Near-Optimal Minimum Cuts in Hypergraphs at Scale

TL;DR

HeiCut tackles the hypergraph minimum cut problem by integrating provably exact reductions that aggressively shrink the instance with guaranteed preservation of the true minimum cut, followed by solving a near-optimal relaxed Binary Integer Linear Program on the reduced hypergraph; it also offers an optional label-propagation–based heuristic to further reduce problem size. The approach yields strong empirical performance, computing the exact minimum cut in over 85% of real-world cases using only the reductions and achieving substantial speedups (up to five orders of magnitude faster) over prior state-of-the-art solvers. Extensive experiments on more than 500 real-world and synthetic hypergraphs demonstrate robust solution quality, scalability, and memory efficiency, with HeiCut solving more instances than baselines and often requiring far less computational resources. The work also introduces a synthetic $(k,2)$-core hypergraph benchmark to stress-test solvers and highlights the practical viability of hypergraph minimum cuts as a subroutine in large-scale optimization tasks.

Abstract

The hypergraph minimum cut problem aims to partition its vertices into two blocks while minimizing the total weight of the cut hyperedges. This fundamental problem arises in network reliability, VLSI design, and community detection. We present HeiCut, a scalable algorithm for computing near-optimal minimum cuts in both unweighted and weighted hypergraphs. HeiCut aggressively reduces the hypergraph size through a sequence of provably exact reductions that preserve the minimum cut, along with an optional heuristic contraction based on label propagation. It then solves a relaxed Binary Integer Linear Program (BIP) on the reduced hypergraph to compute a near-optimal minimum cut. Our extensive evaluation on over 500 real-world hypergraphs shows that HeiCut computes the exact minimum cut in over 85% of instances using our exact reductions alone, and offers the best solution quality across all instances. It solves over twice as many instances as the state-of-the-art within set computational limits, and is up to five orders of magnitude faster.

Figures (9)

• Figure 1: Depictions of our various reduction rules, excluding Rule 1, which is self-evident.
• Figure 2: A sample hypergraph with a minimum cut value of six. When using a non-strict inequality in Reduction Rule \ref{['pr:viecut_2']}, we contract the green hyperedges simultaneously, which increases the minimum cut in the contracted hypergraph.
• Figure 3: Reduction in hypergraph size over each exact reduction (no label propagation) on $M_{HG}$ instances. Each line shows one hypergraph. Reduction is measured as the percentage of remaining edges after each reduction, i.e., the reduced hypergraph size as a percentage of the input.
• Figure 4: $M_{HG}$ Unweighted
• Figure 5: $M_{HG}$ Weighted