Maximum Cutting Research Articles

The Efficiency of Discrete Optimization Algorithm Portfolios

Introduction. Solving large-scale discrete optimization problems requires the processing of large-scale data in a reasonable time. Efficient solving is only possible by using multiprocessor computer systems. However, it is a daunting challenge to adapt existing optimization algorithms to get all the benefits of these parallel computing systems. The available computational resources are ineffective without efficient and scalable parallel methods. In this connection, the algorithm unions (portfolios and teams) play a crucial role in the parallel processing of discrete optimization problems. The purpose. The purpose of this paper is to research the efficiency of the algorithm portfolios by solving the weighted max-cut problem. The research is carried out in two stages using stochastic local search algorithms. Results. In this paper, we investigate homogeneous and non-homogeneous algorithm portfolios. We developed the homogeneous portfolios of two stochastic local optimization algorithms for the weighted max-cut problem, which has numerous applications. The results confirm the advantages of the proposed methods. Conclusions. Algorithm portfolios could be used to solve well-known discrete optimization problems of unprecedented scale and significantly improve their solving time. Further, we propose using communication between algorithms, namely teams and portfolios of algorithm teams. The algorithms in a team communicate with each other to boost overall performance. It is supposed that algorithm communication allows enhancing the best features of the developed algorithms and would improve the computational times and solution quality. The underlying algorithms should be able to utilize relevant data that is being communicated effectively to achieve any computational benefit from communication. Keywords: Discrete optimization, algorithm portfolios, computational experiment.

Delayed improvement local search

Local search is a fundamental tool in the development of heuristic algorithms. A neighborhood operator takes a current solution and returns a set of similar solutions, denoted as neighbors. In best improvement local search, the best of the neighboring solutions replaces the current solution in each iteration. On the other hand, in first improvement local search, the neighborhood is only explored until any improving solution is found, which then replaces the current solution. In this work we propose a new strategy for local search that attempts to avoid low-quality local optima by selecting in each iteration the improving neighbor that has the fewest possible attributes in common with local optima. To this end, it uses inequalities previously used as optimality cuts in the context of integer linear programming. The novel method, referred to as delayed improvement local search, is implemented and evaluated using the travelling salesman problem with the 2-opt neighborhood and the max-cut problem with the 1-flip neighborhood as test cases. Computational results show that the new strategy, while slower, obtains better local optima compared to the traditional local search strategies. The comparison is favourable to the new strategy in experiments with fixed computation time or with a fixed target.

Warm-starting quantum optimization

There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.

The biclique partitioning polytope

Given a bipartite graph G=(V1,V2,E), a biclique of G is a complete bipartite subgraph of G, and a biclique partitioning of G is a set A⊆E such that the bipartite graph G′=(V1,V2,A) is a vertex-disjoint union of bicliques. The biclique partitioning problem (BPP) consists of, given a complete bipartite graph G=(V1,V2,E) with edge weights we∈R for all e∈E (thus, negative weights are allowed), finding a biclique partitioning A⊆E of minimum weight. The bicluster editing problem (BEP) is a variant of the BPP and consists of editing a minimum number of edges of an input bipartite graph G in order to transform its edge set into a biclique partitioning. Editing an edge consists of either adding it to the graph or deleting it from the graph. In addition to the BPP and the BEP, other problems in the literature aim at finding biclique partitionings, and this motivates the study of the biclique partitioning polytope Pnm of the complete bipartite graph Knm (i.e., Pnm is the convex hull of the incidence vectors of the biclique partitionings of Knm). In this paper we develop such a polyhedral study and show that ladder, bellows, and grid inequalities induce facets of Pnm. Our computational results show that these inequalities are very effective in solving the BEP. In particular, they are able to improve the value of the relaxed solution by up to 20%.

Implementation of quantum imaginary-time evolution method on NISQ devices by introducing nonlocal approximation

The imaginary-time evolution method is a well-known approach used for obtaining the ground state in quantum many-body problems on a classical computer. A recently proposed quantum imaginary-time evolution method (QITE) faces problems of deep circuit depth and difficulty in the implementation on noisy intermediate-scale quantum (NISQ) devices. In this study, a nonlocal approximation is developed to tackle this difficulty. We found that by removing the locality condition or local approximation (LA), which was imposed when the imaginary-time evolution operator is converted to a unitary operator, the quantum circuit depth is significantly reduced. We propose two-step approximation methods based on a nonlocality condition: extended LA (eLA) and nonlocal approximation (NLA). To confirm the validity of eLA and NLA, we apply them to the max-cut problem of an unweighted 3-regular graph and a weighted fully connected graph; we comparatively evaluate the performances of LA, eLA, and NLA. The eLA and NLA methods require far fewer circuit depths than LA to maintain the same level of computational accuracy. Further, we developed a “compression” method of the quantum circuit for the imaginary-time steps to further reduce the circuit depth in the QITE method. The eLA, NLA, and compression methods introduced in this study allow us to reduce the circuit depth and the accumulation of error caused by the gate operation significantly and pave the way for implementing the QITE method on NISQ devices.

