Approximating optimization problems in graphs with locational uncertainty

TL;DR

This work studies optimization problems on graphs under locational uncertainty, formalizing robust-$\Pi$ as minimizing the worst-case cost $c(G)=\sum_{\{i,j\}\in E[G]} d(u_i,u_j)$ across uncertainty sets. The authors present two polynomial-time strategies: (i) reduce the robust problem to a deterministic instance using worst-case distances $d^{max}$, with a transfer theorem showing a constant-factor approximation (at most 9 in general, 4 for Ptolemaic distances), and (ii) a dynamic-programming-based FPTAS for the robust shortest path when the uncertainty is restricted to $s$-$t$ paths, enabling $(1+\varepsilon)$-approximate solutions in time polynomial in $n$ and $1/\varepsilon$ for fixed $\sigma=|\mathcal{U}_i|$. The paper also derives tight, structure-specific bounds for the ratio $c_{d^{max}}(G)/c(G)$ across graph families (paths, cycles, cliques, stars, trees) and distance geometries, including improved constants under Euclidean and Ptolemaic assumptions. Collectively, these results show that robust optimization with locational uncertainty can be approximated nearly as efficiently as its deterministic counterpart, and provide practical schemes for robust network design and path problems. The findings advance understanding of how worst-case uncertainty propagates through graph structures and offer actionable algorithms for planners facing locational ambiguity.

Abstract

Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take any position in given uncertainty sets. Then, the cost function to be minimized is the sum of the distances for the worst positions of the vertices in their uncertainty sets. We propose two types of polynomial-time approximation algorithms. The first one relies on solving a deterministic counterpart of the problem where the uncertain distances are replaced with maximum pairwise distances. We study in details the resulting approximation ratio, which depends on the structure of the feasible subgraphs and whether the metric space is Ptolemaic or not. The second algorithm is a fully-polynomial time approximation scheme for the special case of $s-t$ paths.

Figures (1)

• Figure 1: Counter example of Lemma \ref{['lem:triangle']} for non-Ptolemaic distances.

Theorems & Definitions (50)

• Remark 1
• proof
• Theorem 1
• proof
• Theorem 2
• Lemma 1
• proof
• Theorem 3
• proof
• Example 1