Yuval Filmus and Justin Ward

STACS 2012

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 and by Khanna, Motwani, 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 $\frac{n-1}{2n-1-k}$ 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 used by Calinescu et al., resulting in a clear (1−1/*e*)-approximation algorithm running in polynomial time.

We obtained our auxiliary linear objective function using linear programming. This is detailed in Ward’s thesis.

@inproceedings{FW2012a,

author = {Yuval Filmus and Justin Ward},

title = {Maximum coverage over a matroid},

booktitle = {29th Symposium on Theoretical Aspects of Computer

Science ({STACS} 2012)},

year = {2012},

pages = {601--612}

}

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author = {Yuval Filmus and Justin Ward},

title = {Maximum coverage over a matroid},

booktitle = {29th Symposium on Theoretical Aspects of Computer

Science ({STACS} 2012)},

year = {2012},

pages = {601--612}

}