Mimicking Networks for Constrained Multicuts in Hypergraphs

TL;DR

This paper extends the concept of multicut-mimicking networks from graphs to hypergraphs, providing an algorithm that constructs a compact hypergraph preserving certain multicut properties with efficiency related to hypergraph parameters.

Contribution

It introduces a new variant of multicut-mimicking networks for hypergraphs, extending previous graph-based models and providing an algorithm with specific size and time complexity guarantees.

Findings

Proposes a hypergraph multicut-mimicking network with size depending on terminal set and hypergraph rank.

Develops an algorithm with complexity tied to hypergraph parameters and size.

Extends previous graph models to hypergraphs, broadening applicability.

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 [Wahlstr\"om 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.