#### Matroidal Approximations of Independence Systems

##### Sven de Vries, Rakesh V. Vohra

Milgrom (2017) has proposed a heuristic for determining a maximum weight basis of an independence system $\mathcal I$ given that for sets from $\mathcal O \subseteq \mathcal I$ we care in particular about the qualitiy of approximation as we assume, that $\mathcal O$ contains (frequently) the optimal basis. It is based on finding an `inner matroid', one contained in the independence system. We show that without additional assumptions on $\mathcal O$ being different from $\mathcal I$ the worst-case performance of this new heuristic is no better than that of the classical greedy algorithm.

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