A Quantum Inspired Bi-level Optimization Algorithm for the First Responder Network Design Problem

TL;DR

Comparisons with a state-of-the-art exact algorithm for network design problems demonstrate that GAGA offers a promising alternative approach and highlights the need for further exploration of quantum-inspired computing to tackle complex real-world problems.

Abstract

In the aftermath of a sudden catastrophe, First Responders (FR) strive to promptly reach and rescue immobile victims. Simultaneously, other mobile individuals take roads to evacuate the affected region, or access shelters. The escalated traffic congestion significantly hinders critical FR operations if they share some of the same roads. A proposal from the Turkish Ministry of Transportation and Infrastructure being discussed for implementation is to allocate a subset of road segments for use by FRs only, mark them clearly, and pre-communicate them to the citizens. For the FR paths under consideration: (i) there should exist an FR path from designated entry points to each demand point in the network, and (ii) evacuees try to leave the network (through some exit points following the selfish routing principle) in the shortest time possible when they know that certain segments are not available to them. We develop a mixed integer non-linear programming formulation for this First Responder Network Design Problem (FRNDP). We solve FRNDP using a novel hybrid quantum-classical heuristic building on the Graver Augmented Multi-Seed Algorithm (GAMA). Using the flow-balance constraints for the FR and evacuee paths, we use a Quadratic Unconstrained Binary Optimization (QUBO) model to obtain a partial Graver Bases to move between the feasible solutions of FRNDP. To efficiently explore the solution space for high-quality solutions, we develop a novel bi-level nested GAMA within GAMA: GAGA. We test GAGA on random graph instances of various sizes and instances related to an expected Istanbul earthquake. Comparing GAGA against a state-of-the-art exact algorithm for traditional formulations, we find that GAGA offers a promising alternative approach. We hope our work encourages further study of quantum (inspired) algorithms to tackle complex optimization models from other application domains.

Figures (5)

• Figure 1: An example of a $4$-node network.
• Figure 2: The network associated with our case study instances. FR lanes can be reserved along the thick pink edges. The yellow circles stand for exit nodes.
• Figure 3: The evacuation times as a function of time to solution for Turkish Graph Instance-1 computed using Graver walk with varying settings (on evacuee demand normalization and tolerance threshold in inner user equilibrium problem) compared against Branch and Bound solution. For each run, the final FR paths (local optimal solutions for each seed) obtained after the Graver walk are all found to be unique. On average, the inner user-equilibrium level with tolerance threshold is faster by a factor of $\sim 3.85$ than without for the Unnormalized setting and $\sim 2.5$ for a Normalized setting. Top panel: The best solution at the given time with the order of the seeds is the same as in the computational experiment. Bottom panel: The order of seeds is based on the decreasing order of the objective function.
• Figure 4: The evacuation times as a function of time to solution for Turkish Graph Instance-2 computed using Graver walk with varying settings (on demand normalization and tolerance) compared against Branch and Bound solution. For the GAGA run with Unnormalized, only (*M=85) seeds are used due to the time limit. The remaining GAGA runs use 100 seeds. For each run, the final FR paths (local optimal solutions for each seed) obtained after the Graver walk are all found to be unique. On average, the inner user-equilibrium level with tolerance threshold is faster by a factor of $\sim 6.1$ than without for the Unnormalized setting and $\sim 3$ for a Normalized setting.
• Figure 5: The evacuation times as a function of time to solution for Turkish Graph Instance-3 computed using Graver walk with varying settings (on demand normalization and tolerance) compared against Branch and Bound solution. For the GAGA run with Unnormalized setting, only (*M=43) seeds are used due to the time limit. The remaining GAGA runs use 100 seeds. For each run, the final FR paths (local optimal solutions for each seed) obtained after the Graver walk are all found to be unique. On average, the inner user-equilibrium level with tolerance threshold is faster by a factor of $\sim 9.56$ than without for the Unnormalized setting and $\sim 3.44$ for a Normalized setting.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3