The Connected k-Vertex One-Center Problem on Graphs

TL;DR

The paper introduces a generalized connected $k$-vertex one-center problem on graphs, formalizing the objective as $\phi(x,G)=\min_{T^k\in G^k(x)}\max_{v\in V(T^k)} w_v d(v,x)$ and provides algorithms with distinct complexity bounds for weighted graphs, unweighted graphs (with a distance matrix), and trees. The core approach combines a feasibility test for a given $\lambda$ with a line-arrangement search to locate $\lambda^*$, and develops per-edge analyses to determine the largest self-inclusive subtree covered by a point. For unweighted graphs and trees, the authors exploit geometry (k-th level of monotone chains) and centroid/spine decompositions to achieve faster runtimes. The results yield $O(mn\log n\log mn + m^2\log n\log mn)$ time for the weighted graph case, $O(mn\log n)$ for the unweighted graph case with a distance matrix, and $O(n\log^2 n\log k)$ (weighted) or $O(n\log^2 n)$ (unweighted) on trees, highlighting significant improvements and specialized methods in these settings. The work advances the understanding of partial center problems and introduces techniques that may inspire future work on related network-design problems and partial p-center variants.

Abstract

We consider a generalized version of the (weighted) one-center problem on graphs. Given an undirected graph $G$ of $n$ vertices and $m$ edges and a positive integer $k\leq n$, the problem aims to find a point in $G$ so that the maximum (weighted) distance from it to $k$ connected vertices in its shortest path tree(s) is minimized. No previous work has been proposed for this problem except for the case $k=n$, that is, the classical graph one-center problem. In this paper, an $O(mn\log n\log mn + m^2\log n\log mn)$-time algorithm is proposed for the weighted case, and an $O(mn\log n)$-time algorithm is presented for the unweighted case, provided that the distance matrix for $G$ is given. When $G$ is a tree graph, we propose an algorithm that solves the weighted case in $O(n\log^2 n\log k)$ time with no given distance matrix, and improve it to $O(n\log^2 n)$ for the unweighted case.

Figures (7)

• Figure 1: Illustrating the three cases of the (weighted) distance function $D(v,x)$ for $x\in e(r,s)$: As $x$ moves from $r$ to $s$ on $e$, at rate $w_v$, $D(v,x)$ increases, or decreases, or first increases until $v$'s semicircular point $x'$ and then decreases.
• Figure 2: Illustrating that function $f_r(e,x)$ (the heavy line segments) is a piecewise constant function in $x\in e$ that breaks and falls at points where functions $D(v,x)$ increase up to $\lambda$, e.g., $x_2$ and $x_4$, or semicircular points, e.g., $x_1$ and $x_3$.
• Figure 3: Illustrating the $6$-level closure (the heavy chain) of seven functions $D(v,x)$, which has five vertices $p_1, p_2, p_3, p_4, p_5$. It cannot turns at the intersection $p_{1,2}$.
• Figure 6: Illustrating the counting and reporting query of $T_\lambda(x)$ for $T$ being a balanced binary search tree.
• Figure 7: Illustrating the spine decomposition of a tree. On the binary search tree for the root spine from $v_1$ to $v_8$, at an internal node $u$, $S(u)$ is the subspine $\pi(v_5, v_8)$ so $V^t_u = u_5$ and $V^b_u = u_8$, $T(u)$ consists of $S(u)$ and all its hanging subtrees, $E(u) = e(v_6,v_7)$ connects the subtrees in $u$'s two children $L_u$ and $R_u$. At a leaf $u'$, $S(u')=v_2$, $T(u')$ is the left subtree rooted at $S(u')$, $E(u')$ connects $S(u')$ and its hanging subtree.

Theorems & Definitions (33)

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