General Combinatorial Optimization Problems Research Articles

Iterative quantum algorithm for combinatorial optimization based on quantum gradient descent

Combinatorial optimization has wide and high-value applications in many fields of science and industry, but solving general combinatorial optimization problems is non-deterministic polynomial time (NP) hard. Many such problems can be mapped onto the ground-state-search problems of the Ising model. Here, an iterative quantum algorithm based on quantum gradient descent to solve combinatorial optimization problems is introduced, where the initial state of a quantum register evolves over several iterations to a good approximation of the Ising-Hamiltonian ground state. We verified the effectiveness of the proposed algorithm in solving the MaxCut problem for different types of undirected graphs by numerical simulations, and analyzed the robustness of the algorithm to errors by simulating random error and Gaussian error. We compared the performance of the algorithm with the quantum approximate optimization algorithm, and the results indicate that the proposed algorithm has comparable convergence performance. We also verified the feasibility of the algorithm by conducting experiments on a real quantum computer through the quantum cloud platform. Our work provides a potential method for solving combinatorial optimization problems on future quantum devices without the use of complex classical optimization loops.

A Quantum-Inspired Tensor Network Algorithm for Constrained Combinatorial Optimization Problems

Combinatorial optimization is of general interest for both theoretical study and real-world applications. Fast-developing quantum algorithms provide a different perspective on solving combinatorial optimization problems. In this paper, we propose a quantum-inspired tensor-network-based algorithm for general locally constrained combinatorial optimization problems. Our algorithm constructs a Hamiltonian for the problem of interest, effectively mapping it to a quantum problem, then encodes the constraints directly into a tensor network state and solves the optimal solution by evolving the system to the ground state of the Hamiltonian. We demonstrate our algorithm with the open-pit mining problem, which results in a quadratic asymptotic time complexity. Our numerical results show the effectiveness of this construction and potential applications in further studies for general combinatorial optimization problems.

Lower bounds on circuit depth of the quantum approximate optimization algorithm

The quantum approximate optimization algorithm (QAOA) is a method of approximately solving combinatorial optimization problems. While QAOA is developed to solve a broad class of combinatorial optimization problems, it is not clear which classes of problems are best suited for it. One factor in demonstrating quantum advantage is the relationship between a problem instance and the circuit depth required to implement the QAOA method. As errors in noisy intermediate-scale quantum (NISQ) devices increase exponentially with circuit depth, identifying lower bounds on circuit depth can provide insights into when quantum advantage could be feasible. Here, we identify how the structure of problem instances can be used to identify lower bounds for circuit depth for each iteration of QAOA and examine the relationship between problem structure and the circuit depth for a variety of combinatorial optimization problems including MaxCut and MaxIndSet. Specifically, we show how to derive a graph, G, that describes a general combinatorial optimization problem and show that the depth of circuit is at least the chromatic index of G. By looking at the scaling of circuit depth, we argue that MaxCut, MaxIndSet, and some instances of vertex covering and Boolean satisfiability problems are suitable for QAOA approaches while knapsack and traveling salesperson problems are not.

The General Combinatorial Optimization Problem: Towards Automated Algorithm Design

This paper defines a new combinatorial optimization problem, namely General Combinatorial Optimization Problem (GCOP), whose decision variables are a set of parametric algorithmic components, i.e. algorithm design decisions. The solutions of GCOP, i.e. compositions of algorithmic components, thus represent different generic search algorithms. The objective of GCOP is to find the optimal algorithmic compositions for solving the given optimization problems. Solving the GCOP is thus equivalent to automatically designing the best algorithms for optimization problems. Despite recent advances, the evolutionary computation and optimization research communities are yet to embrace formal standards that underpin automated algorithm design. In this position paper, we establish GCOP as a new standard to define different search algorithms within one unified model. We demonstrate the new GCOP model to standardize various search algorithms as well as selection hyperheuristics. A taxonomy is defined to distinguish several widely used terminologies in automated algorithm design, namely automated algorithm composition, configuration and selection. We would like to encourage a new line of exciting research directions addressing several challenging research issues including algorithm generality, algorithm reusability, and automated algorithm design.

A Survey of Quantum Genetic Algorithm for Combinatorial Optimization Problems

Combined with quantum computing and genetic algorithm, quantum genetic algorithm (QGA) shows considerable ability of parallelism. Experiments have shown that QGA performs quite well on TSP, job shop problem and some other typical combinatorial optimization problems. The other problems like nutritional diet which can be transformed into specific combinational optimization problem also can be solved through QGA smoothly. This paper sums up the main points of QGA for general combinatorial optimization problems. These points such as modeling of the problem, qubit decoding and rotation strategy are useful to enhance the convergence speed of QGA and avoid premature at the same time.

