Mimicking Networks for Constrained Multicuts in Hypergraphs

TL;DR

An algorithm for a hypergraph that returns a multicut-mimicking network over terminals $T$ with a parameter $c$ having hyperedges in $p+o(1)+|T|(c^r\log n)^{\tilde{O}(rc)}m$ time, where $p$ and $r$ are the total size and the rank, respectively, of the hypergraph.

Abstract

In this paper, we study a \emph{multicut-mimicking network} for a hypergraph over terminals $T$ with a parameter $c$. It is a hypergraph preserving the minimum multicut values of any set of pairs over $T$ where the value is at most $c$. This is a new variant of the multicut-mimicking network of a graph in [Wahlström ICALP'20], which introduces a parameter $c$ and extends it to handle hypergraphs. Additionally, it is a natural extension of the \emph{connectivity-$c$ mimicking network} introduced by [Chalermsook et al. SODA'21] and [Jiang et al. ESA'22] that is a (hyper)graph preserving the minimum cut values between two subsets of terminals where the value is at most $c$. We propose an algorithm for a hypergraph that returns a multicut-mimicking network over terminals $T$ with a parameter $c$ having $|T|c^{O(r\log c)}$ hyperedges in $p^{1+o(1)}+|T|(c^r\log n)^{\tilde{O}(rc)}m$ time, where $p$ and $r$ are the total size and the rank, respectively, of the hypergraph.

Figures (3)

• Figure 1: The colored square points mark terminals in the graph. (a) The edges $e$ and $e'$ are both non-essential edges. (b) Contraction $\{e,e'\}$ cannot preserve the minimum cut between red terminals and blue terminals.
• Figure 2: Illustration of proof of Lemma \ref{['lem:essential_in_subinstance']}. (a) Illustration of a terminal partition $\mathcal{T}$ and $G\setminus F$. The dotted circle separates $X$ (inside) and $V\setminus X$ (outside). (b) Illustration of $\hat{G}[X]$, terminal partition $\mathcal{T}_X$, and the vertex partition consisting of $\hat{G}[X]\setminus F_X$. Blue and red squared points are anchored terminals $a_{e'}$ and $t_{e'}$, respectively, for $e'\in \partial X$. (c) Illustration of $G\setminus (F_X\cup \bar{F})$. The multiway cut $(F_X\cup \bar{F})$ is a minimum multiway cut of $\mathcal{T}$ and excluding $e$.
• Figure 3: Illustration of the proof of Lemma \ref{['lem:useful_essential']}. (a) Illustration of the terminal partition $\mathcal{T}$ and the vertex partition according to $G\setminus F$. The middle gray area is $X$. The right three red areas form $C$. (b) Illustration of the terminals partition $\mathcal{T}'$ and the partition of $G\setminus F'$. (c) Illustration of the partition $G\setminus (F'\cup \bar{F})$. $F'\cup \bar{F}$ is a multiway cut of $\mathcal{T}$ excluding $e$.

Theorems & Definitions (16)

• Theorem 1
• Lemma 1
• Lemma 1
• Lemma 2
• Lemma 2
• Lemma 2
• Lemma 3
• Lemma 4: long2022near
• Lemma 4
• Theorem 4