The Greedy algorithm of Jain, Mahdian, Markakis, Saberi, and Vazirani was the best Lagrangian-multiplier-preserving (LMP) approximation algorithm for facility location, and thus the engine behind the best $k$-median guarantees, for over two decades. Its $2$-LMP guarantee was only recently broken by a small amount ($(2-η)$-LMP where $η\approx 2.25 \cdot 10^{-7}$) through local-search post-processing. That analysis relies on the structure of worst-case instances, and it appears difficult to obtain substantial further improvements along those lines. We take a different approach and change the algorithm itself: we introduce general bidding functions, replacing the classical linear bid with a richer family of rules and prove a polynomial-time $1.95$-LMP approximation, a large improvement over the long-standing factor of~$2$. We further show that these generalized bidding rules can be made smooth (log-adaptive), building on the recent framework developed for the classical Greedy algorithm by Cohen-Addad, Grandoni, Lee, Schwiegelshohn, and Svensson. This yields a polynomial-time $1.95$-approximation that opens $k+O_\varepsilon(\log n)$ centers. Combined with a $(1.89+\varepsilon)$-approximation algorithm for stable instances in our companion work \cite{stableinstances}, we achieve a better-than-2 approximation for the $k$-median problem in polynomial-time.

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