The k-server conjecture is true
This lightning talk presents the proof of the deterministic k-server conjecture, a decades-old open problem in online algorithms. The work function algorithm (WFA) is shown to be k-competitive on arbitrary metric spaces through an algebraic lift of the work function into determinant valuations and a novel quadratic potential analysis. The proof achieves the conjectured coefficient k exactly, matching the known lower bound and resolving the problem without restrictions on geometry or space structure.Script
For three decades, researchers asked whether k servers moving to satisfy requests on any metric space could be scheduled online within a factor of k times the optimal offline cost. This paper proves they can.
The work function algorithm has been the leading candidate since the problem's formulation. It maintains a cost estimate for reaching every possible configuration, then moves to minimize total cost plus the new work function value. The challenge was proving it achieves competitive ratio k.
The authors lift the work function into a determinantal representation over formal polynomials. Each configuration's cost becomes the valuation of a determinant, where minimum-cost matchings emerge as lowest-exponent terms. This algebraic embedding turns min-plus recurrence into ordinary addition followed by valuation.
The proof constructs a potential in the symmetric square of the column space. Under basis change, the induced transformation on this quadratic space has determinant raised to the power k plus 1. That exponent is the source of the competitive coefficient, and it telescopes perfectly with the terminal work function value to yield exactly k.
The result improves the previous general bound of 2k minus 1 to the conjectured k, matching the lower bound exactly. The additive initial clique term depends only on the starting configuration, not the request sequence, so the competitive ratio is tight.
After thirty years, the deterministic k-server conjecture is resolved through an algebraic proof that every online movement is paid for by a single global potential. To explore the full determinantal construction and see how other open problems might yield to similar techniques, visit EmergentMind.com and create your own video.
