we-solve (beta)

CS-158 · computer science · mathematics · Algorithms and data structures

Randomized k-server with squared-logarithmic ratio

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every metric space with at least k+1 points, $k\geq 2$ and initial labeled server tuple, does there exist one randomized online policy whose expected cost on every oblivious finite request sequence is $\leq C(\log (k+1))^{2}OPT+B$, with a metric-independent absolute C and a finite B independent of sequence and horizon? Can B be zero when initial positions are distinct?

Short English statement

For every metric space with at least k+1 points, $k\geq 2$ and initial labeled server tuple, does there exist one randomized online policy whose expected cost on every oblivious finite request sequence is $\leq C(\log (k+1))^{2}OPT+B$, with a metric-independent absolute C and a finite B independent of sequence and horizon? Can B be zero when initial positions are distinct?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-158-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-158

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 110; 2026-09-24).

Randomized k-server with squared-logarithmic ratio

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 110; 2026-09-24), the asserted result is that randomized $k$-server on every metric has competitive ratio of order $\log^2(k+1)$ against oblivious requests, using one policy for every finite sequence.

Mathematical statement of the claimed result

Manuscript 1.
For each $k\ge2$, arbitrary metric $(X,d)$ with at least $k+1$ points, and initial server tuple $s\in X^k$, the claim provides one randomized online $k$-server policy $A$ and finite $B\ge0$ such that every fixed finite request sequence $\sigma$ satisfies \[ \mathbb E\operatorname{cost}_{A,s}(\sigma) \le C(\log(k+1))^2\operatorname{OPT}_{X,d,k,s}(\sigma)+B, \] with absolute $C$. The adversary is oblivious to policy randomness. Additive $B$ may depend on the metric, $k,s$ but not on the sequence or horizon; distinct initial positions allow $B=0$. No computational-efficiency bound is asserted by this existence statement.

Outline of the argument

For manuscript 1 (family 110; 2026-09-24), a hidden comparator is tracked by its posterior measure, separating changes of true metric mass from allocation costs. Protected anchors and a tree of partition labels control travel and edits, with two telescoping logarithmic budgets accounting for the final ratio.

Formal verification

For manuscript 1 (family 110; 2026-09-24), OpenAI lists Lean statements for this assertion: randomized $k$-server on every metric has competitive ratio of order $\log^2(k+1)$ against oblivious requests, using one policy for every finite sequence.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are KServer.lean, UniformKServer.lean.

Their named declarations include OAI.KServer.main_theorem, OAI.UniformKServer.uniform_squared_logarithmic_main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

Go to contribution →