Submodular Maximization over Many Matroids via Ordered Local Search

2026-07-01

AI Summary

This work addresses the problem of maximizing a monotone submodular function subject to the intersection of $k$ matroids, as well as the weighted $k$-set packing problem. The authors propose a novel ordered local search algorithm that abandons the conventional weight bucketing approach and instead greedily selects elements in decreasing order of marginal value, augmented with threshold-driven feasible exchange operations. By tightly integrating marginal-value ordering with local search, the method achieves an approximation ratio of $k/2 + o(k)$ for matroid $k$-parity matching and $(\ln(4)k)/3 + o(k)$ for weighted $k$-set packing, both improving upon the best previously known results.

Abstract

Given a monotone submodular function, we consider the problem of finding a maximum-valued set in the intersection of $k$ matroids. Our main result is a polynomial time local search based algorithm achieving a $\frac{k}{2} + o(k)$ approximation guarantee. This asymptotically matches the best-known guarantee of $\frac{k}{2} + ε$ 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 $α$ 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).

Problem

• submodular maximization
• matroid intersection
• approximation algorithm
• k-set packing
• local search

Innovation

• Ordered Local Search
• Submodular Maximization
• Matroid Intersection
• k-Set Packing
• Approximation Algorithm