Hard Combinatorial Optimization Problems Research Articles

Real-time solving of computationally hard problems using optimal algorithm portfolios

Various hard real-time systems have a desired requirement which is impossible to fulfill: to solve a computationally hard optimization problem within a short and fixed amount of time T, e.g., T = 0.5 seconds. For such a task, the exact, exponential algorithms, as well as various Polynomial-Time Approximation Schemes, are irrelevant because they can exceed T. What is left in practice is to combine various anytime algorithms in a parallel portfolio. The question is how to build such an optimal portfolio, given a budget of K computing cores. It is certainly not as simple as choosing the K best performing algorithms, because their results are possibly correlated (e.g., there is no point in choosing two good algorithm for the portfolio if they win on a similar set of instances). We prove that the decision variant of this problem is NP-complete, and furthermore that the optimization problem is approximable. On the practical side, our main contribution is a solution of the optimization problem of choosing K algorithms out of n, for a machine with K computing cores, and the related problem of detecting the minimum number of required cores to achieve an optimal portfolio, with respect to a given training set of instances. As a benchmark, we took instances of a hard optimization problem that is prevalent in the real-time industry, in which the challenge is to decide on the best action within time T. We include the results of numerous experiments that compare the various methods. Hence, a side effect of our tests is that it gives the first systematic empirical evaluation of the relative success of various known stochastic-search algorithms in coping with a hard combinatorial optimization problems under a very short and fixed timeout.

Polaritonic XY-Ising machine

AbstractGain-dissipative systems of various physical origin have recently shown the ability to act as analogue minimisers of hard combinatorial optimisation problems. Whether or not these proposals will lead to any advantage in performance over the classical computations depends on the ability to establish controllable couplings for sufficiently dense short- and long-range interactions between the spins. Here, we propose a polaritonic XY-Ising machine based on a network of geometrically isolated polariton condensates capable of minimising discrete and continuous spin Hamiltonians. We elucidate the performance of the proposed computing platform for two types of couplings: relative and absolute. The interactions between the network nodes might be controlled by redirecting the emission between the condensates or by sending the phase information between nodes using resonant excitation. We discuss the conditions under which the proposed machine leads to a pure polariton simulator with pre-programmed couplings or results in a hybrid classical polariton simulator. We argue that the proposed architecture for the remote coupling control offers an improvement over geometrically coupled condensates in both accuracy and stability as well as increases versatility, range, and connectivity of spin Hamiltonians that can be simulated with polariton networks.

Memetic algorithms outperform evolutionary algorithms in multimodal optimisation

Memetic algorithms integrate local search into an evolutionary algorithm to combine the advantages of rapid exploitation and global optimisation. We provide a rigorous runtime analysis of memetic algorithms on the Hurdle problem, a landscape class of tunable difficulty with a “big valley structure”, a characteristic feature of many hard combinatorial optimisation problems. A parameter called hurdle width describes the length of fitness valleys that need to be overcome. We show that the expected runtime of plain evolutionary algorithms like the (1+1) EA increases steeply with the hurdle width, yielding superpolynomial times to find the optimum, whereas a simple memetic algorithm, (1+1) MA, only needs polynomial expected time. Surprisingly, while increasing the hurdle width makes the problem harder for evolutionary algorithms, it becomes easier for memetic algorithms.We further give the first rigorous proof that crossover can decrease the expected runtime in memetic algorithms. A (2+1) MA using mutation, crossover and local search outperforms any other combination of these operators. Our results demonstrate the power of memetic algorithms for problems with big valley structures and the benefits of hybridising multiple search operators.

Smart Predict-and-Optimize for Hard Combinatorial Optimization Problems

Combinatorial optimization assumes that all parameters of the optimization problem, e.g. the weights in the objective function, are fixed. Often, these weights are mere estimates and increasingly machine learning techniques are used to for their estimation. Recently, Smart Predict and Optimize (SPO) has been proposed for problems with a linear objective function over the predictions, more specifically linear programming problems. It takes the regret of the predictions on the linear problem into account, by repeatedly solving it during learning. We investigate the use of SPO to solve more realistic discrete optimization problems. The main challenge is the repeated solving of the optimization problem. To this end, we investigate ways to relax the problem as well as warm-starting the learning and the solving. Our results show that even for discrete problems it often suffices to train by solving the relaxation in the SPO loss. Furthermore, this approach outperforms the state-of-the-art approach of Wilder, Dilkina, and Tambe. We experiment with weighted knapsack problems as well as complex scheduling problems, and show for the first time that a predict-and-optimize approach can successfully be used on large-scale combinatorial optimization problems.

