Deterministic k-server conjecture: a k-competitive online algorithm on every metric space
Difficulty 26: before September 2026 this 1988 conjecture with a factor-2 gap and many failed attempts would sit near 72; a full proof by leading specialists (including a co-author of the 2k−1 bound), with AI assistance acknowledged, is now under review, so the remaining distance is mostly verification. It is discounted slightly less than the APSP/3SUM hypotheses (32) only because the claimed proof is shorter and self-contained; if confirmed it drops to about 5. Application value 4: canonical model for paging and caching, but production systems use heuristics. Leverage 22: the theory of deterministic online algorithms for metrical service systems.
The problem
Is there a rule for moving k servers to requests that is never more than k times costlier than hindsight?
- 1k vans, requests arrive unpredictably
- 2Compare to a planner who knew the future
- 3Conjecture: never worse than k times
- 4Proven only up to 2k−1 for decades
- 52026 proof claimed, awaiting checks
The problem. In the -server problem, servers sit at points of a metric space . Requests arrive online; each must be served by moving some server to it, and the cost is the total distance moved. An online algorithm is if its cost is at most on every sequence, where OPT is the cost of the best offline solution and is a constant.
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Step 2
- Computational complexity theoryTheory · host
- Then
- Deterministic k-server conjecture: a k-competitive online algorithm on every metric spaceThe problem
State of the art
Accepted results. The lower bound (Manasse, McGeoch, Sleator 1990) and the upper bound for the WFA (Koutsoupias–Papadimitriou, J. ACM 1995) were the general bounds for three decades. Special cases are listed in the statement; Huang and Zhang (arXiv:2205.08103, 2022) showed the WFA is 3-competitive on the circle and also found by computer search that no "canonical" potential function of the existing type can settle on general metrics.
Claimed proof (unrefereed). Coester, Koutsoupias and Zbysiński (arXiv:2609.15979, 14 September 2026) claim to prove the conjecture in its strong form: for every metric space and initial configuration , . The proof represents the work function as a matrix whose -column determinants are the work-function values, with min and plus becoming formal addition and multiplication, and uses a potential defined on pairs of coordinates. The acknowledgments state that the authors designed a potential proving without AI assistance, that discussions with ChatGPT 5.5 Pro and Gemini 3.1 Pro led to a symmetric reformulation, and that ChatGPT 6 Astra derived the algebraic proof for general , which the authors adapted and revised. The authors are leading specialists (Koutsoupias co-proved the bound), but as of 15 October 2026 the paper is one month old and no independent verification or refereed publication was found; the status is IN_PROGRESS.
Realistic applications
The -server problem is the canonical model for online caching, paging and movement problems (paging is the uniform-metric case), so a tight bound would complete the theory of deterministic online algorithms for metrical service systems. It would not change systems practice; production caching uses heuristics.
What counts as solved
Resolved when a proof that some deterministic algorithm (for example the WFA) is -competitive on every metric space is accepted by experts, for example the 2026 Coester–Koutsoupias–Zbysiński proof after refereed publication or independent verification; or when a metric space and are found on which every deterministic algorithm has ratio .
How it gets verified
Expert reviewOnly expert review of the argument is possible today.
The claim is that some deterministic algorithm (for example the work function algorithm) is k-competitive on every metric space, or a counterexample metric. The 2026 Coester-Koutsoupias-Zbysinski proof needs refereed or independent verification, and Mathlib has no online-algorithm framework to check it formally. A counterexample metric space could be verified by exhaustive computation for small k.
Claims under verification
+ add a claimWhere the work is
Combinatorial/potential-function proof or construction; computer search over small metrics informs conjectures.
Bottlenecks
ranked by votes- #1TheoryVerifying the 2026 determinant-potential proof
The claimed proof replaces metric-specific potentials with an algebraic matrix representation of work functions and a potential on pairs of coordinates. Experts must check the change-of-basis and row-replacement steps for every request type and confirm that the AI-derived general- argument has no gaps.
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Progress
- Sep 14, 2026Claimed proof of the deterministic conjecturesource
Coester, Koutsoupias and Zbysiński posted a proof that the work function algorithm is k-competitive on every metric space, with AI assistance acknowledged for the general-k algebraic argument. Unrefereed.
- Nov 10, 2022Randomised conjecture refutedsource
Bubeck, Coester and Rabani proved an Ω(log² k) lower bound for randomised algorithms in some metrics.
- Sep 1, 1995Work function algorithm is (2k−1)-competitivesource
Koutsoupias and Papadimitriou proved the upper bound.
- Jun 1, 1990k-server conjecture and lower boundsource
Manasse, McGeoch and Sleator posed the conjecture and proved the k lower bound (J. Algorithms 1990).