Better Approximation for Weighted $k$-Matroid Intersection

TL;DR

This work addresses the weighted $k$-Matroid Intersection problem, generalizing the classical matching-like tasks to the intersection of $k$ matroids on a common ground set. The authors introduce a randomized, interval-based local-search framework that partitions the instance into almost unweighted weight classes and solves them iteratively, leveraging the $k/2$-approximation for the unweighted case. The main technical contribution is a refined matroid-exchange construction coupled with randomness to avoid bad local minima, yielding a $\frac{k+1}{2\ln 2}$-approximation (approximately $0.722\,(k+1)$) for weighted $k$-Matroid Intersection and its matroid $k$-Parity generalization; this improves upon the previous $k-1$-approximation. The method provides a blueprint for turning unweighted insights into weighted gains and suggests new avenues for improving local-search analyses in matroid systems. The results have potential implications for related problems like $k$-Set Packing and other matroid parity variants, offering a concrete pathway to tighter approximation guarantees. All results are stated with formal $\mathcal{O}$-style algorithmic guarantees and polynomial-time computability for fixed $k$, including randomized high-probability bounds.

Abstract

We consider the problem of finding an independent set of maximum weight simultaneously contained in $k$ matroids over a common ground set. This $k$-matroid intersection problem appears naturally in many contexts, for example in generalizing graph and hypergraph matching problems. In this paper, we provide a $(k+1)/(2 \ln 2)$-approximation algorithm for the weighted $k$-matroid intersection problem. This is the first improvement over the longstanding $(k-1)$-guarantee of Lee, Sviridenko and Vondrák (2009). Along the way, we also give the first improvement over greedy for the more general weighted matroid $k$-parity problem. Our key innovation lies in a randomized reduction in which we solve almost unweighted instances iteratively. This perspective allows us to use insights from the unweighted problem for which Lee, Sviridenko, and Vondrák have designed a $k/2$-approximation algorithm. We analyze this procedure by constructing refined matroid exchanges and leveraging randomness to avoid bad local minima.

Theorems & Definitions (41)

• Theorem 1
• Definition 1
• Theorem 2
• Theorem 3
• Definition 2: $(A, I, J)$-swap
• Definition 3: Markers
• Remark 1
• Proposition 1
• Theorem 4
• Lemma 1