Solving LPs/SDPs via Multiplicative Weighs

From the previous lecture, we have proved the following lemma. Lemma 1. If the cost vector c t ∈ [−ρ, ρ] n , and T ≥ 4ρ 2 ln n 2 , then average loss 1 T T t=1 p t c t ≤ 1 T T t=1 c t (i) + , ∀i ∈ [n] In this lecture, we are going to see the application of multi-weight algorithm for solving LPs and SDPs. For illustration purpose, we will be working on an LP example (the LP relaxation for set cover) and an SDP example (the SDP relaxation for maximum cut). The algorithms should be able to extended to general LPs and SDPs with a few modifications. 1 Set cover LP First, we are going to show how to use multi-weight algorithm to solve LPs. We will use set-cover LP as an example. Let U denotes the universe set, |U | = n. And there is a family of sets F = {S 1 , S 2 , · · · , S m }. We want to choose as less as possible sets from F to cover the whole set. We could interpret the problem as such a IP: min S∈F χ S