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# A Faster Algorithm for Recognizing Directed Graphs Invulnerable to Braess’s Paradox

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## Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments.

## Cite As

Akira Matsubayashi and Yushi Saito. A Faster Algorithm for Recognizing Directed Graphs Invulnerable to Braess’s Paradox. In 23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023). Open Access Series in Informatics (OASIcs), Volume 115, pp. 12:1-12:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/OASIcs.ATMOS.2023.12

## Abstract

Braess’s paradox is a counterintuitive and undesirable phenomenon, in which for a given graph with prescribed source and sink vertices and cost functions for all edges, removal of edges decreases the cost of a Nash flow from source to sink. The problem of deciding if the phenomenon occurs is generally NP-hard. In this paper, we consider the problem of deciding if, for a given graph with prescribed source and sink vertices, Braess’s paradox does not occur for any cost functions. It is known that this problem can be solved in O(nm²) time for directed graphs, where n and m are the numbers of vertices and edges of the input graph, respectively. In this paper, we propose a faster O(m²) time algorithm solving this problem for directed graphs. Our approach is based on a simple implementation of a known characterization that the subgraph of a given graph induced by all source-sink paths is series-parallel. The faster running time is achieved by speeding up the simple implementation using another characterization that a certain structure is embedded in the given graph. Combined with a known technique, the proposed algorithm can also be used to design a faster O(km²) time algorithm for directed graphs with k source-sink pairs, which improves the previous O(knm²) time algorithm.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Graph algorithms analysis
• Theory of computation → Network flows
• Mathematics of computing → Graph algorithms
• Mathematics of computing → Network flows
##### Keywords
• series-parallel graph
• route-induced graph
• Nash flow

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