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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1292_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">P</mi> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1292_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="159" /> </InlineMediaObject> <EquationSource Format="TEX">$O(|G|+ m^{2}n^{2}\log mn)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>G</mi> <mo stretchy="false">|</mo> <mo>+</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <msup> <mi>n</mi> <mn>2</mn> </msup> <mo>log</mo> <mi>m</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> time.

The two-center problem of uncertain points on cactus graphs

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2025年4月21日

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Haitao Xu,
Jingru Zhang

摘要

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 $\mathcal {P}$ $P$ 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 $O(|G|+ m^{2}n^{2}\log mn)$ $O (| G | + m^{2} n^{2} log m n)$ time.