The 
-server
problem is an online algorithm problem in which
mobile servers occupy points of a metric space 
. Requests arrive one at a time at points of 
, and each request must be served by moving a server to the
requested point before the next request is revealed. The cost is the total distance
traveled by the servers. An optimal offline algorithm
knows the entire request sequence in advance (Manasse et al. 1990).
The deterministic 
-server
conjecture (Manasse et al. 1990) asserts that every metric
space admits an online algorithm with competitive ratio 
. For a metric space with at
least 
points, no deterministic online algorithm can
have a smaller competitive ratio. Koutsoupias
and Papadimitriou (1995) proved that the work
function algorithm has competitive ratio
at most 
.
Coester et al. (2026) reported that the work function algorithm is 
-competitive on every metric space.
For an initial configuration 
and every finite request sequence 
, their claimed bound is
 |
where 
and 
denote the costs of the work function algorithm
and the optimal offline algorithm, respectively. The
additive term depends only on the initial configuration, so the claimed bound would
establish the deterministic 
-server conjecture.
Coester et al. (2026) credit ChatGPT-5.5 Pro and Gemini-3.1 Pro with helping symmetrize a human-designed potential, and GPT-6 Astra with deriving the algebraic
argument for general 
and adapting it to a representation supplied by the authors. The authors revised
the resulting proof. As of Sep. 18, 2026, independent specialist verification
and external peer review of the complete proof had not been reported.
-Server Conjecture Is True." 14 Sep
2026. 
-Server Conjecture." J. ACM 42, 971-983,
1995.