Existence of a true constant-factor approximation for capacitated k-median

Determine whether the capacitated k-median problem admits a true O(1)-approximation algorithm without cardinality or capacity violations.

Background

The paper discusses bicriteria approximation algorithms for the capacitated k-median problem that achieve constant-factor cost guarantees while allowing violations in either the number of opened facilities or the facility capacities. Prior results establish (1+ϵ)(1+\epsilon)-violation factors in both settings, but their approximation ratios depend on ϵ\epsilon. A true constant-factor approximation would satisfy the original cardinality and capacity constraints exactly, and its existence remains unresolved.

References

Together, these results establish optimal $(1+\epsilon)$-violation factors in both settings, leaving the existence of a true $O(1)$-approximation as the primary remaining open problem.

— Simpler Algorithms for Capacitated $k$-Median with $(1 + ε)$-Violations Using New Inequalities  (2610.23348 - Dai et al., 16 Oct 2026) in Introduction, paragraph beginning “Li~\cite{li2017uniform} was the first to break this factor-$2$ barrier”