Partitioning Hypergraphs is Hard: Models, Inapproximability, and Applications

TL;DR

This paper proves the computational hardness of balanced hypergraph partitioning, explores its parameterized complexity, and extends the problem to hyperDAGs and hierarchical partitioning for practical applications in manycore scheduling.

Contribution

It establishes strong inapproximability results for balanced hypergraph partitioning and introduces new models like hyperDAGs and hierarchical partitioning for real-world systems.

Findings

Partitioning hypergraphs is hard to approximate within n^{1/poly log log n}.

HyperDAGs effectively model precedence constraints in computations.

Ignoring hierarchy in partitioning weakens solution quality.

Abstract

We study the balanced $k$-way hypergraph partitioning problem, with a special focus on its practical applications to manycore scheduling. Given a hypergraph on $n$ nodes, our goal is to partition the node set into $k$ parts of size at most $(1+\epsilon)\cdot \frac{n}{k}$ each, while minimizing the cost of the partitioning, defined as the number of cut hyperedges, possibly also weighted by the number of partitions they intersect. We show that this problem cannot be approximated to within a $n^{1/\text{poly} \log\log n}$ factor of the optimal solution in polynomial time if the Exponential Time Hypothesis holds, even for hypergraphs of maximal degree 2. We also study the hardness of the partitioning problem from a parameterized complexity perspective, and in the more general case when we have multiple balance constraints. Furthermore, we consider two extensions of the partitioning problem that are motivated from practical considerations. Firstly, we introduce the concept of hyperDAGs to model precedence-constrained computations as hypergraphs, and we analyze the adaptation of the balanced partitioning problem to this case. Secondly, we study the hierarchical partitioning problem to model hierarchical NUMA (non-uniform memory access) effects in modern computer architectures, and we show that ignoring this hierarchical aspect of the communication cost can yield significantly weaker solutions.