Accuracy-enhanced coherent Ising machine using the quantum adiabatic theorem.

The coherent Ising machine (CIM) implemented by degenerate optical parametric oscillator (DOPO) networks is a novel optical platform to accelerate computation of hard combinatorial optimization problems. Nevertheless, with the increase of the problem size, the probability of the machine being trapped by local minima increases exponentially. According to the quantum adiabatic theorem, a physical system will remain in its instantaneous ground state if the time-dependent Hamiltonian varies slowly enough. Here, we propose a method to help the machine partially avoid getting stuck in local minima by introducing quantum adiabatic evolution to the ground-state-search process of the CIM, which we call A-CIM. Numerical simulation results demonstrate that A-CIM can obtain improved solution accuracy in solving MAXCUT problems of vertices ranging from 10 to 2000 than CIM. The proposed machine that is based on quantum adiabatic theorem is expected to solve optimization problems more correctly.

On the minimal degree condition of graphs implying equality of the largest Kr-free subgraphs and (r − 1)-partite subgraphs

Erdős posed the problem of finding conditions on a graph G which imply that the largest number of edges in a triangle-free subgraph is equal to the largest number of edges in a bipartite subgraph. We consider a more general problem by defining δr to be the least number so that any graph G on n vertices with minimum degree δrn has the property that the largest number of edges in an (r−1)-partite subgraph equals to the largest number of edges in a Kr-free subgraph. We show that 3r−43r−1<δr≤4(3r−7)(r−1)+14(r−2)(3r−4) when r≥4. We also show δ4≤0.9415.

Pareto Optimization for Subset Selection with Dynamic Partition Matroid Constraints

In this study, we consider the subset selection problems with submodular or monotone discrete objective functions under partition matroid constraints where the thresholds are dynamic. We focus on POMC, a simple Pareto optimization approach that has been shown to be effective on such problems. Our analysis departs from singular constraint problems and extends to problems of multiple constraints. We show that previous results of POMC's performance also hold for multiple constraints. Our experimental investigations on random undirected maxcut problems demonstrate POMC's competitiveness against the classical GREEDY algorithm with restart strategy.

Approximate optimization of the MaxCut problem with a local spin algorithm

Local tensor methods are a class of optimization algorithms that was introduced in [Hastings,arXiv:1905.07047v2][1] as a classical analogue of the quantum approximate optimization algorithm (QAOA). These algorithms treat the cost function as a Hamiltonian on spin degrees of freedom and simulate the relaxation of the system to a low energy configuration using local update rules on the spins. Whereas the emphasis in [1] was on theoretical worst-case analysis, we here investigate performance in practice through benchmarking experiments on instances of the MAXCUT problem.Through heuristic arguments we propose formulas for choosing the hyperparameters of the algorithm which are found to be in good agreement with the optimal choices determined from experiment. We observe that the local tensor method is closely related to gradient descent on a relaxation of maxcut to continuous variables, but consistently outperforms gradient descent in all instances tested. We find time to solution achieved by the local tensor method is highly uncorrelated with that achieved by a widely used commercial optimization package; on some MAXCUT instances the local tensor method beats the commercial solver in time to solution by up to two orders of magnitude and vice-versa. Finally, we argue that the local tensor method closely follows discretized, imaginary-time dynamics of the system under the problem Hamiltonian.

Qubit-efficient encoding schemes for binary optimisation problems

We propose and analyze a set of variational quantum algorithms for solving quadratic unconstrained binary optimization problems where a problem consisting of nc classical variables can be implemented on O(log⁡nc) number of qubits. The underlying encoding scheme allows for a systematic increase in correlations among the classical variables captured by a variational quantum state by progressively increasing the number of qubits involved. We first examine the simplest limit where all correlations are neglected, i.e. when the quantum state can only describe statistically independent classical variables. We apply this minimal encoding to find approximate solutions of a general problem instance comprised of 64 classical variables using 7 qubits. Next, we show how two-body correlations between the classical variables can be incorporated in the variational quantum state and how it can improve the quality of the approximate solutions. We give an example by solving a 42-variable Max-Cut problem using only 8 qubits where we exploit the specific topology of the problem. We analyze whether these cases can be optimized efficiently given the limited resources available in state-of-the-art quantum platforms. Lastly, we present the general framework for extending the expressibility of the probability distribution to any multi-body correlations.

