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Cosigning Crossing Families and Outer-Planar Gadgets

Ahmad Abdi, Mahsa Dalirrooyfard, Meike Neuwohner

Abstract

Let $F$ be a crossing family over ground set $V$, that is, for any two sets $U,W\in{F}$ with nonempty intersection and proper union, both sets $U\cap{W},U\cup{W}$ are in $F$. Let $σ:V\to \{+,-\}$ be a signing. We call $σ$ a "cosigning" if every set includes a positive element and excludes a negative element. It is "$\cap\cup$-closed" if every pairwise nonempty intersection and co-intersection include positive and negative elements, respectively. We characterize the existence of ($\cap\cup$-closed) cosignings $σ$ through necessary and sufficient conditions. Our proofs are algorithmic and lead to elegant `forcing' algorithms for finding $σ$, reminiscent of the Cameron-Edmonds algorithm for bicoloring balanced hypergraphs. We prove that the algorithms run in polynomial time, and further, the cosigning algorithm can be run in oracle polynomial time through an application of submodular function minimization. Cosigned crossing families arise naturally in digraphs with vertex set $V$ comprised of sources and sinks, where every set in $F$ is "covered" by an incoming arc. Under mild and necessary conditions, we build an outer-planar arc covering of $F$ when the vertices are placed around a circle. These gadgets are then used to find disjoint dijoins in $0,1$-weighted planar digraphs when the weight-$1$ arcs form a connected component that is not necessarily spanning.

Cosigning Crossing Families and Outer-Planar Gadgets

Abstract

Let be a crossing family over ground set , that is, for any two sets with nonempty intersection and proper union, both sets are in . Let be a signing. We call a "cosigning" if every set includes a positive element and excludes a negative element. It is "-closed" if every pairwise nonempty intersection and co-intersection include positive and negative elements, respectively. We characterize the existence of (-closed) cosignings through necessary and sufficient conditions. Our proofs are algorithmic and lead to elegant `forcing' algorithms for finding , reminiscent of the Cameron-Edmonds algorithm for bicoloring balanced hypergraphs. We prove that the algorithms run in polynomial time, and further, the cosigning algorithm can be run in oracle polynomial time through an application of submodular function minimization. Cosigned crossing families arise naturally in digraphs with vertex set comprised of sources and sinks, where every set in is "covered" by an incoming arc. Under mild and necessary conditions, we build an outer-planar arc covering of when the vertices are placed around a circle. These gadgets are then used to find disjoint dijoins in -weighted planar digraphs when the weight- arcs form a connected component that is not necessarily spanning.
Paper Structure (22 sections, 28 theorems, 1 equation, 10 figures)

This paper contains 22 sections, 28 theorems, 1 equation, 10 figures.

Table of Contents

  1. Introduction
  2. The Circle Problem.
  3. Disjoint Dijoins.
  4. Outline of The Paper.
  5. Cosigning a Crossing Family
  6. The Cosigning Forcing Algorithm
  7. Running Time Analysis
  8. $\cap\cup$-Closed Cosignings of a Crossing Family
  9. The $\cap\cup$-Closed Cosigning Forcing Algorithm
  10. Running Time Analysis
  11. The Circle Problem
  12. Preliminaries and Notation
  13. Necessity Examples
  14. Algorithm
  15. Running Time Analysis.
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $V$ be a finite set, and let $F\subseteq 2^V\backslash \{\emptyset,V\}$ be a crossing family. Then $F$ admits a cosigning if and only if every set in $F$ includes an element $u$ and excludes an element $v$ such that $V\backslash\{u\}$ and $\{v\}$ do not belong to $F$.

Figures (10)

  • Figure 1: (Left) A weight-$0$ vertex $v$ and its in- and out-neighbors in a planar $0,1$-weighted digraph. (Right) A naive reduction that does not preserve planarity. \ref{['thm:circle']} tells us how to retain an outer-planar subset of these arcs such that every dicut has weight at least $2$. Filled-in and non-filled-in vertices are the in and out neighbors of $v$, respectively.
  • Figure 2: An example for \ref{['thm:circle']} with the outer-planar arc subset from $V^-$ to $V^+$
  • Figure 3: An example to show that \ref{['P0']} cannot be dropped from the conditions of \ref{['thm:circle']}
  • Figure 4: An example to show that \ref{['P1']} cannot be dropped from the conditions of \ref{['thm:circle']}
  • Figure 5: An example to show that \ref{['P3']} cannot be dropped from the conditions of \ref{['thm:circle']}
  • ...and 5 more figures

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 4.0
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 58 more