Contention Resolution Schemes

We wish to approximately maximize f , a nonnegative submodular function on 2 . We consider the space [0, 1] , where a set S corresponds to the indicator point xS (with 1s at the coordinates of elements in S and 0s elsewhere). We can think of a point x ∈ [0, 1] as representing a distribution on 2 , and let R(x) denote a random set obtained by including each i independently with probability xi. The multilinear extension of f is denoted F : [0, 1] → R, and is defined by F (x) = ES∼x f(S), where S ∼ x means to draw S randomly by including each element i ∈ N independently with probability xi. In other words, F (x) = ∑ S f(S) ∏ i∈S xi ∏ i 6∈S(1− xi). Last time, Eric covered the continuous greedy algorithm of Vondrák [2008], Calinescu et al. [2011]. In that setting, f was monotone and we had matroid constraints. The idea was to find a point x such that F (x) ≥ ( 1− 1e ) maxy F (y), noting that maxy F (y) ≥ maxS f(S). We could then round this point x into a set S such that f(S) ≥ F (x) using the pipage rounding technique covered by Jean, of Ageev and Sviridenko [2004].