A multi-objective evolutionary approach for planning and optimal condition restoration of secondary distribution networks

A secondary distribution network (SDN), corresponding to the final user low voltage distribution circuit, is continuously growing due to a persistent increase in load demand. Consequently, the performance of any optimized design will inevitably degrade over time. To avoid the associated repercussions such as faults, congestion, voltage drops, and other major quality issues, we are eventually prompted to redesign this part of the grid. To do so, we propose a Two-Stage Multi-Objective Evolutionary Approach (TS-MOEAP), which is able to find a new optimal network configuration, circumventing the associated quality issues. The proposed approach is oriented to improve the performance of SDNs by combining the concepts of network reconfiguration (NR) and optimal placement of distribution transformers (DTs). Due to the large and complex topology of SDNs, we deal with a hard combinatorial, non-convex, and nonlinear optimization problem. Consequently, to facilitate the resolution of the problem, the proposal is divided into two stages: (1) optimal placement and sizing of distribution transformers, as well as conductor sizing and branch routing, and (2) optimal network reconfiguration. For the first stage, an improved particle swarm optimization technique (IPSO) combined with a greedy algorithm is used, and for the second stage, an improved nondominated sorting genetic algorithm with a heuristic mutation operator (NSGA-HO) is implemented. The approach redesigns SDNs by minimizing total power loss and investment costs while satisfying quality issues and technical constraints. The proposed approach is validated by improving a real-life SDN with critical quality and technical issues. We also compare the results with respect to other state-of-the-art algorithms.

Lagrangian bounds for large‐scale multicommodity network design: a comparison between Volume and Bundle methods

AbstractThe Bundle Method and the Volume Algorithm are among the most efficient techniques to obtain accurate Lagrangian dual bounds for hard combinatorial optimization problems. We propose here to compare their performance on very large scale Fixed‐Charge Multicommodity Capacitated Network Design problems. The motivation is not only the quality of the approximation of these bounds as a function of the computational time but also the ability to produce feasible primal solutions and thus to reduce the gap for very large instances for which optimal solutions are out of reach. Feasible solutions are obtained through the use of Lagrangian information in constructive and improving heuristic schemes. We show in particular that, if the Bundle implementation has provided great quality bounds in fewer iterations, the Volume Algorithm is able to reduce the gaps of the largest instances, taking profit of the low computational cost per iteration compared to the Bundle Method.

Speed-up coherent Ising machine with a squeezed feedback system

As a solver for non-deterministic polynomial time (NP)-hard combinatorial optimization problems, the coherent Ising machine (CIM) is in the early stages of research, and the potential of this innovative physical system will be developed. Here, we propose a speed-up coherent Ising machine with a squeezed feedback system, which we call S-CIM. We couple squeezed feedback pulses generated by the squeezed feedback system into the degenerate optical parametric oscillator (DOPO) network. Simulations indicate that quantum inseparability of the coupled DOPO network is further enhanced during the whole optimization process, and quantum fluctuations are significantly smaller around the oscillation threshold. Computation experiments are performed on MAX-CUT problems of order between 4 and 20000. Numerical results demonstrate that S-CIM increases the optimal normalized output by 2.27% and significantly reduces the optimal computation time by 75.12%.

An Effective Regression Test Case Selection Using Hybrid Whale Optimization Algorithm

Test suite optimization is an ever-demanded approach for regression test cost reduction. Regression testing is conducted to identify any adverse effects of maintenance activity on previously working versions of the software. It consumes almost seventy percent of the overall software development lifecycle budget. Regression test cost reduction is therefore of vital importance. Test suite optimization is the most explored approach to reduce the test suite size to re-execute. This article focuses on test suite optimization as a regression test case selection, which is a proven N-P hard combinatorial optimization problem. The authors have proposed an almost safe regression test case selection approach using a Hybrid Whale Optimization Algorithm and empirically evaluated the same on subject programs retrieved from the Software Artifact Infrastructure Repository with Bat Search and ACO-based regression test case selection approaches. The analyses of the obtained results indicate an improvement in the fault detection ability of the proposed approach over the compared ones with significant reduction in test suite size.

