We investigate the problem of finding a maximum-valued set in the intersection of $k$ matroids, given a monotone submodular function. Our main result is a polynomial-time local search algorithm achieving a $\frac{k}{2} + o(k)$ approximation guarantee. This asymptotically matches the best-known guarantee of $\frac{k}{2} + \epsilon$ in the unweighted setting by Lee, Sviridenko, and Vondrák (2009).
Prior to this work, the state-of-the-art was a $\frac{\ln(4)k}{1+\ln(2)} + o(k)$-approximation algorithm obtained by Feldman and Ward (2026). Our approach extends to Matroid $k$-Parity yielding the same approximation guarantee.
In contrast to the weight bucketing approach underlying the recent advances of Singer and Thiery (2025) and Feldman and Ward (2026), our algorithm processes elements greedily in decreasing order of marginal value and searches for sufficiently profitable swaps, whose gain exceeds a parameter $\alpha$ given as a function of $k$. We further combine this idea with the weight bucketing approach to obtain improved guarantees for weighted $k$-Set Packing.
Our second main result is a $\frac{\ln(4)k}{3} + o(k)$-approximation algorithm for weighted $k$-Set Packing, improving on the state of the art $\frac{k}{2.00561} + O(1)$-approximation by Neuwohner (2023).
Blogger's Review: The algorithm proposed in this paper achieves new approximation guarantees for submodular maximization under multiple matroids, showcasing the effectiveness of local search strategies, particularly in tackling complex combinatorial optimization problems. The innovative combination of greedy methods and weight bucketing offers fresh directions for future research.