Instructor : Nikhil Bansal Scribe : Tim Oosterwijk 1 Independent set problem for bipartite graphs

Last lecture, we started a discussion about the independent set (“stable set”) problem. For a given graph G = (V,E), an independent set is a subset of vertices that are mutually non-adjacent. The independent set problem asks to find the maximum cardinality of such a vertex set. We saw this problem is notoriously NP-hard: it is NP-hard to approximate within a factor n1−ε for any ε > 0. However, we saw that for planar graphs the problem is still NP-hard but less difficult: for any ε > 0 one can find an independent set of size (1− ε)α(G) in time 2 1 ε2 n, where α(G) denotes the maximum size of an independent set in G. This lecture, we will concentrate on the independent set problem on bipartite graphs. We denote a bipartite graph as G = (V,E) = (L ∪R,E), where L and R are the two vertex classes of the bipartite graphs such that all edges are of the form (u, v) where u ∈ L and v ∈ R. We will look at this problem from the view of an integer programming formulation (IP):