The Densest SWAMP problem: subhypergraphs with arbitrary monotonic partial edge rewards

TL;DR

This paper generalizes the densest subhypergraph problem to include arbitrary monotonic partial edge rewards, providing hardness results and approximation algorithms, along with empirical validation on real-world hypergraphs.

Contribution

It introduces a broader setting for the densest subhypergraph problem with arbitrary monotonic rewards, and proposes novel approximation algorithms with empirical evaluation.

Findings

The problem is NP-hard for non-convex rewards.

A 1/k-approximation algorithm based on convex projection is developed.

A faster peeling-based 1/k-approximation algorithm is proposed.

Abstract

We consider a generalization of the densest subhypergraph problem where nonnegative rewards are given for including partial hyperedges in a dense subhypergraph. Prior work addressed this problem only in cases where reward functions are convex, in which case the problem is poly-time solvable. We consider a broader setting where rewards are monotonic but otherwise arbitrary. We first prove hardness results for a wide class of non-convex rewards, then design a 1/k-approximation by projecting to the nearest set of convex rewards, where k is the maximum hyperedge size. We also design another 1/k-approximation using a faster peeling algorithm, which (somewhat surprisingly) differs from the standard greedy peeling algorithm used to approximate other variants of the densest subgraph problem. Our results include an empirical analysis of our algorithm across several real-world hypergraphs.