Beyond 2-Approximation for k-Center in Graphs

2025-03-12

ACM-SIAM Symposium on Discrete Algorithms

AI Summary

This paper studies approximation algorithms for the undirected graph $k$-Center problem, breaking the long-standing belief that a 2-approximation is optimal. We present the first sub-2 multiplicative approximation algorithm with small additive error, establishing precise trade-offs between approximation ratio and running time under fine-grained complexity assumptions. Assuming the Strong Exponential Time Hypothesis (SETH), we prove that a $(3/2, O(1))$-approximation is optimal. Technically, our approach integrates structural analysis of graph distances, combinatorial optimization, and fast matrix multiplication (e.g., achieving $n^{omega/3}$ speedups). For any $k geq 2$, we achieve a $(2 - 1/(2k-1),, 1 - 1/(2k-1))$-bicriteria approximation in $O(mn + n^{k/2+1})$ time. In particular, for $k=2$, we obtain a $(5/3, 2/3)$-approximation in $ ilde{O}(mn^{omega/3})$ time—faster than computing all-pairs shortest paths (APSP).

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We consider the classical $k$-Center problem in undirected graphs. The problem is known to have a polynomial-time 2-approximation. There are even $(2+varepsilon)$-approximations running in near-linear time. The conventional wisdom is that the problem is closed, as $(2-varepsilon)$-approximation is NP-hard when $k$ is part of the input, and for constant $kgeq 2$ it requires $n^{k-o(1)}$ time under SETH. Our first set of results show that one can beat the multiplicative factor of $2$ in undirected unweighted graphs if one is willing to allow additional small additive error, obtaining $(2-varepsilon,O(1))$ approximations. We provide several algorithms that achieve such approximations for all integers $k$ with running time $O(n^{k-delta})$ for $delta>0$. For instance, for every $kgeq 2$, we obtain an $O(mn + n^{k/2+1})$ time $(2 - frac{1}{2k-1}, 1 - frac{1}{2k-1})$-approximation to $k$-Center. For $2$-Center we also obtain an $ ilde{O}(mn^{omega/3})$ time $(5/3,2/3)$-approximation algorithm. Notably, the running time of this $2$-Center algorithm is faster than the time needed to compute APSP. Our second set of results are strong fine-grained lower bounds for $k$-Center. We show that our $(3/2,O(1))$-approximation algorithm is optimal, under SETH, as any $(3/2-varepsilon,O(1))$-approximation algorithm requires $n^{k-o(1)}$ time. We also give a time/approximation trade-off: under SETH, for any integer $tgeq 1$, $n^{k/t^2-1-o(1)}$ time is needed for any $(2-1/(2t-1),O(1))$-approximation algorithm for $k$-Center. This explains why our $(2-varepsilon,O(1))$ approximation algorithms have $k$ appearing in the exponent of the running time. Our reductions also imply that, assuming ETH, the approximation ratio 2 of the known near-linear time algorithms cannot be improved by any algorithm whose running time is a polynomial independent of $k$, even if one allows additive error.

Problem

• Improving k-Center approximation beyond 2-factor in graphs.
• Developing faster algorithms with additive error for k-Center.
• Establishing fine-grained lower bounds for k-Center approximations.

Innovation

• Achieves (2-ε, O(1)) approximations for k-Center
• Provides O(mn + n^(k/2+1)) time algorithms
• Establishes strong fine-grained lower bounds