On $G^p$-unimodality of radius functions in graphs: structure and algorithms

TL;DR

This work introduces and exploits the notion of $G^p$-unimodality for radius functions $r_pi$ to tackle the center problem on broad graph classes. By proving $G^2$-unimodality for bridged, CB-graphs, cube-free median, and bipartite Helly graphs, the authors enable robust local-search strategies (sample-select-descent and ImproveEccentricity) that run subquadratically, e.g., $\tilde{O}(\sqrt{n}m)$ for several classes. They also present a deterministic $O(n\log^2 n)$ method for cube-free median graphs and extend unimodality analyses to δ-hyperbolic and α-coarse Helly graphs, yielding $O(\delta m)$ or related bounds in special regimes. Moreover, the paper offers polynomial-time recognition of $G^p$-unimodality and conditional lower bounds under the Hitting Set conjecture, framing the computational landscape of center problems in metric graphs. Collectively, the results advance subquadratic center-finding algorithms across key graph families and illuminate the geometric–combinatorial structure that enables efficient local search in metric graphs.

Abstract

For every weight assignment $π$ to the vertices in a graph $G$, the radius function $r_π$ maps every vertex of $G$ to its largest weighted distance to the other vertices. The center problem asks to find a center, i.e., a vertex of $G$ that minimizes $r_π$. We here study some local properties of radius functions in graphs, and their algorithmic implications; our work is inspired by the nice property that in Euclidean spaces every local minimum of every radius function $r_π$ is a center. We study a discrete analogue of this property for graphs, which we name $G^p$-unimodality: specifically, every vertex that minimizes the radius function in its ball of radius $p$ must be a central vertex. While it has long been known since Dragan (1989) that graphs with $G$-unimodal radius functions $r_π$ are exactly the Helly graphs, the class of graphs with $G^2$-unimodal radius functions has not been studied insofar. We prove the latter class to be much larger than the Helly graphs, since it also comprises (weakly) bridged graphs, graphs with convex balls, and bipartite Helly graphs. Recently, using the $G$-unimodality of radius functions $r_π$, a randomized $\widetilde{\mathcal{O}}(\sqrt{n}m)$-time local search algorithm for the center problem on Helly graphs was proposed by Ducoffe (2023). Assuming the Hitting Set Conjecture (Abboud et al., 2016), we prove that a similar result for the class of graphs with $G^2$-unimodal radius functions is unlikely. However, we design local search algorithms (randomized or deterministic) for the center problem on many of its important subclasses.

Figures (4)

• Figure 1: ImproveEccentricity$(G,\pi,v)$ for bridged graphs.
• Figure 2: Computing $S_1$ in the case of bipartite Helly graphs.
• Figure 3: ReduceConvexRegion for cube-free median graphs. Starting from a vertex $v$, its star $\mathop{\mathrm{St}}\nolimits(v)$ is in blue and given an improving neighbor $z$ of $v$, the parallel with $vz$ edges defining the halfspace $H(z,v)$ are drawn in bold red, the fiber $\Psi_{\mathop{\mathrm{St}}\nolimits(v)}(z)$ is drawn in stripped yellow, and its boundary $\Psi_{\mathop{\mathrm{St}}\nolimits(v)}(z)$ in orange.
• Figure 4: An example showing that the halfspace $H(z,v)$ for an i-neighbor $z$ of $v$ does not necessarily intersect $C_{\pi}(G)$.

Theorems & Definitions (108)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Lemma 2.1
• proof
• Lemma 3.1: Du_Helly
• Lemma 4.1
• proof