[Paper Review] Approximating the shortest path problem with scenarios

Adam Kasperski, Paweł Zieliński|arXiv (Cornell University)|Jun 23, 2018

Complexity and Algorithms in Graphs29 references3 citations

TL;DR

This paper presents deterministic approximation algorithms for the min-max shortest path and minimum spanning tree problems under multiple cost scenarios. It achieves an $ O(igtriangledown{n\log K/\log\log K}) $-approximation for shortest path and $ O(\log n \log K / \log\log K) $ for spanning tree in bounded-degree or planar graphs, with a randomized $ O(\log n) $-approximation for general graphs, improving prior bounds.

ABSTRACT

This paper discusses the shortest path problem in a general directed graph with $n$ nodes and $K$ cost scenarios (objectives). In order to choose a solution, the min-max criterion is applied. The min-max version of the problem is hard to approximate within $Ω(\log^{1-ε} K)$ for any $ε>0$ unless NP$\subseteq ext{DTIME}(n^{ ext{polylog} \,n})$ even for arc series-parallel graphs and within $Ω(\log n/\log\log n)$ unless NP$\subseteq ext{ZPTIME}(n^{\log\log n})$ for acyclic graphs. The best approximation algorithm for the min-max shortest path problem in general graphs, known to date, has an approximation ratio of~$K$. In this paper, an $\widetilde{O}(\sqrt{n})$ flow LP-based approximation algorithm for min-max shortest path in general graphs is constructed. It is also shown that the approximation ratio obtained is close to an integrality gap of the corresponding flow LP relaxation.

Motivation & Objective

• Address the computational hardness of min-max network problems under multiple cost scenarios.
• Develop improved approximation algorithms for the min-max shortest path and minimum spanning tree problems when the number of scenarios $ K $ is unbounded.
• Close the gap between known inapproximability bounds and existing approximation ratios for these robust optimization problems.
• Provide algorithms with provable performance guarantees that scale efficiently with $ n $ and $ K $.

Proposed method

• Formulate the min-max shortest path and minimum spanning tree problems as robust optimization problems with $ K $ distinct cost scenarios.
• Use linear programming relaxations with scenario-based constraints to model the worst-case cost across all scenarios.
• Design a deterministic rounding procedure based on randomized rounding of LP solutions, leveraging Chernoff-Hoeffding bounds to control failure probability.
• Apply a multi-round edge sampling process to construct a spanning tree, ensuring connectivity with high probability via component reduction analysis.
• Use concentration inequalities to bound the deviation of sampled solution cost from the LP optimum across all scenarios.
• Introduce a threshold-based sampling scheme where edge inclusion probabilities are derived from an LP solution to balance cost and connectivity.

Experimental results

Research questions

• RQ1 Can a better deterministic approximation ratio be achieved for the min-max shortest path problem than the known $ K $-approximation?
• RQ2 What is the best possible approximation ratio for the min-max spanning tree problem in general graphs, and can it be achieved deterministically?
• RQ3 How do the approximation ratios relate to the integrality gaps of the corresponding LP relaxations?
• RQ4 Can randomized rounding techniques be adapted to ensure high-probability connectivity in spanning tree construction under worst-case scenario costs?

Key findings

• A deterministic $ O(\sqrt{n\log K/\log\log K}) $-approximation algorithm is developed for the min-max shortest path problem.
• For min-max spanning tree, a deterministic $ O(\log n \log K / \log\log K) $-approximation is achieved for graphs with bounded average degree or planar graphs.
• A randomized $ O(\log n) $-approximation algorithm is constructed for min-max spanning tree in general graphs, with high probability of success.
• The approximation ratios achieved are shown to be close to the integrality gaps of the underlying LP relaxations.
• The analysis confirms that the failure probability of constructing a valid spanning tree is at most $ 1/n^2 $, ensuring high reliability.
• The results suggest that the $ \sqrt{n} $-factor in approximation ratios may be inherent for such robust network problems, based on reductions to related problems.