Estimation of Distribution Algorithm for Optimization of System Reliability

As an important branch of evolutionary algorithms, an Estimation of Distribution Algorithm (EDA) uses some selected individuals to build a probability model to produce offspring by sampling the probability model, which is a process of statistically learning from the selected individuals and then probabilistically sampling the probability model. A redundancy optimization model is given and many optimization methods to solve optimum model and their advantages and shortages are analyzed. A new method based on estimation distribution algorithm for optimization of system reliability is put forward and better population selection proportions are analyzed. Compared with simulated annealing algorithm, genetic algorithm and particle swarm algorithm, the estimation distribution algorithm is more effective through result. This method has good scalability, and it can be modified solve general combinatorial optimization problems.

On the computational complexity of combinatorial flexibility problems

The concept of flexibility – originated in the context of heat exchanger networks design – is associated with a substructure which allows the same optimal value on the substructure (for example an optimal flow) as in the whole structure, for all the costs in a given range of costs. In this work, we extend the concept of flexibility to general combinatorial optimization problems, and prove several computational complexity results in this new framework. Under some monotonicity conditions, we prove that a combinatorial optimization problem can be polynomially reduced to its associated flexibility problem. However, the minimum cut, maximum weighted matching and shortest path problems have NP-complete associated flexibility problems. In order to obtain polynomial flexibility problems, we have to restrict ourselves to combinatorial optimization problems on matroids.

Minmax regret approach and optimality evaluation in combinatorial optimization problems with interval and fuzzy weights

This paper deals with a general combinatorial optimization problem in which closed intervals and fuzzy intervals model uncertain element weights. The notion of a deviation interval is introduced, which allows us to characterize the optimality and the robustness of solutions and elements. The problem of computing deviation intervals is addressed and some new complexity results in this field are provided. Possibility theory is then applied to generalize a deviation interval and a solution concept to fuzzy ones.

On [formula omitted]-max-optimization

We generalize bottleneck objectives in combinatorial optimization by minimizing the k th largest cost coefficient in a feasible solution. A bisection algorithm is presented which is based on iteratively solving an associated sum objective problem with binary cost coefficients. This algorithm is applicable to general combinatorial optimization problems.

Binary-coding-based ant colony optimization and its convergence

Ant colony optimization (ACO for short) is a meta-heuristics for hard combinatorial optimization problems. It is a population-based approach that uses exploitation of positive feedback as well as greedy search. In this paper, genetic algorithm's (GA for short) ideas are introduced into ACO to present a new binary-coding based ant colony optimization. Compared with the typical ACO, the algorithm is intended to replace the problem's parameter-space with coding-space, which links ACO with GA so that the fruits of GA can be applied to ACO directly. Furthermore, it can not only solve general combinatorial optimization problems, but also other problems such as function optimization. Based on the algorithm, it is proved that if the pheromone remainder factor ρ is under the condition of ρ ≥ 1, the algorithm can promise to converge at the optimal, whereas if 0 < ρ < 1, it does not.

Hereditary systems and greedy-type algorithms

We define a hereditary system on a finite set U as a partition of the family 2 U of all subsets of U into disjoint families A and D satisfying (A∈ A,A′⊆A)⇒A′∈ A and (D∈ D,D⊆D′)⇒D′∈ D , respectively. The members of A are called independent sets, the sets D∈ D are called dependent. We consider two important special cases of hereditary systems, matroids and comatroids, and study the structure of these objects. Two general combinatorial optimization problems on a hereditary system, the maximum independent set problem max{f(X) : X∈ A} and the minimum dependent set problem min{f(X) : X∈ D} , are considered. Jenkyns, Korte and Hausmann obtained a performance guarantee of the greedy heuristic (‘best in’) for the maximum independent set problem. We present a greedy-type approximation algorithm for solving the minimum dependent set problem, the steepest descent heuristic (‘worst out’), study interconnections between the above-mentioned problems and derive performance guarantees of the steepest descent algorithm. Finally, we apply our results to obtain performance guarantees of the steepest descent algorithm for some known special combinatorial minimization problems.

Microcanonical Optimization Applied to the Traveling Salesman Problem

Optimization strategies based on simulated annealing and its variants have been extensively applied to the traveling salesman problem (TSP). Recently, there has appeared a new physics-based metaheuristic, called the microcanonical optimization algorithm (μO), which does not resort to annealing, and which has proven a superior alternative to the annealing procedures in various applications. Here we present the first performance evaluation of μO as applied to the TSP. When compared to three annealing strategies (simulated annealing, microcanonical annealing and Tsallis annealing), and to a tabu search algorithm, the microcanonical optimization has yielded the best overall results for several instances of the euclidean TSP. This confirms μO as a competitive approach for the solution of general combinatorial optimization problems.