Integer Traffic Assignment Problem: Algorithms and Insights on Random Graphs

TL;DR

The paper investigates the Integer Traffic Assignment Problem (ITAP), a discrete variant of TAP where edge flows must be integer. By analyzing both repulsive (convex φ) and attractive (concave φ) interaction regimes, the authors compare multiple algorithms—Greedy, CBP, Simulated Annealing, and the Relaxed ITAP solver (RITAP)—on large sparse random graphs with random origin–destination pairs. They show that in the attractive case, simulated annealing and CBP excel, while in the repulsive case many methods perform similarly with greedy being particularly efficient; moreover, relaxing ITAP to TAP and increasing the number of paths reveals two scaling regimes (ρ and η) where TAP converges to ITAP, making ITAP easier to solve in the high-flow limit. Real-world graph validations confirm the two-scale phenomena, with TAP energies approaching ITAP as η grows and degeneracy of supporting paths plateauing. Overall, the work provides practical algorithms and theoretical insight into when continuous relaxations suffice and how path interactions shape routing efficiency in ITAP.

Abstract

Path optimization is a fundamental concern across various real-world scenarios, ranging from traffic congestion issues to efficient data routing over the internet. The Traffic Assignment Problem (TAP) is a classic continuous optimization problem in this field. This study considers the Integer Traffic Assignment Problem (ITAP), a discrete variant of TAP. ITAP involves determining optimal routes for commuters in a city represented by a graph, aiming to minimize congestion while adhering to integer flow constraints on paths. This restriction makes ITAP an NP-hard problem. While conventional TAP prioritizes repulsive interactions to minimize congestion, this work also explores the case of attractive interactions, related to minimizing the number of occupied edges. We present and evaluate multiple algorithms to address ITAP, including a message passing algorithm, a greedy approach, simulated annealing, and relaxation of ITAP to TAP. Inspired by studies of random ensembles in the large-size limit in statistical physics, comparisons between these algorithms are conducted on large sparse random regular graphs with a random set of origin-destination pairs. Our results indicate that while the simplest greedy algorithm performs competitively in the repulsive scenario, in the attractive case the message-passing-based algorithm and simulated annealing demonstrate superiority. We then investigate the relationship between TAP and ITAP in the repulsive case. We find that, as the number of paths increases, the solution of TAP converges toward that of ITAP, and we investigate the speed of this convergence. Depending on the number of paths, our analysis leads us to identify two scaling regimes: in one the average flow per edge is of order one, and in another the number of paths scales quadratically with the size of the graph, in which case the continuous relaxation solves the integer problem closely.

Figures (15)

• Figure 1: Relative energy difference between the paths obtained using each algorithm and the shortest paths. The higher the better. Algorithms used are: greedy, CBP, simulated annealing, and RITAP. The CBP line is semi-transparent when CBP converges on less than 80% of the instances. Results are averages over 200 instances of ITAP, with $d=3, \,N=500, \,\gamma=1/2$.
• Figure 2: Relative energy difference between the paths obtained using each algorithm and the shortest paths. The higher the better. The algorithms used are: (a) greedy, (b) CBP, (c) simulated annealing, and (d) RITAP. The CBP line is semi-transparent when CBP converges on less than 80% of the instances. We fix $\gamma = 1/2$, and consider graphs of degree $d=3$ with different sizes $N\in\{100,200,400,1000\}$. Results are averages over 200 instances of ITAP.
• Figure 3: Relative energy difference between the paths obtained using each algorithm and the shortest paths. The higher, the better. The algorithms used are (a) greedy algorithm, (b) CBP, (c) simulated annealing, and (d) RITAP. Transparent parts of a line correspond to values for which the algorithm converges only for less than 80% of instances. Results are averages over 200 instances of ITAP, with $N=200,\gamma=1/2$ and $d\in\{ 3,6,12\}$.
• Figure 4: Relative energy difference between the paths obtained using each algorithm and the shortest paths. The higher the better. Algorithms used are: greedy, CBP, simulated annealing, RITAP and, as an upper bound, the TAP optimal energy. Results are averaged over 200 instances of ITAP, with $d=3, \,N=500, \,\gamma=2$.
• Figure 5: Relative energy difference between the paths obtained using each algorithm and the shortest paths. The algorithms used are: (a) greedy , (b) CBP, (c) simulated annealing, and (d) RITAP. The higher the better. Graphs of degree $d=3$ with different sizes $N\in\{100,200,400,1000\}$ are used. Results shown are averages over 200 realizations with $\gamma=2$.

Theorems & Definitions (18)

• Theorem 1.1
• Definition 1.2: TAP local minimum
• Proposition 1.3
• proof
• Proposition 3.1: Wardrop's second principle
• Conjecture 3.2
• Proposition H.1: Wardrop's second principle
• proof
• Proposition H.2
• proof