Optimization of a new Golay sequences shaping for low sidelobe radar ambiguity function

This work addresses the radar waveform design problem using phase-coded waveforms especially Golay complementary pair (GCP). Thus, a new shaping of GCP is proposed and the good performance of this shaping in terms of ambiguity function sidelobe (SL) rejection is shown. With the new shaping, a gain of at least 10 dB is easily obtained in comparison with the standard GCP. In order to improve the performance of the proposed waveform, two optimization algorithms are proposed to deal with low sidelobe ambiguity function problem, a very hard combinatorial optimization problem. The first one is the simulated annealing algorithm, a local search algorithm. The second represents the genetic algorithm-based global search strategy. Simulation results are provided and demonstrate that both types of algorithms decrease the SL rejection of the ambiguity function for all tested instances.

An Improved Heuristic Based on Clustering and Genetic Algorithm for Solving the Multi-Depot Vehicle Routing Problem

Abstract: This paper introduces the multi depot vehicle routing problem (MDVRP) with location depot, a hard combinatorial optimization problem arising in several applications. The MDVRP with location depot based on two well-known NP-hard problems: the facility location problem (FLP) and the multi depot vehicle routing problem (MDVRP) with location depot. In the first phase, the problem consists of selecting on which sites to install a warehouse and assigning one and only one warehouse to each customer. In the second phase, for each depot, we search to solve a vehicle routing problem (VRP), multiple vehicles homogenous leave from a single depot and return to the same one. Each route must respect the capacity of each vehicle. The goal of this problem is to minimize the total distances to the performed routes. The MDVRP with location depots is classified as an NP-hard problem. Hence, the use of exact optimization methods may be difficult to solve this problem. In this work, we propose a method to solve it to optimality. The proposed procedure initially uses K-means algorithm to optimize the location selection and customer assignment, then planning the routes from the selected warehouses to a set of customers using Clarke and Wright saving method. The routes from the resulting multi-depot vehicle-routing problem (MDVRP) are improved using a Genetic Algorithm (GA). The proposed approach is tested and compared on a set of twelve benchmark instances from the MDVRP literature. The computational experiments confirm that our heuristic is able to find best solutions.

Facets from gadgets

We present a new tool for generating cutting planes for $$\mathcal{NP}$$ -hard combinatorial optimisation problems. It is based on the concept of gadgets—small subproblems that are “glued” together to form hard problems—which we borrow from the literature on computational complexity. Using gadgets, we are able to derive huge (exponentially large) new families of strong (and sometimes facet-defining) cutting planes, accompanied by efficient separation algorithms. We illustrate the power of this approach on the asymmetric traveling salesman, stable set and clique partitioning problems.

A branch-and-price algorithm for capacitated hypergraph vertex separation

We exactly solve the $${\mathcal {NP}}$$-hard combinatorial optimization problem of finding a minimum cardinality vertex separator with k (or arbitrarily many) capacitated shores in a hypergraph. We present an exponential size integer programming formulation which we solve by branch-and-price. The pricing problem, an interesting optimization problem on its own, has a decomposable structure that we exploit in preprocessing. We perform an extensive computational study, in particular on hypergraphs coming from the application of re-arranging a matrix into single-bordered block-diagonal form. Our experimental results show that our proposal complements the previous exact approaches in terms of applicability for larger k, and significantly outperforms them in the case $$k=\infty $$.

Disassembly sequence planning using a fast and effective precedence-based disassembly subset-generation method

Disassembly sequence planning plays a crucial role in the reuse and remanufacturing of end-of-life products. However, it is challenging to obtain optimal disassembly sequences as it is a difficult non-deterministic polynomial-time–hard combinatorial optimisation problem. This study proposes a precedence-based disassembly subset-generation method to solve the disassembly sequence-planning problem. The proposed method first generates disassembly subsets based on disassembly precedence relationships. Subsequently, multiple high-level feasible subsets are generated, and some of them are selected to generate higher level subsets using specially designed operators. After several iterations, different optimal disassembly sequences can be obtained. The principal merit of this method is that high-quality disassembly sequences can be generated in the solution construction process. Numerical experiments were conducted by applying the proposed method to three case studies, and the performance of the proposed method for finding optimal solutions and reducing computational time was evaluated in comparison with state-of-the-art methods. The parameters employed in this method were also analysed to explain the mechanism adopted.

