Solving Hypergraph Laplacian Systems in Almost-Linear Time

2026-04-30

AI Summary

This work addresses the efficient solution of Poisson problems involving cut-based hypergraph Laplacians. By reformulating the Fenchel dual as a convex flow optimization problem on an auxiliary graph, the authors introduce a novel primal recovery theorem that exactly reconstructs the original potentials using only a single non-negative scalar per hyperedge, thereby circumventing the traditional reliance on detailed flow routing. Combining finite-precision rounding, minimum-cost flow algorithms, and a base-point reduction for regularized objectives, the proposed method outputs an approximately optimal solution in time $P^{1+o(1)}$ with high probability, where $P$ denotes the input size. The solution achieves an additive duality gap no larger than $\exp(-\log^C P)$, yielding nearly linear time complexity.

Abstract

For a connected weighted hypergraph, we give a randomized almost-linear-time solver for the Poisson problem for the cut-based hypergraph Laplacian in the natural input size $P=\sum_{e\in E}|e|$, the sum of hyperedge sizes. For every fixed constant $C>0$, our randomized algorithm runs in $P^{1+o(1)}$ time and, with high probability over its internal randomness, returns a primal point and a dual certificate, with additive optimality gap at most $\exp(-\log^C P)$. A key step is to rewrite the Fenchel dual as a convex-flow problem on an auxiliary $O(P)$-arc graph, yielding a near-optimal dual flow. The main difficulty is primal recovery, because this flow does not by itself determine a primal potential. Our main new ingredient is a recovery theorem showing that, for primal recovery, the detailed routing of the dual flow inside each hyperedge gadget can be discarded: one nonnegative scalar per hyperedge is enough. After the necessary finite-precision rounding, these scalars define a linear-cost min-cost-flow instance on the auxiliary graph, and solving it exactly recovers a primal potential. Finally, a ground-vertex reduction from regularized objectives to the Poisson solver gives randomized almost-linear-time resolvent/proximal primitives for the same cut-based hypergraph Laplacian.

Problem

• hypergraph Laplacian
• Poisson problem
• almost-linear time
• cut-based
• primal recovery

Innovation

• hypergraph Laplacian
• almost-linear-time solver
• primal recovery
• convex-flow
• min-cost flow