Hybrid quantum annealing via molecular dynamics

A novel quantum–classical hybrid scheme is proposed to efficiently solve large-scale combinatorial optimization problems. The key concept is to introduce a Hamiltonian dynamics of the classical flux variables associated with the quantum spins of the transverse-field Ising model. Molecular dynamics of the classical fluxes can be used as a powerful preconditioner to sort out the frozen and ambivalent spins for quantum annealers. The performance and accuracy of our smooth hybridization in comparison to the standard classical algorithms (the tabu search and the simulated annealing) are demonstrated by employing the MAX-CUT and Ising spin-glass problems.

The impacts of COVID-19 pandemic on retailing in Vietnam: current situations and recommendations

The COVID-19 pandemic seriously impacted almost the fields of the Vietnam economy in the shortterm and long-term, including the retail industry. While, retailing is regarded as one of the most strongly-impacted fields, but it also expresses the quick capability of recovery in comparison to other sectors. This study has been carried out to evaluate and analyze holistically the major effects of the pandemic on the Vietnam retail industry with nine critical impacts: the number of shoppers decreases; great volatility in the rental market; closed shops, maximum staff cuts; retail sales decline seriously; the explosion of e-commerce and home delivery services; change the campaign and decrease marketing budget; invest in building new online sales channels; convert consumption from offline to online and increase stockpiles of goods in the epidemic. Based on these analyses, the author recommends several significant suggestions for the Vietnam government including (1) directly support by in cash for the closed-stores during the pandemic for the traditional retailing; (2) quickly implement liquidity support packages and offer credit packages with preferential interest rates for the modern retailers; (3) solving problems of labor shortages in businesses that produce essential products; (4) increasing the essential stockpile source and manage efficiently the market supply. While some suggestions are delivered for retail businesses to overcome the recession and boost growth after the pandemic: (1) understand customers' problems; (2) reshape business strategy; (3) keep in touch with stakeholders and (4) expand new sales channels. The above suggestions are recommended on the basis of the bright outlooks from both macro factors and the internal retailing market.

A Quantum Approximate Optimization Algorithm with Metalearning for MaxCut Problem and Its Simulation via TensorFlow Quantum

A quantum approximate optimization algorithm (QAOA) is a polynomial-time approximate optimization algorithm used to solve combinatorial optimization problems. However, the existing QAOA algorithms have poor generalization performance in finding an optimal solution from a feasible solution set of combinatorial problems. In order to solve this problem, a quantum approximate optimization algorithm with metalearning for the MaxCut problem (MetaQAOA) is proposed. Specifically, a quantum neural network (QNN) is constructed in the form of the parameterized quantum circuit to detect different topological phases of matter, and a classical long short-term memory (LSTM) neural network is used as a black-box optimizer, which can quickly assist QNN to find the approximate optimal QAOA parameters. The experiment simulation via TensorFlow Quantum (TFQ) shows that MetaQAOA requires fewer iterations to reach the threshold of the loss function, and the threshold of the loss value after training is smaller than comparison methods. In addition, our algorithm can learn parameter update heuristics which can generalize to larger system sizes and still outperform other initialization strategies of this scale.

On even-cycle-free subgraphs of the doubled Johnson graphs

The generalized Turán number ex(G,H) is the maximum number of edges in an H-free subgraph of a graph G. It is an important extension of the classical Turán number ex(n,H), which is the maximum number of edges in a graph with n vertices that does not contain H as a subgraph. In this paper, we consider the maximum number of edges in an even-cycle-free subgraph of the doubled Johnson graphs J(n;k,k+1), which are bipartite subgraphs of hypercube graphs. We give an upper bound for ex(J(n;k,k+1),C2r) with any fixed k∈Z+ and any n∈Z+ with n≥2k+1. We also give an upper bound for ex(J(2k+1;k,k+1),C2r) with any k∈Z+, where J(2k+1;k,k+1) is known as doubled Odd graph O˜k+1. This bound induces that the number of edges in any C2r-free subgraph of O˜k+1 is o(e(O˜k+1)) for r≥6, which also implies a Ramsey-type result.

Ultrafast Ising Machines using spin torque nano-oscillators

Combinatorial optimization problems are known for being particularly hard to solve on traditional von Neumann architectures. This has led to the development of Ising Machines (IMs) based on quantum annealers and optical and electronic oscillators, demonstrating speed-ups compared to central processing unit (CPU) and graphics processing unit (GPU) algorithms. Spin torque nano-oscillators (STNOs) have shown GHz operating frequency, nanoscale size, and nanosecond turn-on time, which would allow their use in ultrafast oscillator-based IMs. Here, we show using numerical simulations based on STNO auto-oscillator theory that STNOs exhibit fundamental characteristics needed to realize IMs, including in-phase/out-of-phase synchronization and second harmonic injection locking phase binarization. Furthermore, we demonstrate numerically that large STNO network IMs can solve Max-Cut problems on nanosecond timescales.

