Approximation algorithm for weighted set cover, using multiplicative weights
Yes, fractional weighted set cover and, indeed, all explicitly given packing/covering LP's can be solved using multiplicative-weights algorithms in poly time independent of the width. There are many papers in this line of work. The earliest are probably due to Garg and Konemann. See e.g. https://cstheory.stackexchange.com/questions/4697/toy-examples-for-plotkin-shmoys-tardos-and-arora-kale-solvers/14388#14388 .