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Data Reductions for the Strong Maximum Independent Set Problem in Hypergraphs

Ernestine Großmann, Christian Schulz, Darren Strash, Antonie Wagner

TL;DR

The paper tackles the strong Maximum Independent Set problem on hypergraphs and introduces nine exact data-reduction rules as a preprocessing step to shrink instances before solving. By integrating these reductions with ILP-based and graph-based solvers (via clique expansion), the authors demonstrate substantial empirical gains: on average reducing instance size to 22% in 6.76 seconds and achieving average speedups up to $3.84\times$ (with individual cases up to $53\times$) on the best configurations, as well as enabling one new solvable instance. The experimental evaluation across diverse hypergraph datasets shows consistent improvements across solvers, highlighting the practical value of reduction rules in accelerating MIS on hypergraphs. The work lays groundwork for further reductions, weighted generalizations, and heuristic strategies that can leverage preprocessing in large-scale hypergraph settings.

Abstract

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.

Data Reductions for the Strong Maximum Independent Set Problem in Hypergraphs

TL;DR

The paper tackles the strong Maximum Independent Set problem on hypergraphs and introduces nine exact data-reduction rules as a preprocessing step to shrink instances before solving. By integrating these reductions with ILP-based and graph-based solvers (via clique expansion), the authors demonstrate substantial empirical gains: on average reducing instance size to 22% in 6.76 seconds and achieving average speedups up to (with individual cases up to ) on the best configurations, as well as enabling one new solvable instance. The experimental evaluation across diverse hypergraph datasets shows consistent improvements across solvers, highlighting the practical value of reduction rules in accelerating MIS on hypergraphs. The work lays groundwork for further reductions, weighted generalizations, and heuristic strategies that can leverage preprocessing in large-scale hypergraph settings.

Abstract

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.
Paper Structure (14 sections, 1 theorem, 2 equations, 8 figures, 3 tables)

This paper contains 14 sections, 1 theorem, 2 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Related Work
  4. Methods for the MIS Problem on Hypergraphs
  5. Optimal Solutions
  6. Data Reductions
  7. Experimental Evaluation
  8. Reduction Impact on Instance Size
  9. Reduction Impact on Solver Performance
  10. Reduction Impact - Solver Comparison
  11. Reduction Impact - Focus on ILP and ILP$_G$
  12. Reduction Impact - Instance Comparison
  13. Solver Comparison
  14. Conclusion and Future Work

Key Result

Lemma 4

Let $S$ be an independent set that is contained in all maximum independent sets of $H$. Then, any maximum independent set has to contain one vertex $v \in N(u) \setminus N[S]$ for each child $u\in N(S)$.

Figures (8)

  • Figure 1: Left: A twin set $T = \{t_1,t_2,t_3\}$ with $\Delta_T = 1$, s.t. $|T| > \Delta_T$. Thus, all vertices in $T$ can be added to the solution $\mathcal{I}$ and $N[T]$ can be removed. Right: The twin set $T = \{t_1,t_2\}$ is of size $2$, but $\Delta_T = 3$. In this case, we do not add the vertices in $T$ to the solution.
  • Figure 2: A $3$-sunflower before (left) and after (right) the sunflower reduction.
  • Figure 3: An example of Reduction \ref{['isolatedcliquereduction']} with the original instance on the left, reduced to the instance on the right.
  • Figure 4: For each instance, we plot the remaining hypergraph size $|H|$ (in percent) and its reduction time $t$ (in seconds).
  • Figure 5: Solver-wise speedup achieved when using reductions. We present the geometric mean speedup over all instances solvable by all algorithms.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Definition 1: Sunflower
  • Definition 2: Simplicial Vertex
  • Definition 3: Child
  • Lemma 4