A Probabilistic Compute Fabric Based on Coupled Ring Oscillators for Solving Combinatorial Optimization Problems

Nondeterministic polynomial time hard (NP-hard) combinatorial optimization problems (COPs) are intractable to solve using a traditional computer as the time to find a solution increases very rapidly with the number of variables. An efficient alternative computing method uses coupled spin networks to solve COP. This work presents a first-of-its-kind coupled ring oscillator (ROSC)-based scalable probabilistic Ising computer to solve NP-hard COPs. An integrated coupled oscillator network was designed with 560 ROSCs that mimic a coupled spin network. Each ROSC can be coupled to any of its neighbors using programmable back-to-back (B2B) inverter-based coupling mechanism. The ROSC-based spins and B2B inverter-based coupling were optimized to work under a wide range of system noise as well as voltage and temperature variations. Randomly generated 1000 max-cut problems were mapped and solved in the hardware. The integrated Ising computer produced satisfactory solutions of max-cut problems when compared with commercial software running on a CPU. Experiments show that the integrated CMOS-based Ising computer can find the solution to NP-hard problems with an accuracy of 82%-100%. In addition, the repeated measurements of the same problem showed that the Ising computer can traverse through several local minima to find high-quality solutions under various voltage and temperature variation conditions. The experimental results show that ROSCs are a potential candidate for a dedicated hardware accelerator aiming to solve a wide range of COPs.

Greedy Differencing Edge-Contraction heuristic for the Max-Cut problem

The Max-Cut problem is a classical NP-hard problem where the objective is to partition the nodes of an edge-weighted graph in a way that maximizes the sum of edges between the parts. We present a greedy heuristic for solving Max-Cut that combines an Edge-Contraction heuristic with the Differencing Method. We compare the heuristic’s performance to other greedy heuristics using a large and diverse set of problem instances.

Graph Neural Networks for Maximum Constraint Satisfaction.

Many combinatorial optimization problems can be phrased in the language of constraint satisfaction problems. We introduce a graph neural network architecture for solving such optimization problems. The architecture is generic; it works for all binary constraint satisfaction problems. Training is unsupervised, and it is sufficient to train on relatively small instances; the resulting networks perform well on much larger instances (at least 10-times larger). We experimentally evaluate our approach for a variety of problems, including Maximum Cut and Maximum Independent Set. Despite being generic, we show that our approach matches or surpasses most greedy and semi-definite programming based algorithms and sometimes even outperforms state-of-the-art heuristics for the specific problems.

Exact Facetial Odd-Cycle Separation for Maximum Cut and Binary Quadratic Optimization

The exact solution of the NP-hard (nondeterministic polynomial-time hard) maximum cut problem is important in many applications across, for example, physics, chemistry, neuroscience, and circuit layout—which is also due to its equivalence to the unconstrained binary quadratic optimization problem. Leading solution methods are based on linear or semidefinite programming and require the separation of the so-called odd-cycle inequalities. In their groundbreaking research, F. Barahona and A. R. Mahjoub have given an informal description of a polynomial-time algorithm for this problem. As pointed out recently, however, additional effort is necessary to guarantee that the inequalities obtained correspond to facets of the cut polytope. In this paper, we shed more light on a so enhanced separation procedure and investigate experimentally how it performs in comparison with an ideal setting where one could even employ the sparsest, most violated, or geometrically most promising facet-defining odd-cycle inequalities. Summary of Contribution: This paper aims at a better capability to solve binary quadratic optimization or maximum cut problems and their various applications using integer programming techniques. To this end, the paper describes enhancements to a well-known algorithm for the central separation problem arising in this context; it is demonstrated experimentally that these enhancements are worthwhile from a computational point of view. The linear relaxations of the aforementioned problems are typically solved using fewer iterations and cutting planes than with a nonenhanced approach. It is also shown that the enhanced procedure is only slightly inferior to an ideal, enumerative, and, in practice, intractable global cutting-plane selection.

Symmetric reduction of high-multiplicity one-loop integrals and maximal cuts

We derive useful reduction formulae which express one-loop Feynman integrals with a large number of external momenta in terms of lower-point integrals carrying easily derivable kinematic coefficients which are symmetric in the external momenta. These formulae apply for integrals with at least two more external legs than the dimension of the external momenta, and are presented in terms of two possible bases: one composed of a subset of descendant integrals with one fewer external legs, the other composed of the complete set of minimally-descendant integrals with just one more leg than the dimension of external momenta. In 3+1 dimensions, particularly compact representations of kinematic invariants can be computed, which easily lend themselves to spinor-helicity or trace representations. The reduction formulae have a close relationship with D-dimensional unitarity cuts, and thus provide a path towards computing full (all-ϵ) expressions for scattering amplitudes at arbitrary multiplicity.