Iterative Randomized Rounding for Better-Than-2 k-Median Approximation

Abstract

We study the classic $k$-median problem and refine the iterative rounding framework recently introduced by Byrka, Guo, Hu, Li, Wan, and Wang [FOCS'26], which yields a $(2+\varepsilon)$-approximation. Our algorithm opens $k+O(1)$ facilities and achieves an expected connection cost of at most $1.9999$ times the value of the standard LP relaxation. Although the numerical improvement over $2$ is small, it establishes a conceptual result: the integrality gap of the standard relaxation is strictly below $2$ when a constant number of extra facilities is allowed. This partially answers an open question of Li and Svensson [STOC'13]. Combined with their reduction, our result yields a $1.9999$-approximation algorithm for $k$-median. Our approach builds on several technical ingredients. First, we develop a preprocessing routine that makes the LP solution $\mathrm{poly}(\varepsilon)$-integral, at the cost of a $(1+\varepsilon)$ factor in the connection cost and $O(1)$ extra facilities. Unlike the preprocessing of Byrka et al. [FOCS'26], our conversion is unconditional: it requires no special treatment of clients whose fractional facilities are concentrated. Next, we show that the main rounding algorithm can open facilities one at a time, eliminating the batch processing used by Byrka et al. [FOCS'26]. Finally, the improvement over 2 comes from a tighter analysis for clients whose distances from to its facilities are not all equal, and a complementary algorithm for the case in which these distances are nearly uniform.