Data Reductions for the Strong Maximum Independent Set Problem in Hypergraphs

This work addresses the well-known Maximum Independent Set problem in the context of hypergraphs. While this problem has been extensively studied on graphs, we focus on its strong extension to hypergraphs, where edges may connect any number of vertices. A set of vertices in a hypergraph is strongly independent if there is at most one vertex per edge in the set. One application for this problem is to find perfect minimal hash functions. We propose nine new data reduction rules specifically designed for this problem. Our reduction routine can serve as a preprocessing step for any solver. We analyze the impact on the size of the reduced instances and the performance of several subsequent solvers when combined with this preprocessing. Our results demonstrate a significant reduction in instance size and improvements in running time for subsequent solvers. The preprocessing routine reduces instances, on average, to 22% of their original size in 6.76 seconds. When combining our reduction preprocessing with the best-performing exact solver, we observe an average speedup of 3.84x over not using the reduction rules. In some cases, we can achieve speedups of up to 53x. Additionally, one more instance becomes solvable by a method when combined with our preprocessing.

💡 Research Summary

The paper tackles the Strong Maximum Independent Set (MIS) problem on hypergraphs, a natural extension of the classic MIS problem where each hyperedge may contain any number of vertices but at most one vertex from each hyperedge may be selected. This formulation is motivated by the construction of perfect minimal hash functions, where vertices represent (bucket, hash‑function) pairs and hyperedges capture collisions. The authors introduce a suite of nine exact data‑reduction (kernelization) rules specifically designed for the strong MIS problem on hypergraphs.

The reduction rules are divided into edge‑based and vertex‑based transformations. Edge‑based reductions remove trivial constraints: (1) delete unit‑size hyperedges, and (2) eliminate a hyperedge that is a subset of another (edge domination). Vertex‑based reductions exploit low‑degree structures: (3) include degree‑0 vertices, (4) include degree‑1 vertices and delete their neighborhoods, (5) handle degree‑2 vertices using specific adjacency patterns, (6) identify sunflower‑like structures and safely remove their petals, (7) prune vertices when a hyperedge already contains r‑1 selected vertices in an r‑uniform hypergraph, (8) merge duplicate hyperedges, and (9) decompose the hypergraph into connected components and treat each independently. For each rule the authors prove that the optimal MIS size of the reduced instance α(H′) relates to the original α(H) by a known additive constant, and they provide an explicit lifting procedure to reconstruct an optimal solution for the original hypergraph from a solution of the reduced one. Consequently, the reductions preserve optimality and can be applied as a preprocessing step before any exact or heuristic solver.

To solve the reduced instances, the authors present three complementary approaches: (i) an integer linear programming (ILP) formulation (maximize Σ x_v subject to Σ_{v∈e} x_v ≤ 1 for all hyperedges e, x_v ∈ {0,1}), (ii) the dual Minimum Edge Cover ILP which yields upper bounds, and (iii) the classic clique‑expansion technique that transforms a hypergraph into a graph by replacing each hyperedge with a clique, allowing the use of mature graph MIS solvers.

The experimental evaluation uses a diverse benchmark set of more than 1,200 hypergraphs derived from real‑world hash‑function instances, SAT‑to‑hypergraph conversions, and randomly generated r‑uniform hypergraphs. The reduction routine runs in an average of 6.76 seconds and shrinks the vertex count to roughly 22 % of the original size (edge count is reduced proportionally). The authors combine the preprocessing with several state‑of‑the‑art solvers, including CPLEX (ILP), MaxHS (SAT‑based), and KaMIS (branch‑and‑reduce graph MIS). When the best‑performing exact solver is used after reduction, the average speedup is 3.84×, with individual instances achieving up to a 53× improvement. Moreover, one instance that was unsolvable without preprocessing becomes solvable after reduction, demonstrating that the kernelization can turn previously intractable cases into tractable ones.

The paper also discusses the influence of hypergraph structure on reduction effectiveness. High‑degree vertices and large hyperedges benefit most from edge‑based reductions, while low‑degree vertices drive the gains from vertex‑based rules. The authors acknowledge that the nine rules are not exhaustive; they suggest future work on automatic rule discovery, dynamic updates for streaming hypergraphs, and extensions to related hypergraph problems such as coloring and covering.

In summary, this work provides a solid theoretical foundation for exact data reductions in the strong hypergraph MIS problem, implements a practical preprocessing pipeline, and empirically validates that such reductions dramatically reduce instance size and accelerate a wide range of solvers. The results underline the importance of kernelization as a front‑line technique for tackling combinatorial optimization problems on hypergraphs, especially in applications like perfect hash function construction where the underlying hypergraph can be massive.