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Covering a supermodular-like function in a mixed hypergraph

Hui Gao

Abstract

In this paper, we solve a conjecture by Szigeti in [Matroid-rooted packing of arborescences, submitted], which characterizes a mixed hypergraph $\mathcal{F}=(V, \mathcal{E} \cup \mathcal{A})$ having an orientation $\overrightarrow{\mathcal{E}}$ of $\mathcal{E}$ such that $e_{\overrightarrow{\mathcal{E}} \cup \mathcal{A}} (\mathcal{P}) \geq \sum_{X \in \mathcal{P}}h(X) -b(\cup \mathcal{P})$ for every subpartition $\mathcal{P}$ of $V$, where $h$ is an integer-valued, intersecting supermodular function on $V$ and $b$ a submodular function on $V$. As a corollary, another conjecture in the same paper is confirmed, which characterizes a mixed hypergraph having a packing of mixed hyperarborescences such that their roots form a basis in a given matroid, each vertex $v$ belongs to exactly $k$ of them and is the root of at least $f(v)$ and at most $g(v)$ of them.

Covering a supermodular-like function in a mixed hypergraph

Abstract

In this paper, we solve a conjecture by Szigeti in [Matroid-rooted packing of arborescences, submitted], which characterizes a mixed hypergraph having an orientation of such that for every subpartition of , where is an integer-valued, intersecting supermodular function on and a submodular function on . As a corollary, another conjecture in the same paper is confirmed, which characterizes a mixed hypergraph having a packing of mixed hyperarborescences such that their roots form a basis in a given matroid, each vertex belongs to exactly of them and is the root of at least and at most of them.
Paper Structure (3 sections, 7 theorems, 23 equations)

This paper contains 3 sections, 7 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Proof of Conjecture \ref{['conj-orient']}
  3. Acknowledgement

Key Result

Theorem 1.1

Let $D = (V, A)$ be a digraph and $S$ a multiset of vertices in $V$. There exists a packing of spanning $s$-arborescences $(s \in S)$ in $D$ if and only if

Theorems & Definitions (11)

  • Theorem 1.1: E-73
  • Theorem 1.2: E-73
  • Theorem 1.3: C-83F-78
  • Theorem 1.4: DNS-13
  • Theorem 1.5: S-24
  • Theorem 1.6: S-24
  • Theorem 1.7: S-24
  • Conjecture 1.8: S-24
  • Conjecture 1.9: S-24
  • Claim 2.1
  • ...and 1 more