Deep Reinforcement Learning for Online Computation Offloading in Wireless Powered Mobile-Edge Computing Networks

Wireless powered mobile-edge computing (MEC) has recently emerged as a promising paradigm to enhance the data processing capability of low-power networks, such as wireless sensor networks and internet of things (IoT). In this paper, we consider a wireless powered MEC network that adopts a binary offloading policy, so that each computation task of wireless devices (WDs) is either executed locally or fully offloaded to an MEC server. Our goal is to acquire an online algorithm that optimally adapts task offloading decisions and wireless resource allocations to the time-varying wireless channel conditions. This requires quickly solving hard combinatorial optimization problems within the channel coherence time, which is hardly achievable with conventional numerical optimization methods. To tackle this problem, we propose a Deep Reinforcement learning-based Online Offloading (DROO) framework that implements a deep neural network as a scalable solution that learns the binary offloading decisions from the experience. It eliminates the need of solving combinatorial optimization problems, and thus greatly reduces the computational complexity especially in large-size networks. To further reduce the complexity, we propose an adaptive procedure that automatically adjusts the parameters of the DROO algorithm on the fly. Numerical results show that the proposed algorithm can achieve near-optimal performance while significantly decreasing the computation time by more than an order of magnitude compared with existing optimization methods. For example, the CPU execution latency of DROO is less than $0.1$ second in a $30$-user network, making real-time and optimal offloading truly viable even in a fast fading environment.

GPU-based Ising computing for solving max-cut combinatorial optimization problems

In VLSI physical design, many algorithms require the solution of difficult combinatorial optimization problems such as max/min-cut, max-flow problems etc. Due to the vast number of elements typically found in this problem domain, these problems are computationally intractable leading to the use of approximate solutions. In this work, we explore the Ising spin glass model as a solution methodology for hard combinatorial optimization problems using the general purpose GPU (GPGPU). The Ising model is a mathematical model of ferromagnetism in statistical mechanics. Ising computing finds a minimum energy state for the Ising model which essentially corresponds to the expected optimal solution of the original problem. Many combinatorial optimization problems can be mapped into the Ising model. In our work, we focus on the max-cut problem as it is relevant to many VLSI design automation problems. Our method is inspired by the observation that Ising annealing process is very amenable to fine-grain massive parallel GPU computing. We will illustrate how the natural randomness of GPU thread scheduling can be exploited during the annealing process to create random update patterns and allow better GPU resource utilization. Furthermore, the proposed GPU-based Ising computing can handle any general Ising graph with arbitrary connections, which was shown to be difficult for existing FPGA and other hardware based implementation methods. Numerical results show that the proposed GPU Ising max-cut solver can deliver more than 2000X speedup over the CPU version of the algorithm on some large examples, which shows huge performance improvement for addressing many hard optimization algorithms for solving practical VLSI design automation problems.

An improved sexual genetic algorithm for solving 0/1 multidimensional knapsack problem

PurposeThe purpose of this paper is to provide an improved genetic algorithm to solve 0/1 multidimensional knapsack problem (0/1 MKP), by proposing new selection and crossover operators that cooperate to explore the search space.Design/methodology/approachThe authors first present a new sexual selection strategy that significantly improves the one proposed by (Varnamkhasti and Lee, 2012), while working in phenotype space. Then they propose two variants of the two-stage recombination operator of (Aghezzaf and Naimi, 2009), while they adapt the latter in the context of 0/1 MKP. The authors evaluate the efficiency of both proposed operators on a large set of 0/1 MKP benchmark instances. The obtained results are compared against that of conventional selection and crossover operators, in terms of solution quality and computing time.FindingsThe paper shows that the proposed selection respects the two major factors of any metaheuristic: exploration and exploitation aspects. Furthermore, the first variant of the two-stage recombination operator pushes the search space towards exploitation, while the second variant increases the genetic diversity. The paper then demonstrates that the improved genetic algorithm combining the two proposed operators is a competitive method for solving the 0/1 MKP.Practical implicationsAlthough only 0/1 MKP standard instances were tested in the empirical experiments in this paper, the improved genetic algorithm can be used as a powerful tool to solve many real-world applications of 0/1 MKP, as the latter models several industrial and investment issues. Moreover, the proposed selection and crossover operators can be incorporated into other bio-inspired algorithms to improve their performance. Furthermore, the two proposed operators can be adapted to solve other binary combinatorial optimization problems.Originality/valueThis research study provides an effective solution for a well-known non-deterministic polynomial-time (NP)-hard combinatorial optimization problem; that is 0/1 MKP, by tackling it with an improved genetic algorithm. The proposed evolutionary mechanism is based on two new genetic operators. The first proposed operator is a new and deeply different variant of the so-called sexual selection that has been rarely addressed in the literature. The second proposed operator is an adaptation of the two-stage recombination operator in the 0/1 MKP context. This adaptation results in two variants of the two-stage recombination operator that aim to improve the quality of encountered solutions, while taking advantage of the sexual selection criteria to prevent the classical issue of genetic algorithm that is premature convergence.

