An SDP Primal-Dual Approximation Algorithm for Directed Hypergraph Expansion and Sparsest Cut with Product Demands

TL;DR

This paper introduces a polynomial-time approximation algorithm for directed hypergraph expansion and sparsest cut problems using SDP primal-dual methods, extending previous graph models and simplifying the algorithmic framework.

Contribution

It develops a new SDP-based primal-dual approximation algorithm for directed hypergraph problems, unifying and extending prior graph-based approaches with a clearer presentation.

Findings

Achieves approximation ratios matching previous algorithms for restricted models.

Provides a simplified and cleaner algorithmic framework.

Extends applicability to directed hypergraphs and product demands.

Abstract

We give approximation algorithms for the edge expansion and sparsest cut with product demands problems on directed hypergraphs, which subsume previous graph models such as undirected hypergraphs and directed normal graphs. Using an SDP formulation adapted to directed hypergraphs, we apply the SDP primal-dual framework by Arora and Kale (JACM 2016) to design polynomial-time algorithms whose approximation ratios match those of algorithms previously designed for more restricted graph models. Moreover, we have deconstructed their framework and simplified the notation to give a much cleaner presentation of the algorithms.