Unconventional GVNS for Solving the Garbage Collection Problem with Time Windows

The Distributed Kolkata Paise Restaurant Game

The Kolkata Paise Restaurant Problem is a challenging game in which n agents decide where to have lunch during their break. The game is not trivial because there are exactly n restaurants, and each restaurant can accommodate only one agent. We study this problem from a new angle and propose a novel strategy that results in greater utilization. Adopting a spatially distributed approach where the restaurants are uniformly distributed in the entire city area makes it possible for every agent to visit multiple restaurants. For each agent, the situation resembles that of the iconic traveling salesman, who must compute an optimal route through n cities. We rigorously prove probabilistic formulas that confirm the advantages of this policy and the increase in utilization. The derived equations generalize formulas that were previously known in the literature, which can be seen as special cases of our results.

Experimental Analysis of Quantum Annealers and Hybrid Solvers Using Benchmark Optimization Problems

This paper studies the Hamiltonian cycle problem (HCP) and the traveling salesman problem (TSP) on D-Wave quantum systems. Motivated by the fact that most libraries present their benchmark instances in terms of adjacency matrices, we develop a novel matrix formulation for the HCP and TSP Hamiltonians, which enables the seamless and automatic integration of benchmark instances in quantum platforms. We also present a thorough mathematical analysis of the precise number of constraints required to express the HCP and TSP Hamiltonians. This analysis explains quantitatively why, almost always, running incomplete graph instances requires more qubits than complete instances. It turns out that QUBO models for incomplete graphs require more quadratic constraints than complete graphs, a fact that has been corroborated by a series of experiments. Moreover, we introduce a technique for the min-max normalization for the coefficients of the TSP Hamiltonian to address the problem of invalid solutions produced by the quantum annealer, a trend often observed. Our extensive experimental tests have demonstrated that the D-Wave Advantage_system4.1 is more efficient than the Advantage_system1.1, both in terms of qubit utilization and the quality of solutions. Finally, we experimentally establish that the D-Wave hybrid solvers always provide valid solutions, without violating the given constraints, even for arbitrarily big problems up to 120 nodes.

A QUBO Model for the Traveling Salesman Problem with Time Windows

This work focuses on expressing the TSP with Time Windows (TSPTW for short) as a quadratic unconstrained binary optimization (QUBO) problem. The time windows impose time constraints that a feasible solution must satisfy. These take the form of inequality constraints, which are known to be particularly difficult to articulate within the QUBO framework. This is, we believe, the first time this major obstacle is overcome and the TSPTW is cast in the QUBO formulation. We have every reason to anticipate that this development will lead to the actual execution of small scale TSPTW instances on the D-Wave platform.