An Integer Interior Point Method for Min-Cost Flow Using Arc Contractions and Deletions

TL;DR

This paper introduces an integer interior point method for min-cost flow problems that uses arc contractions and deletions, enabling exact arithmetic and avoiding numerical issues, with a randomized expected runtime of  O(m^{3/2}).

Contribution

The paper presents a novel integer arithmetic interior point algorithm for min-cost flow that guarantees numerical stability and efficiency, bridging the gap between combinatorial and numerical methods.

Findings

Expected  O(m^{3/2}) runtime for the algorithm.

Explicit bounds on the size of numbers during computations.

Guarantees of numerical stability and exactness in solutions.

Abstract

We present an interior point method for the min-cost flow problem that uses arc contractions and deletions to steer clear from the boundary of the polytope when path-following methods come too close. We obtain a randomized algorithm running in expected $\tilde O( m^{3/2} )$ time that only visits integer lattice points in the vicinity of the central path of the polytope. This enables us to use integer arithmetic like classical combinatorial algorithms typically do. We provide explicit bounds on the size of the numbers that appear during all computations. By presenting an integer arithmetic interior point algorithm we avoid the tediousness of floating point error analysis and achieve a method that is guaranteed to be free of any numerical issues. We thereby eliminate one of the drawbacks of numerical methods in contrast to combinatorial min-cost flow algorithms that still yield the most efficient implementations in practice, despite their inferior worst-case time complexity.