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The Two-Center Problem of Uncertain Points on Cactus Graphs

Haitao Xu, Jingru Zhang

TL;DR

This work solves the two-center problem for uncertain demands on cactus graphs by reducing to a vertex-constrained instance and exploiting a tree representation T that contracts cycles to nodes. A threshold-based decision algorithm on T identifies peripheral-center edges or cycles and tests feasibility, with the cycle case governing the leading cost at O(m^2 n^2). The overall complexity is O(|G| + m^2 n^2 log mn), and the authors also provide a λ* computation framework via line-arrangement ideas. This first result for cactus graphs with m>1 advances facility-location under uncertainty and opens questions about extending to general k-center problems on such graphs.

Abstract

We study the two-center problem on cactus graphs in facility locations, which aims to place two facilities on the graph network to serve customers in order to minimize the maximum transportation cost. In our problem, the location of each customer is uncertain and may appear at $O(m)$ points on the network with probabilities. More specifically, given are a cactus graph $G$ and a set $\calP$ of $n$ (weighted) uncertain points where every uncertain point has $O(m)$ possible locations on $G$ each associated with a probability and is of a non-negative weight. The problem aims to compute two centers (points) on $G$ so that the maximum (weighted) expected distance of the $n$ uncertain points to their own expected closest center is minimized. No previous algorithms are known for this problem. In this paper, we present the first algorithm for this problem and it solves the problem in $O(|G|+ m^{2}n^{2}\log mn)$ time.

The Two-Center Problem of Uncertain Points on Cactus Graphs

TL;DR

This work solves the two-center problem for uncertain demands on cactus graphs by reducing to a vertex-constrained instance and exploiting a tree representation T that contracts cycles to nodes. A threshold-based decision algorithm on T identifies peripheral-center edges or cycles and tests feasibility, with the cycle case governing the leading cost at O(m^2 n^2). The overall complexity is O(|G| + m^2 n^2 log mn), and the authors also provide a λ* computation framework via line-arrangement ideas. This first result for cactus graphs with m>1 advances facility-location under uncertainty and opens questions about extending to general k-center problems on such graphs.

Abstract

We study the two-center problem on cactus graphs in facility locations, which aims to place two facilities on the graph network to serve customers in order to minimize the maximum transportation cost. In our problem, the location of each customer is uncertain and may appear at points on the network with probabilities. More specifically, given are a cactus graph and a set of (weighted) uncertain points where every uncertain point has possible locations on each associated with a probability and is of a non-negative weight. The problem aims to compute two centers (points) on so that the maximum (weighted) expected distance of the uncertain points to their own expected closest center is minimized. No previous algorithms are known for this problem. In this paper, we present the first algorithm for this problem and it solves the problem in time.

Paper Structure

This paper contains 8 sections, 13 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Approach
  4. Preliminaries
  5. The Decision Algorithm
  6. Proofs of Lemmas
  7. Computing $\lambda^*$
  8. Conclusion

Key Result

lemma 1

Any general instance of the two-center problem can be reduced to a vertex-constrained instance in $O(|G| + mn)$ time.

Figures (1)

  • Figure 1: (a) Illustrating a cactus graph that consists of three cycles $\{C_1, C_2, C_3\}$, two hinges $\{h_1, h_2\}$ (squares), and six G-vertices (disks). Cycle $C_2$ has four split subgraphs $\{G_1, G_2, G_3, G_4\}$, and four hanging subgraphs $\{C_1, C_3, G_3\cup\{h_2\}, G_4\cup\{h_2\}\}$. (b) Illustrating the tree representation of the cactus graph in (a) where each cycle is uniquely represented by a (circular) node. Specifically, node $u_1$ is for cycle $C_1$, node $u_2$ for cycle $C_2$, and node $u_3$ for cycle $C_3$. Additionally, $u_2$ and its all adjacent hinge nodes induce its $H$-subtree $T^H_{u_2}$. Every hanging subgraph of $C_2$ is uniquely represented by a hanging subtree of $T^H_{u_2}$.

Theorems & Definitions (22)

  • lemma 1
  • proof
  • lemma 2
  • lemma 3
  • lemma 4
  • lemma 5
  • proof
  • lemma 6
  • proof
  • lemma 7
  • ...and 12 more