The Two-Center Problem of Uncertain Points on Trees

TL;DR

This work tackles the two-center problem for uncertain points on a tree, where each uncertain point $P_i$ has $m$ location possibilities with probabilities and weight $w_i$, and the objective $\phi(x_1,x_2)$ minimizes the maximum weighted expected distance to the nearer center. The authors develop an $O(mn)$-time decision algorithm that tests feasibility for a given threshold $\lambda$ by locating a peripheral-center edge and performing a structured pruning to cover all $P_i$; they then compute the optimal centers $q^*_1,q^*_2$ by recursively identifying two critical edges and using a line-arrangement approach to find $\lambda^*$, ultimately achieving $O(|T|+ mn\log mn)$ total time after reducing to the vertex-constrained case. The method leverages convexity of $\mathsf{E}\mathrm{d}(P_i,x)$ on paths, centroid-based pruning, and edge-detection lemmas to efficiently decompose the tree and solve the problem with provable efficiency. The results closely approach the path-case complexity and significantly improve prior general-$k$-center algorithms for uncertain points on trees, enabling practical applications in robust facility-location under uncertainty on tree-like networks.

Abstract

In this paper, we consider the (weighted) two-center problem of uncertain points on a tree. Given are a tree $T$ and a set $\calP$ of $n$ (weighted) uncertain points each of which has $m$ possible locations on $T$ associated with probabilities. The goal is to compute two points on $T$, i.e., two centers with respect to $\calP$, so that the maximum (weighted) expected distance of $n$ uncertain points to their own expected closest center is minimized. This problem can be solved in $O(|T|+ n^{2}\log n\log mn + mn\log^2 mn \log n)$ time by the algorithm for the general $k$-center problem. In this paper, we give a more efficient and simple algorithm that solves this problem in $O(|T| + mn\log mn)$ time.

Figures (3)

• Figure 1: The point $x$ has three split subtrees $T_1$, $T_2$ and $T_3$.
• Figure 2: Illustrating the tree $T_h$ for the case $C=2$ where $V =\{v_1, v_2\}$ and $\Gamma(V)=\{T^1_h, \cdots, T^6_h\}$.
• Figure 3: Illustrating an example for the peripheral-center detecting problem: $Y=\{y_1, y_2, \cdots, y_6\}$ and $\Gamma = \{T^+_1, T^+_2, \cdots, T^+_8\}$ shown with triangles.

Theorems & Definitions (17)

• lemma thmcounterlemma
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• lemma thmcounterlemma
• proof