Deterministic online algorithms achieve best possible ratio on all metric spaces
The $k$-server conjecture is true
Data Structures and Algorithms
Summary
The paper proves a long-standing math puzzle called the k-server conjecture, which is about how well certain computer programs can perform when making decisions step-by-step without knowing the future. The authors show that a particular method, the work function algorithm, does as well as theoretically possible across all spaces where distances matter. They use a clever new way to represent the problem with matrices and formulas to track all possible moves efficiently. This proof settles a decades-old question about how good online decision-making algorithms can be.
What this means in practice
- •For network schedulers: Design online network routing protocols with guaranteed best worst-case efficiency for handling requests in arbitrary topologies.
- •For cloud resource managers: Improve decision-making methods for allocating servers to tasks dynamically with provable efficiency limits in general environments.
A theory result. No direct application yet.
Authors
Christian Coester, Elias Koutsoupias, Marek Zbysiński
Abstract
The $k$-server conjecture states that a deterministic online algorithm can achieve competitive ratio $k$ on every metric space. We prove the conjecture. Specifically, we show that the work function algorithm satisfies it. Our proof uses a natural algebraic representation of the work function as a matrix, which encodes all feasible paths to reach a configuration. In this representation, the minimum and addition operations arising in the definition of optimal costs correspond to addition and multiplication of formal expressions, and each work function value corresponds to the determinant of $k$ columns of the matrix. A request arrival updates the representation via a change of basis and row replacement. The amortized analysis is based on a potential function defined in terms of a larger matrix whose coordinates are pairs of coordinates of the original matrix representation.