A Finite Time Combinatorial Algorithm for Instantaneous Dynamic Equilibrium Flows

TL;DR

This paper introduces the first finite time combinatorial algorithm for computing instantaneous dynamic equilibrium flows in traffic models, addressing longstanding computational challenges and establishing NP-hardness for certain cases.

Contribution

It presents a novel finite time algorithm for IDE flow computation and proves NP-hardness of computing IDE with natural properties.

Findings

First finite time combinatorial algorithm for IDE

Convergence in finitely many phases

NP-hardness results for natural IDE computations

Abstract

Instantaneous dynamic equilibrium (IDE) is a standard game-theoretic concept in dynamic traffic assignment in which individual flow particles myopically select en route currently shortest paths towards their destination. We analyze IDE within the Vickrey bottleneck model, where current travel times along a path consist of the physical travel times plus the sum of waiting times in all the queues along a path. Although IDE have been studied for decades, several fundamental questions regarding equilibrium computation and complexity are not well understood. In particular, all existence results and computational methods are based on fixed-point theorems and numerical discretization schemes and no exact finite time algorithm for equilibrium computation is known to date. As our main result we show that a natural extension algorithm needs only finitely many phases to converge leading to the first finite time combinatorial algorithm computing an IDE. We complement this result by several hardness results showing that computing IDE with natural properties is NP-hard.