Solving Travelling Salesman Problem Using Ant Systems: A Programmer’s Approach

For an engineer, optimization is the process of determining the best design. In this work, a computer-based approach to a design optimization is investigated. This enables us to evaluate many design combinations that cannot be done manually. Travelling salesman problem (TSP) is leading problem among NP hard combinatorial optimization problems. It is widely used in many engineering applications. Ant system is a heuristic approach used to solve combinatorial optimization problems. In this work, a MATLAB program to solve TSP using Ant Search method is developed and is explained through the perspective of a programmer. Mathematical model of the problem is presented along with a detailed flowchart to describe the algorithm. The proposed program is described in accordance with the flow chart. Results obtained for two cases are summarized using graphs and CPU times. Finally a brief study on tuning of parameters of the method is also presented.

Performance Evaluation of Continuous and Discrete Particle Swarm Optimization in Job-Shop Scheduling Problems

The Particle Swarm Optimization (PSO) is an optimization method that was modeled based on the social behavior of organisms, such as bird flocks or swarms of bees. It was initially applied for cases defined over continuous spaces, but it can also be modified to solve problems in discrete spaces. Such problems include scheduling problems, where the Job-shop Scheduling Problem (JSP) is among the hardest combinatorial optimization problems. Although the JSP is a discrete problem, the continuous version of PSO has been able to handle the problem through a suitable mapping. Subsequently, its modified model, namely the discrete PSO, has also been proposed to solve it. In this paper, the performance of continuous and discrete PSO in solving JSP are evaluated and compared. The benchmark tests used are FT06 and FT10 problems available in the OR-library, where the goal is to minimize the maximum completion time of all jobs, i.e. the makespan. The experimental results show that the discrete PSO outperforms the continuous PSO for both benchmark problems.

Improved Hunting Search Algorithm for the Quadratic Assignment Problem

<p>Nowadays, the metaheuristics are the most studied methods used to solve the hard optimization problems. Hunting Search algorithm is a metaheuristic inspired by the method of group hunting of predatory animals like wolves. Created for solving continuous optimization problems, recently, it is adapted and evaluated to solve hard combinatorial optimization problems. This paper proposes an improved hunting search algorithm to solve the quadratic assignment problem. No local search method is used. To evaluate the performances of this work, the improved Hunting Search is checked on a set of 36 instances of QAPLib and it outperforms the results obtained by the well-known metaheuristics.</p>

Branch-and-Cut for the Split Delivery Vehicle Routing Problem with Time Windows

The split delivery vehicle routing problem with time windows (SDVRPTW) is a notoriously hard combinatorial optimization problem. First, it is hard to find a useful compact mixed-integer programming (MIP) formulation for the SDVRPTW. Standard modeling approaches either suffer from inherent symmetries (mixed-integer programs with a vehicle index) or cannot exactly capture all aspects of feasibility. Because of the possibility to visit customers more than once, the standard mechanisms to propagate load and time along the routes fail. Second, the lack of useful formulations has rendered any direct MIP-based approach impossible. Up to now, the most effective exact algorithms for the SDVRPTW have been branch-and-price-and-cut approaches using path-based formulations. In this paper, we propose a new and tailored branch-and-cut algorithm to solve the SDVRPTW. It is based on a new, relaxed compact model, in which some integer solutions are infeasible for the SDVRPTW. We use known and introduce some new classes of valid inequalities to cut off such infeasible solutions. One new class is path-matching constraints that generalize infeasible-path constraints. However, even with the valid inequalities, some integer solutions to the new compact formulation remain to be tested for feasibility. For a given integer solution, we build a generally sparse subnetwork of the original instance. On this subnetwork, all time-window-feasible routes can be enumerated, and a path-based residual problem then solved to decide on the selection of routes, the delivery quantities, and thereby the overall feasibility. All infeasible solutions need to be cut off. For this reason, we derive some strengthened feasibility cuts exploiting the fact that solutions often decompose into clusters. Computational experiments show that the new approach is able to prove optimality for several previously unsolved instances from the literature. The e-companion is available at https://doi.org/10.1287/trsc.2018.0825 .