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Scalable Algorithms for 2-Packing Sets on Arbitrary Graphs

Jannick Borowitz, Ernestine Großmann, Christian Schulz, Dominik Schweisgut

TL;DR

This work tackles the NP-hard maximum 2-packing set problem on arbitrary graphs by introducing red2pack, a three-phase framework that interleaves problem-specific data reductions, a graph transformation to a square graph, and maximum independent set (MIS) solving on the transformed instance. The authors prove an equivalence between MIS on the square graph G² and 2-packing on the original graph G, enabling the use of state-of-the-art MIS solvers after kernelization. They present both an exact variant (red2pack_b&r) and a heuristic variant (red2pack_heuristic), and show that the approach dramatically outperforms existing methods for arbitrary graphs in solution quality and speed, solving a large share of instances optimally within seconds and many previously unsolved large instances. On planar graphs, the method remains competitive, approaching the performance of specialized solvers while offering substantial speed advantages in many cases. The contributions include novel data-reduction rules, a robust graph-transformation pipeline, and publicly available code, with future work targeting additional reductions, weighted and larger-k packing variants, and hypergraph-based motifs.

Abstract

A 2-packing set for an undirected graph $G=(V,E)$ is a subset $\mathcal{S} \subset V$ such that any two vertices $v_1,v_2 \in \mathcal{S}$ have no common neighbors. Finding a 2-packing set of maximum cardinality is a NP-hard problem. We develop a new approach to solve this problem on arbitrary graphs using its close relation to the independent set problem. Thereby, our algorithm red2pack uses new data reduction rules specific to the 2-packing set problem as well as a graph transformation. Our experiments show that we outperform the state-of-the-art for arbitrary graphs with respect to solution quality and also are able to compute solutions multiple orders of magnitude faster than previously possible. For example, we are able to solve 63% of the graphs in the tested data set to optimality in less than a second while the competitor for arbitrary graphs can only solve 5% of these graphs to optimality even with a 10 hour time limit. Moreover, our approach can solve a wide range of large instances that have previously been unsolved.

Scalable Algorithms for 2-Packing Sets on Arbitrary Graphs

TL;DR

This work tackles the NP-hard maximum 2-packing set problem on arbitrary graphs by introducing red2pack, a three-phase framework that interleaves problem-specific data reductions, a graph transformation to a square graph, and maximum independent set (MIS) solving on the transformed instance. The authors prove an equivalence between MIS on the square graph G² and 2-packing on the original graph G, enabling the use of state-of-the-art MIS solvers after kernelization. They present both an exact variant (red2pack_b&r) and a heuristic variant (red2pack_heuristic), and show that the approach dramatically outperforms existing methods for arbitrary graphs in solution quality and speed, solving a large share of instances optimally within seconds and many previously unsolved large instances. On planar graphs, the method remains competitive, approaching the performance of specialized solvers while offering substantial speed advantages in many cases. The contributions include novel data-reduction rules, a robust graph-transformation pipeline, and publicly available code, with future work targeting additional reductions, weighted and larger-k packing variants, and hypergraph-based motifs.

Abstract

A 2-packing set for an undirected graph is a subset such that any two vertices have no common neighbors. Finding a 2-packing set of maximum cardinality is a NP-hard problem. We develop a new approach to solve this problem on arbitrary graphs using its close relation to the independent set problem. Thereby, our algorithm red2pack uses new data reduction rules specific to the 2-packing set problem as well as a graph transformation. Our experiments show that we outperform the state-of-the-art for arbitrary graphs with respect to solution quality and also are able to compute solutions multiple orders of magnitude faster than previously possible. For example, we are able to solve 63% of the graphs in the tested data set to optimality in less than a second while the competitor for arbitrary graphs can only solve 5% of these graphs to optimality even with a 10 hour time limit. Moreover, our approach can solve a wide range of large instances that have previously been unsolved.
Paper Structure (18 sections, 2 theorems, 6 figures, 6 tables, 1 algorithm)

This paper contains 18 sections, 2 theorems, 6 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Related Work
  4. 2-Packing Set Algorithms
  5. Independent Set Algorithms
  6. Data Reduction Rules
  7. The Algorithm red2pack
  8. Data Reduction Rules
  9. Miscellaneous
  10. Data Structure
  11. Graph Transformation
  12. MIS Solver
  13. Experimental Evaluation
  14. Impact of Data Reductions
  15. Comparison against State-of-the-Art
  16. ...and 3 more sections

Key Result

Lemma 1

Let $u,v \in V$ be neighbors in $\mathcal{G}$ with $N[v]\subseteq N[u]$ such that $\textnormal{deg}^2(v) +\deg(v) \leq \deg(u)$. Then, $N_2(v) = N[u]\setminus N[v]$, i.e. all 2-neighbors of the vertex $v$, are also neighbors of the vertex $u$.

Figures (6)

  • Figure 1: Original graph on the left reduced to the graph on the right. Due to clarity reasons, we only display the necessary edges from $\mathcal{E}$ (dashed) regarding the 2-neighborhood information.
  • Figure 2: Demonstration of the vital role of retaining 2-neighborhood information for non-reduced vertices in maintaining a valid solution to the M2S problem. Reduced vertices and edges are light gray shaded. Green vertices are included, red vertices are excluded from the solution.
  • Figure 3: The graph ${\mathcal{G}=(V,E,\mathcal{E})}$ on the left with edges in $\mathcal{E}$ marked as dashed lines. This graph is transformed to ${G^2=(V,E^2)}$ on the right. The green vertices present the M2S (left) and the MIS (right).
  • Figure 4: Solution quality (left) and running time (right) for red2pack_b&r on planar and social graphs comparing our different reduction variants.
  • Figure 5: State of the art comparison on solution quality (top) and running time (bottom) for different graph classes. For planar graphs (right) the competitor gen2pack is not able to solve the instances. To this end we add the algorithm Apx-2p + Imp2p with $h=50$ and $h=100$ as well as red2pack_heuristic for comparison.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Theorem 2