New Algorithm from MIT, Yale, and USC Tackles Max Flow Challenge

• By Dian Schaffhauser
• 01/09/14

Researchers from MIT, Yale University, and the University of Southern California have developed what they are labeling the "fastest known algorithm" for solving the problem of "maximum flow."

The max flow problem, as it's called, refers to the process of figuring out how much capacity is available between any given starting point and any given end point. Capacity could refer to trucks on roads, water in pipes, or packets on a cable. The challenge first surfaced in 1954 in an attempt to model the behavior of Soviet railway traffic. The first algorithm to solve it came the following year. Since then numerous algorithms have surfaced to address the calculation of max flow more efficiently.

Typically, the network has been represented as a graph with a series of nodes or "vertices"; the lines between the vertices are called "edges." Each edge has a given maximum capacity, and the algorithms set out to calculate the most efficient way to use the capacity represented in the graph without exceeding the constraints.

As the network grows, however, the use of traditional algorithms may become too time-consuming to be practical. "There has recently been an explosion in the sizes of graphs being studied," explained Jonathan Kelner, an associate professor of applied mathematics at MIT and a member of MIT's Computer Science and Artificial Intelligence Laboratory (CSAIL). "For example, if you wanted to route traffic on the Internet, study all the connections on Facebook, or analyze genomic data, you could easily end up with graphs with millions, billions or even trillions of edges."

The algorithm developed by Kelner and his four colleagues chooses to solve the max-flow problem by analyzing all potential paths at the same time. According to university coverage of the technique, the researchers viewed the graph as a collection of electrical resistors and then imagined connecting a battery to node A and a ground to node B, and allowing the current to flow through the network. "Electrical current doesn't pick just one path, it will send a little bit of current over every resistor on the network," Kelner said. "So it probes the whole graph globally, studying many paths at the same time."

This approach introduced definite efficiencies. Then the team added an additional twist. Their algorithm also identifies network routes that pose bottlenecks. "Our algorithm figures out which parts of the graph can easily route what they need to and which parts are the bottlenecks. This allows you to focus on the problem areas and the high-level structure instead of spending a lot of time making unimportant decisions, which means you can use your time a lot more efficiently," Kelner noted.

As the number of nodes increases so does the amount of running time required to solve the problem. However, the paper stated, rather than an exponential increase, the computation scales "near-linearly."