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.

翻译：路径优化是众多现实场景中的核心问题，涵盖从交通拥堵到互联网高效数据路由等领域。交通分配问题（TAP）是该领域经典的连续优化问题。本研究考虑整数交通分配问题（ITAP），即TAP的离散变体。ITAP旨在为以图表示的城市中的通勤者确定最优路径，在满足路径整数流约束的同时最小化拥堵。这一限制使ITAP成为NP-hard问题。传统TAP侧重于通过排斥交互最小化拥堵，而本文亦探索吸引交互情形（与最小化占用边数相关）。我们提出并评估了多种解决ITAP的算法，包括消息传递算法、贪心方法、模拟退火以及将ITAP松弛为TAP的方法。受统计物理中大规模极限下随机系综研究的启发，我们在包含随机起点-终点对集合的大型稀疏随机正则图上对这些算法进行了比较。结果表明，在排斥情形中，最简单的贪心算法具有竞争力；而在吸引情形中，基于消息传递的算法与模拟退火表现出优越性。随后，我们研究了排斥情形下TAP与ITAP的关系。发现随着路径数量增加，TAP的解趋近于ITAP的解，并探讨了这一收敛速度。根据路径数量的不同，我们识别出两种标度区域：一种区域中每条边的平均流量为阶一，另一种区域中路径数量随图规模二次增长，此时连续松弛能精确求解整数问题。