Multiobjective Hypergraph Min-Cut in Quasi-Polynomial Time
TL;DR
A quasi-polynomial time approximation scheme (QPTAS) for hypergraph connectivity interdiction for hypergraph connectivity interdiction is obtained and AI tools were used to iterate and refine the algorithmic ideas underlying this work.
Abstract
We study the multiobjective hypergraph min-cut problem: Given a hypergraph $H=(V,E)$ and $k$ cost functions $c_1, c_2, \ldots, c_k:E\to\mathbb{Z}_{\ge 0}$, the goal is to find a non-empty proper subset $U\subsetneq V$ of vertices with minimum $\max_{i\in [k]} c_i(\delta(U))$. When $k$ is part of input, the problem is NP-hard (even in graphs). We focus on fixed-constant $k$ setting (e.g., $k=1, 2, 3, \ldots$). Single-objective hypergraph min-cut as well as multiobjective graph min-cut for a constant number of objectives admit polynomial-time algorithms. In contrast to these special cases, the complexity of multiobjective hypergraph min-cut remains open even for $k=2$. Known techniques fail to extend due to structural differences between graphs and hypergraphs. For $k$-objective hypergraph min-cut when $k$ is a fixed constant, we design a randomized PTAS, and two different randomized quasi-polynomial time algorithms. As an application of our $2$-objective hypergraph min-cut results, we obtain a quasi-polynomial time approximation scheme (QPTAS) for hypergraph connectivity interdiction. AI tools were used to iterate and refine the algorithmic ideas underlying this work.