Packing of mixed hyperarborescences with flexible roots via matroid intersection

Independent and maximal branching packing in infinite matroid-rooted digraphs

We prove a common generalization of the maximal independent arborescence packing theorem of Cs. Király and two of our earlier works about packing branchings in infinite digraphs.

On the complexity of packing rainbow spanning trees

One of the most important questions in matroid optimization is to find disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases. Bérczi and Schwarcz showed that the problem is hard in general, therefore identifying the borderline between tractable and intractable instances is of interest. In the present paper, we study the special case when one of the matroids is a partition matroid while the other one is a graphic matroid. This setting is equivalent to the problem of packing rainbow spann…

Matroid intersection, base packing and base covering for infinite matroids

As part of the recent developments in infinite matroid theory, there have been a number of conjectures about how standard theorems of finite matroid theory might extend to the infinite setting. These include base packing, base covering, and matroid intersection and union. We show that several of these conjectures are equivalent, so that each gives a perspective on the same central problem of infinite matroid theory. For finite matroids, these equivalences give new and simpler proofs for the finite theorems corresponding to these conjectures. This new point of view also allows us to extend, and…

On the Packing/Covering Conjecture of Infinite Matroids

The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem by Edmonds and Fulkerson. A packing for a family $ (M_i: i\inΘ) $ of matroids on the common edge set $ E $ is a system $ (S_i: i\inΘ) $ of pairwise disjoint subsets of $ E $ where $ S_i $ is panning in $ M_i $. Similarly, a covering is a system $ (I_i: i\inΘ) $ with $\bigcup_{i\inΘ} I_i=E $ where $ I_i $ is independent in $ M_i $. The conjecture states that for every matroid family on $ E $ there is a partition $E=E_p \sqcup E_c$ such that $ (M_i \upharpoonright E_p: i\in Θ) $ admit…

A note on packing spanning trees in graphs and bases in matroids

We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We provide descriptions of such graphs and matroids, showing that such a graph (or matroid) has a unique decomposition. In the case of graphs, our results are relevant for certain communication protocols.

拟阵交与打包/覆盖猜想在有限拟阵类中成立

Matroid intersection and packing/covering conjectures are true in the class of finitary matroids

该研究证明了拟阵交猜想和打包/覆盖猜想在有限拟阵类中成立，主要利用非标准分析中的迭代非标准扩张技术，将无限拟阵嵌入超有限结构并转移结果。

Basic Packing of Arborescences

We provide the directed counterpart of a slight extension of Katoh and Tanigawa's result on rooted-tree decompositions with matroid constraints. Our result characterises digraphs having a packing of arborescences with matroid constraints. It is a proper extension of Edmonds' result on packing of spanning arborescences and implies - using a general orientation result of Frank - the above result of Katoh and Tanigawa. We also give a complete description of the convex hull of the incidence vectors of the basic packings of arborescences and prove that the mimimum cost version of the problem can be…

Packing of maximal independent mixed arborescences

Király in [On maximal independent arborescence packing, SIAM J. Discrete. Math. 30 (4) (2016), 2107-2114] solved the following packing problem: Given a digraph $D = (V, A)$, a matroid $M$ on a set $S = \{s_{1}, \ldots,s_{k} \}$ along with a map $π: S \rightarrow V$, find $k$ arc-disjoint maximal arborescences $T_{1}, \ldots ,T_{k}$ with roots $π(s_{1}), \ldots ,π(s_{k})$, such that, for any $v \in V$, the set $\{s_{i} : v \in V(T_{i})\}$ is independent and its rank reaches the theoretical maximum. In this paper, we give a new characterization for packing of maximal independent mixed arborescen…

覆盖混合超图中的类超模函数

Covering a supermodular-like function in a mixed hypergraph

本文解决Szigeti关于混合超图定向覆盖超模函数的猜想，并推论证实混合超树形图打包的根构成拟阵基的刻画。

$k$-Set Packing 与 $k$-Matroid Intersection 的渐近最优硬度

Asymptotically Optimal Hardness for $k$-Set Packing and $k$-Matroid Intersection

本文证明了 $k$-Dimensional Matching 及相关独立系统优化问题的近似硬度下界，将其从 $\Omega(k/\log k)$ 提升至常数级别，缩小了与 $O(k)$ 近似算法的差距，并基于新颖的保近似 gadget 归约提供了理论解释。

一种更快的Mader's $\mathcal{S}$-路径打包确定性算法

A Faster Deterministic Algorithm for Mader's $\mathcal{S}$-Path Packing

针对Mader's $\mathcal{S}$-路径打包问题，提出一种新的确定性算法，通过归约到线性拟阵奇偶性，利用增广路径算法并优化矩阵运算，时间复杂度达$O(mnk)$，优于现有结果。

delta-正则集对于delta-正则集和Mader delta-正则集的陈述

Representative set statements for delta-matroids and the Mader delta-matroid

本文提出delta-正则集的代表集陈述，并展示了Mader网络的稀疏化结果，扩展了核化工具箱。