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Polylogarithmic Approximations for Robust s-t Path

Authors Shi Li , Chenyang Xu , Ruilong Zhang

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Author Details

Shi Li
  • Department of Computer Science and Technology, Nanjing University, Jiangsu, China
Chenyang Xu
  • Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, China
Ruilong Zhang
  • Department of Computer Science, City University of Hong Kong, Hong Kong, China

Funding

Chenyang Xu is supported by the National Key Research Project of China under Grant No. 2023YFA1009402, the National Natural Science Foundation of China (No. 62302166), the Dean’s Fund of Shanghai Key Laboratory of Trustworthy Computing, ECNU, and the Key Laboratory of Interdisciplinary Research of Computation and Economics (SUFE), Ministry of Education. Shi Li was supported by the State Key Laboratory for Novel Software Technology, and the New Cornerstone Science Laboratory.

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Shi Li, Chenyang Xu, and Ruilong Zhang. Polylogarithmic Approximations for Robust s-t Path. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 106:1-106:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.106

Abstract

The paper revisits the Robust s-t Path problem, one of the most fundamental problems in robust optimization. In the problem, we are given a directed graph with n vertices and k distinct cost functions (scenarios) defined over edges, and aim to choose an s-t path such that the total cost of the path is always provable no matter which scenario is realized. Viewing each cost function as an agent, our goal is to find a fair s-t path, which minimizes the maximum cost among all agents. The problem is NP-hard to approximate within a factor of o(log k) unless NP ⊆ DTIME(n^{polylog n}), and the best-known approximation ratio is Õ(√n), which is based on the natural flow linear program. A longstanding open question is whether we can achieve a polylogarithmic approximation for the problem; it remains open even if a quasi-polynomial running time is allowed. 
Our main result is a O(log n log k) approximation for the Robust s-t Path problem in quasi-polynomial time, solving the open question in the quasi-polynomial time regime. The algorithm is built on a novel linear program formulation for a decision-tree-type structure, which enables us to overcome the Ω(√n) integrality gap for the natural flow LP. Furthermore, we show that for graphs with bounded treewidth, the quasi-polynomial running time can be improved to a polynomial. We hope our techniques can offer new insights into this problem and other related problems in robust optimization.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Approximation Algorithm
  • Randomized LP Rounding
  • Robust s-t Path

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