DIRECT: A Scalable Approach for Route Guidance in Selfish Orienteering Problems

We address the problem of crowd congestion at venues liketheme parks, museums and world expos by providing routeguidance to multiple selfish users (with budget constraints)moving through the venue simultaneously. To representthese settings, we introduce the Selfish Orienteering Problem(SeOP) that combines two well studied problems from liter-ature, namely Orienteering Problem (OP) and Selfish Rout-ing (SR). OP is a single agent routing problem where thegoal is to minimize latency (or maximize reward) in travers-ing a subset of nodes while respecting budget constraints.SR is a game between selfish agents looking for minimumlatency routes from source to destination along edges of anetwork available to all agents. Thus, SeOP is a multi-agentplanning problem where agents have selfish interests and in-dividual budget constraints. As with Selfish Routing, weemploy Nash Equilibrium as the solution concept in solvingSeOP. A direct mathematical program formulation to find aNash equilibrium in SeOP cannot scale because the numberof constraints is quadratic in the number of paths, which it-self is an exponential quantity. To address scalability issues,we make two key contributions. First, we provide a compactnon-pairwise formulation with linear number of constraintsin the number of paths to enforce the equilibrium condition.Second, we introduce DIRECT, an incremental and iter-ative master-slave decomposition approach to compute anapproximate equilibrium solution. Similar to existing flowbased approaches, DIRECT is scale invariant in the numberof agents. We also provide a theoretical discussion of ourapproximation quality and present extensive empirical re-sults on synthetic and real-world graphs demonstrating thescalability of combining DIRECT with our non-pairwise for-mulation.