Beyond 2-approximation for k-Center in Graphs

TL;DR

This work breaks the long-standing barrier of a 2-approximation for k-Center on graphs by introducing fast (2−ε,β)−approximations with β=O(1) and by establishing tight fine-grained lower bounds under SETH and ETH. It provides a spectrum of algorithmic techniques, including sampling-based methods, fast matrix multiplication, and careful gadget-based reductions, to achieve (3/2,1/2) and (2−1/(2k−1),1−1/(2k−1)) guarantees with runtimes that scale favorably with k and the graph size. The paper also demonstrates strong hardness: under SETH, improving the multiplicative factor below 2 without sacrificing additive error is infeasible, and it details a runtime/approximation tradeoff streamlining the exponent in k. A notable practical contribution is the near-linear-time improvement for 2-Center (e.g., a (5/3,2/3) approximation in Õ(m n^{ω/3})) and a generalized framework for approximating 3-Center with subquadratic performance in many graph regimes. Overall, the results illuminate fundamental limits and unlock faster mixed-approximation algorithms that are relevant for large-scale graph clustering tasks.

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 $k\geq 2$ 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 $O(n^{k-δ})$ for $δ>0$. For instance, for every $k\geq 2$, 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 $\tilde{O}(mn^{ω/3})$ 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 $t\geq 1$, $n^{k/t^2-1-o(1)}$ 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.

Figures (6)

• Figure 1: The $k$-center instance $G'=(V',E')$ constructed in the proof of \ref{['thm:simplelb']} (for distinguishing radius $\le 2\ell$ or $\ge 3\ell$). In this example, $\ell=3$. The dashed green box contains the base gadget graph $\mathrm{Gad}(A',B',c',\ell)$. Given a Set Cover solution of size $k$, picking their copies in $A'$ gives a $k$-center solution of radius $2\ell$.
• Figure 2: The $(k+1)$-center instance $G'=(V',E')$ constructed in the proof of \ref{['thm:simplelb']} (for distinguishing radius $\le 3\ell$ or $\ge 5\ell$). Here we use a thick segment to denote a path of prescribed number of edges. Given a Set Cover solution of size $k$ (this example has $k=2$ attained by $\{a_1,a_2\}$), picking their copies in $A'$ together with $c'$ gives a $(k+1)$-center solution of radius $3\ell$. In this example, on the $2\ell$-edge path from $a'_3\in A'$ to $b'_3\in B'$, the first half of nodes are covered by center $c'$, and the second half are covered by center $a_2'$.
• Figure 3: The $(2k+2)$-center instance $G'=(V',E')$ constructed in the proof of \ref{['thm:betterlbwarmup2']} (distinguishing between radius $\le 5\ell$ or $\ge 9\ell$). For example, the $4\ell$-edge path between $a'$ and $b'$ can be covered within radius $5\ell$ by centers $a'_{\star}, \bar{a}_{\star}, c'$.
• Figure 4: The "skeleton graph" of the graph $G'=(V',E')$ in \ref{['fig2:95lb']}. Here, each super-node represents a subset of nodes in $G'$, and the lengths denote shortest distances between subsets of nodes.
• Figure 5: The "skeleton graph" of graph $G'$ constructed by \ref{['alg:lbrecursion']} for $t=7$ (for distinguishing between radius $\le 15\ell$ or $\ge 29\ell$ for $f(7)\cdot (k+1)$-center). To avoid clutter, we do not draw the graph $G'$ itself; one can refer to \ref{['fig2:95lb']} and \ref{['fig3:95simple']} to understand the relation between $G'$ and its skeleton graph by analogy.

Theorems & Definitions (95)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 3.1: Equivalent form of \ref{['thm:fulllb']}
• Theorem 3.2: Williams05PatrascuW10
• Lemma 3.3: SETH-hardness of Gap Set Cover, implied by SLM19
• Corollary 3.4: Small-$B$ version of \ref{['templem:approxsetcoverlb']}
• proof