A Polynomial-time Algorithm for Detecting the Possibility of Braess Paradox in Directed Graphs
Pietro Cenciarelli, Daniele Gorla, Ivano Salvo
TL;DR
This work addresses the problem of Braess paradox in directed multigraphs by seeking a graph-theoretic criterion for vulnerability. The authors prove that a directed $st$-connected multigraph is vulnerable if and only if it contains a homeomorphic copy of the Wheatstone graph $\\mathcal{W}$, and they provide a constructive polynomial-time algorithm to decide vulnerability with runtime $O(|V|\cdot|E|^2)$. This characterization also yields a polynomial-time approach to the directed subgraph homeomorphism problem for the fixed pattern $\\mathcal{W}$, by testing all $(s,t)$ pairs; the overall complexity for the containment test is $O(|V|^3\cdot|E|^2)$. The paper additionally establishes NP-hardness for related irredundancy problems, clarifying the computational limits of prior reductions and guiding practical algorithm design. Overall, the results advance understanding of vulnerability in directed networks and connect Braess paradox analysis to a concrete, actionable graph pattern, with implications for network design and analysis of congestion games.
Abstract
A directed multigraph is said vulnerable if it can generate Braess paradox in Traffic Networks. In this paper, we give a graph-theoretic characterisation of vulnerable directed multigraphs; analogous results appeared in the literature only for undirected multigraphs and for a specific family of directed multigraphs. The proof of our characterisation also provides an algorithm that checks if a multigraph is vulnerable in O(|V| |E|^2); this is the first polynomial time algorithm that checks vulnerability for general directed multigraphs. The resulting algorithm also contributes to another well known problem, i.e. the directed subgraph homeomorphism problem without node mapping, by providing another pattern graph for which a polynomial time algorithm exists.