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Advances in Exact and Approximate Group Closeness Centrality Maximization

Christian Schulz, Jakob Ternes, Henning Woydt

Abstract

In the NP-hard \textsc{Group Closeness Centrality Maximization} problem, the input is a graph $G = (V,E)$ and a positive integer $k$, and the task is to find a set $S \subseteq V$ of size $k$ that maximizes the reciprocal of group farness $f(S) = \sum_{v \in V} \min_{s \in S} \text{dist}(v,s)$. A widely used greedy algorithm with previously unknown approximation guarantee may produce arbitrarily poor approximations. To efficiently obtain solutions with quality guarantees, known exact and approximation algorithms are revised. The state-of-the-art exact algorithm iteratively solves ILPs of increasing size until the ILP at hand can represent an optimal solution. In this work, we propose two new techniques to further improve the algorithm. The first technique reduces the size of the ILPs while the second technique aims to minimize the number of needed iterations. Our improvements yield a speedup by a factor of $3.6$ over the next best exact algorithm and can achieve speedups by up to a factor of $22.3$. Furthermore, we add reduction techniques to a $1/5$-approximation algorithm, and show that these adaptations do not compromise its approximation guarantee. The improved algorithm achieves mean speedups of $1.4$ and a maximum speedup of up to $2.9$ times.

Advances in Exact and Approximate Group Closeness Centrality Maximization

Abstract

In the NP-hard \textsc{Group Closeness Centrality Maximization} problem, the input is a graph and a positive integer , and the task is to find a set of size that maximizes the reciprocal of group farness . A widely used greedy algorithm with previously unknown approximation guarantee may produce arbitrarily poor approximations. To efficiently obtain solutions with quality guarantees, known exact and approximation algorithms are revised. The state-of-the-art exact algorithm iteratively solves ILPs of increasing size until the ILP at hand can represent an optimal solution. In this work, we propose two new techniques to further improve the algorithm. The first technique reduces the size of the ILPs while the second technique aims to minimize the number of needed iterations. Our improvements yield a speedup by a factor of over the next best exact algorithm and can achieve speedups by up to a factor of . Furthermore, we add reduction techniques to a -approximation algorithm, and show that these adaptations do not compromise its approximation guarantee. The improved algorithm achieves mean speedups of and a maximum speedup of up to times.

Paper Structure

This paper contains 15 sections, 3 theorems, 6 equations, 7 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Related Work
  4. Exact Algorithms
  5. Heuristic and Approximate Algorithms
  6. Grover
  7. Reducing Iterations
  8. Absorbed Vertices
  9. The Complete Algorithm
  10. Approximation Results
  11. Experimental Evaluation
  12. Absorbed Vertices and Improved $d(v)$
  13. Comparison to SOTA
  14. Approximation
  15. Conclusion and Future Work

Key Result

Theorem 1

Let a reduced optimized ILP be sufficient, i.e., after optimizing the ILP it holds that $x_{v,\widetilde{d}(v)} = 0$ with $\widetilde{d}(v) < \text{ecc}(v)$ or $x_{v, \widetilde{d}(v)} = 1$ with $\widetilde{d}(v) \geq \text{ecc}(v)$. Then $S^* = \{v \in V \setminus D \mid x_{v, 0} = 1\}$ is a global

Figures (7)

  • Figure 1: Figure showing absorbed vertices. In case (a) and (b) the red vertices can be absorbed by their neighboring blue vertex. In case (b) any combination of the dashed edges results in the blue vertex absorbing its neighbors. In case (c) no vertex can be absorbed.
  • Figure 3: Speedup of different optimizations over Base. Base uses dominating vertices and $d(v) = 2$. Absorbing vertices AV and $\widetilde{d}(v)$ are evaluated separately. Grover uses both.
  • Figure 4: Speedup of Grover initialized with an optimal solution when compared to the default initialization with GS-LS-Cangriman2023algorithms. Speedups are determined per instance and then sorted.
  • Figure 5: Number of solved instances for Grover (ours), ILPindstaus2023exact, DVindstaus2023exact and SubModSTWoydt24. Grover solves the most instances in the least amount of time.
  • Figure 6: Distribution of number of iterations that Grover (ours) and ILPindstaus2023exact need to solve all instances. Grover solves more instances in fewer iterations.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Corollary 3