The k-server conjecture is true

[Submitted connected 14 Sep 2026]

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Abstract:The $k$-server conjecture states that a deterministic online algorithm tin execute competitory ratio $k$ connected each metric space. We beryllium the conjecture. Specifically, we show that the activity usability algorithm satisfies it.
Our impervious uses a earthy algebraic practice of the activity usability arsenic a matrix, which encodes each feasible paths to scope a configuration. In this representation, the minimum and summation operations arising successful the meaning of optimal costs correspond to summation and multiplication of general expressions, and each activity usability worth corresponds to the determinant of $k$ columns of the matrix. A petition presence updates the practice via a alteration of ground and statement replacement. The amortized study is based connected a imaginable usability defined successful position of a larger matrix whose coordinates are pairs of coordinates of the original matrix representation.