization result by Goemans et al. (2017)