A finite time combinatorial algorithm for instantaneous dynamic equilibrium flows

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.

A Finite Time Combinatorial Algorithm for Instantaneous Dynamic Equilibrium Flows

“…is the maximum out-degree of any edge, T the termination time of the IDE and P is the number of intervals with constant network inflow rates. A formal deduction of this bound can be found in the full version of this paper [8].…”

“…such that ∞ i=1 α i converges to some point strictly before the IDE's termination time. In fact, in the full version of this paper [8], we provide an example of a rather simple network wherein extension phases may indeed become arbitrarily small, provided a long enough lasting network inflow rate. 3 However, this is not a counter example to the finiteness of the extension algorithm, as the shrinking of the extension phases is slow enough to still allow for a finite number of phases to span any fixed time horizon.…”

“…On the other hand, in a flow corresponding to a satisfying interpretation of the formula this does not happen, as only one of the two types of flow is present for every variable. The detailed construction of the described gadgets as well as the formal proof of the reduction's correctness can be found in the full version of this paper [8]. For an illustration of the reduction see Fig.…”

Convergence of a Packet Routing Model to Flows over Time

“…It asks that for every node in the network that an agent uses in their path, 1 Other equilibria notions distinct from Nash equilibria have been considered in the literature, in which agents make decisions without full information of the overall traffic situation. We refer to Graf, Harks and Sering [GHS20] and references therein for a discussion of instantaneous dynamic equilibria (see also [GH23a,GH23b]), where agents make decisions as they traverse the network based on current queues; and to Graf, Harks, Kollias and Merkl [GHKM22] for a very interesting approach to a much more general information framework where users use predictions of future congestion patterns. not just the sink, the agent's departure time from that node is at most δ later than its earliest possible arrival time (considering all possible routes to the node).…”

Convergence of Approximate and Packet Routing Equilibria to Nash Flows Over Time

“…1 Other equilibria notions distinct from Nash equilibria have been considered in the literature, in which agents make decisions without full information of the overall traffic situation. We refer to Graf, Harks and Sering [GHS20] and references therein for a discussion of instantaneous dynamic equilibria (see also [GH23a], [GH23b]), where agents make decisions as they traverse the network based on current queues; and to Graf, Harks, Kollias and Merkl [GHKM22] for a very interesting approach to a much more general information framework where users use predictions of future congestion patterns.…”