Computing the Center of Uncertain Points on Cactus Graphs

TL;DR

This paper addresses the weighted one-center problem for n uncertain points on a cactus graph, where each uncertain point has m possible locations with probabilities. It introduces a linear-time reduction to a vertex-constrained instance and leverages a skeleton tree decomposition of the cactus to guide a binary-search-like center localization, resolving a center-detecting subproblem on blocks, including a cycle case handled by dynamic convex hulls. The main result is an O(|G| + mn log mn) time algorithm, near-optimal given the input size, which extends prior tree-centric and path-centric results to cactus graphs. The approach provides a general framework for efficient center computation under uncertainty on complex graph families and suggests directions for further optimization of cycle subproblems and broader uncertain-graph center problems.

Abstract

In this paper, we consider the (weighted) one-center problem of uncertain points on a cactus graph. Given are a cactus graph $G$ and a set of $n$ uncertain points. Each uncertain point has $m$ possible locations on $G$ with probabilities and a non-negative weight. The (weighted) one-center problem aims to compute a point (the center) $x^*$ on $G$ to minimize the maximum (weighted) expected distance from $x^*$ to all uncertain points. No previous algorithm is known for this problem. In this paper, we propose an $O(|G| + mn\log mn)$-time algorithm for solving it. Since the input is $O(|G|+mn)$, our algorithm is almost optimal.

Figures (3)

• Figure 1: (a) Illustrating a cactus $G$ that consists of $3$ cycles, $5$ hinges (squares) and $6$ G-vertices (disks); (b) Illustrating $G$'s skeleton $T$ where circular and disk nodes represent cycles and grafts of $G$ (e.g., nodes $u$, $u_C$ and $u_B$ respectively representing hinge $h$, cycle $C$ and graft $B$ on $G$).
• Figure 2: (a) Cycle $C$ on $G$ has $7$ split subgraphs (blue dash doted lines) and accordingly $7$ hanging subgraphs (red dashed lines); (b) on $T$, the H-subtree of node $u_c$ representing cycle $C$ has $7$ split subtrees each of which represents a distinct hanging subgraph of $C$ on $G$.
• Figure 3: Consider $y=\mathsf{E}\mathrm{d}(P_i,x)$ in $x,y$-coordinate system by projecting cycle $G$ onto $x$-axis; $\mathsf{E}\mathrm{d}(P_i,x)$ of each $P_i\in\mathcal{P}$ is linear in $x$ on any edge of $G$; center $x^*$ is decided by the projection on $x$-axis of the lowest point on the upper envelope of all $y=\mathsf{E}\mathrm{d}(P_i,x)$'s.

Theorems & Definitions (20)

• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• corollary thmcountercorollary
• corollary thmcountercorollary
• lemma thmcounterlemma
• proof
• corollary thmcountercorollary
• theorem thmcountertheorem