The power of local search: Maximum coverage over a matroid

We present an optimal, combinatorial 1-1/e approximation algorithm for Maximum Coverage over a matroid constraint, using non-oblivious local search. Calinescu, Chekuri, Pál and Vondrák have given an optimal 1-1/e approximation algorithm for the more general problem of monotone submodular maximization over a matroid constraint. The advantage of our algorithm is that it is entirely combinatorial, and in many circumstances also faster, as well as conceptually simpler. Following previous work on satisfiability problems by Alimonti, as well as by Khanna, Mot-wani, Sudan and Vazirani, our local search algorithm is non-oblivious. That is, our algorithm uses an auxiliary linear objective function to evaluate solutions. This function gives more weight to elements covered multiple times. We show that the locality ratio of the resulting local search procedure is at least 1-1/e. Our local search procedure only considers improvements of size 1. In contrast, we show that oblivious local search, guided only by the problem's objective function, achieves an approximation ratio of only (n - 1)/(2n - 1 - k) when improvements of size k are considered. In general, our local search algorithm could take an exponential amount of time to converge to an exact local optimum. We address this situation by using a combination of approximate local search and the same partial enumeration techniques as Calinescu et al., resulting in a clean (1-1/e)-approximation algorithm running in polynomial time.