Polynomial Kernels with Reachability for Weighted $d$-Matroid Intersection

2026-03-18

AI Summary

The existence of a polynomial kernel for the weighted $d$-matroid intersection problem under general matroid combinations remains an open question. This work proposes a novel reachability-based kernelization technique that, for the first time, yields a polynomial-size kernel for weighted $d$-matroid intersection when one matroid is arbitrary and the remaining $d-1$ are partition matroids—or more generally, belong to classes such as graphic, cographic, or transversal matroids. The resulting kernel has size $\tilde{O}(k^d)$, matching the best-known bound for $d$-dimensional matching. This advance not only broadens the scope of problems admitting polynomial kernels but also enables the design of parameterized streaming algorithms and fast EPTASes, achieving near-optimal kernel sizes across diverse matroid combinations.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

This paper studies randomized polynomial kernelization for the weighted $d$-matroid intersection problem. While the problem is known to have a kernel of size $O(d^{(k - 1)d})$ where $k$ is the solution size, the existence of a polynomial kernel is not known, except for the cases when either all the given matroids are partition matroids~(i.e., the $d$-dimensional matching problem) or all the given matroids are linearly representable. The main contribution of this paper is to develop a new kernelization technique for handling general matroids. We first show that the weighted $d$-matroid intersection problem admits a polynomial kernel when one matroid is arbitrary and the other $d-1$ matroids are partition matroids. Interestingly, the obtained kernel has size $\tilde{O}(k^d)$, which matches the optimal bound~(up to logarithmic factors) for the $d$-dimensional matching problem. This approach can be adapted to the case when $d-1$ matroids in the input belong to a more general class of matroids, including graphic, cographic, and transversal matroids. We also show that the problem has a kernel of pseudo-polynomial size when given $d-1$ matroids are laminar. Our technique finds a kernel such that any feasible solution of a given instance can reach a better solution in the kernel, which is sufficiently versatile to allow us to design parameterized streaming algorithms and faster EPTASs.

Problem

• polynomial kernel
• weighted d-matroid intersection
• matroid
• kernelization
• parameterized complexity

Innovation

• polynomial kernelization
• weighted d-matroid intersection
• reachability
• parameterized streaming algorithms
• EPTAS