Before revision:  There is a similar situation in the distributed setting: While there are \text{poly} -round LP-based O(\alpha)-approximation algorithms [KMW06, DKM19], the best non-LP-based algorithm by Lenzen and Wattenhofer [LW10] is an implementation of their centralized algorithm, providing an O(\alpha^2)-approximation within O(\log n) rounds with high probability.
After revision:  There is a similar situation in the distributed setting: While there are \text{poly} is an O(\log^2 n)-round LP-based O(\alpha)-approximation algorithms [KMW06, DKM19], the best non-LP-based algorithm by Lenzen and Wattenhofer [LW10] is an implementation of their centralized algorithm, providing an O(\alpha^2)-approximation within O(\log n) rounds with high probability.