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

Chien-Chung Huang, Naonori Kakimura, Yusuke Kobayashi, Tatsuya Terao

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.

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

Abstract

This paper studies randomized polynomial kernelization for the weighted -matroid intersection problem. While the problem is known to have a kernel of size where 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 -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 -matroid intersection problem admits a polynomial kernel when one matroid is arbitrary and the other matroids are partition matroids. Interestingly, the obtained kernel has size , which matches the optimal bound~(up to logarithmic factors) for the -dimensional matching problem. This approach can be adapted to the case when 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 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.
Paper Structure (23 sections, 18 theorems, 23 equations, 7 algorithms)

This paper contains 23 sections, 18 theorems, 23 equations, 7 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Advantages of Reachable Kernel
  4. Additional Related Work
  5. Preliminaries
  6. Basic Notation
  7. Technical Tools
  8. Partition Matroids and Extensions
  9. Kernelization Algorithm for Simple Partition Matroids
  10. Kernelization Algorithm for $g(k)$-coverable Matroids
  11. Examples of $g(k)$-coverable Matroids
  12. Kernelization Algorithm for Transversal Matroids
  13. Kernelization Algorithm for Laminar Matroids
  14. Matching with a Matroid Constraint
  15. Deterministic Reachable Kernel
  16. ...and 8 more sections

Key Result

Theorem 1

For the weighted $d$-matroid intersection problem such that, out of the $d$ given matroids, $d-1$ are simple partition matroids, we can construct a reachable kernel of sizeThe notation $\tilde{O}$ hides a polylogarithmic function of $k$.$\tilde{O}(k^d)$ in polynomial time.

Theorems & Definitions (40)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 1
  • ...and 30 more