Maximum Cutting Research Articles

Maximum bipartite subgraphs in graphs without short cycles

Given a graph G, let f(G) denote the maximum number of edges in a bipartite subgraph of G. Given a set ℋ of graphs and a positive integer m, let f(m,ℋ) denote the minimum possible cardinality of f(G), as G ranges over all graphs on m edges that contains no member of ℋ as a subgraph. Suppose that r≥10 is an even integer and k≥2 is an integer. In this paper, we prove that there is a constant c(r)>0 such that fm,{C6,C7,…,Cr−1}≥m/2+c(r)mr/(r+1) and there is a constant c(k)>0 such that fm,{C4,C6,…,C2k,C2k+1}≥m/2+c(k)m(2k+2)/(2k+3), both of which improve a result of Alon, Bollobás, Krivelevich and Sudakov.

Generalised 2-circulant inequalities for the max-cut problem

The max-cut problem is a fundamental combinatorial optimisation problem, with many applications. Poljak and Turzik found some facet-defining inequalities for the associated polytope, which we call 2-circulant inequalities. We present a more general family of facet-defining inequalities, an exact separation algorithm that runs in polynomial time, and some computational results.

Creating electronic oscillator-based Ising machines without external injection locking

Coupled electronic oscillators have recently been explored as a compact, integrated circuit- and room temperature operation-compatible hardware platform to design Ising machines. However, such implementations presently require the injection of an externally generated second-harmonic signal to impose the phase bipartition among the oscillators. In this work, we experimentally demonstrate a new electronic autaptic oscillator (EAO) that uses engineered feedback to eliminate the need for the generation and injection of the external second harmonic signal to minimize the Ising Hamiltonian. Unlike conventional relaxation oscillators that typically decay with a single time constant, the feedback in the EAO is engineered to generate two decay time constants which effectively helps generate the second harmonic signal internally. Using this oscillator design, we show experimentally, that a system of capacitively coupled EAOs exhibits the desired bipartition in the oscillator phases without the need for any external second harmonic injection, and subsequently, demonstrate its application in solving the computationally hard Maximum Cut (MaxCut) problem. Our work not only establishes a new oscillator design aligned to the needs of the oscillator Ising machine but also advances the efforts to creating application specific analog computing platforms.

Quantum Analog Annealing of Gain‐Dissipative Ising Machine Driven by Colored Gaussian Noise

AbstractGain‐dissipative Ising machines (GIMs) are a type of quantum analog equipment that can rapidly determine the optimal solution for combinatorial optimization problems. When the noise intensity is significantly lower than the fixed point of the system, the performance of a GIM is not influenced by the fluctuation of the noise intensity. However, the noise in this study is limited to Gaussian white noise. The influence of prevalent colored noise on GIMs has not been researched. In this study, the influence of common‐colored noise on the performance of GIMs is numerically investigated. The results of a domain clustering dynamics analysis reveal that red noise can better suppress the generation of the noise‐induced irregular temporary domain. Furthermore, several prevalent MAXCUT problem topologies, including the Moebius ladder, random Moebius ladder, and 2D random lattice, are adopted as test benchmarks. The results reveal that GIMs influenced by white, blue, and violet noise perform better at low‐intensity noise condition. In contrast, pink and red noise‐injected GIMs demonstrate higher performances when applied to MAXCUT topologies with both ferromagnetic and antiferromagnetic connections under larger noise intensity conditions. This indicates that the noise dispersion can be used as an additional hyperparameter to optimize the performance of GIMs.

Synthesis of combinational circuits by means of bi-decomposition of Boolean functions

O b j e c t i v e s . The problem of synthesis of combinational circuits in the basis of two-input gates is considered. Those gates are AND, OR, NAND and NOR. The objective of the paper is to investigate the possibilities of application of bi-decomposition of Boolean functions to the synthesis of combinational circuits.M e t h o d s . The method for bi-decomposition is reduced to the search in a graph for a weighted two-block cover with complete bipartite subgraphs (bi-cliques).R e s u l t s . The initial Boolean function is given as two ternary matrices, one of which represents the domain of Boolean space where the function has the value 1, and the other is the domain of Boolean space where the function has the value 0. The orthogonality graph of rows of ternary matrices representing the given function is considered. The method for two-bi-clique covering the orthogonality graph of rows of ternary matrices is described. Every bi-clique in the obtained cover is assigned in a certain way with а set of variables that are the arguments of the function. This set is the weight of the bi-clique. Each of those bi-cliques defines a Boolean function whose arguments are the variables assigned to it. The functions obtained in such a way constitute the required decomposition.Co n c l u s i o n . The process of synthesis of a combinational circuit consists of a successive application of bi-decomposition to obtained functions. The suggested method allows obtaining the circuits with short delay.

Geometrical configuration of hydrocyclone for improving the separation performance

The inherent defect of particle misplacement in traditional hydrocyclones is the main reason for the deterioration of separation accuracy and has obtained wide attentions. This paper presents three novel hydrocyclones based on the traditional cylindrical-conical type and cylindrical type. The effect of the hydrocyclone structure on the flow field characteristics and separation performance is investigated by a validated two fluid model. The numerical results show that a higher turbulence intensity deteriorates the separation performance of CCB type (cylindrical-conical-flat bottom type) and C type (cylindrical type), while a greater tangential velocity and velocity gradient improve the separation accuracy of MS type (multi-stage cylindrical type). Hydrocyclones with a flat-bottom structure reduces the misplaced fine particles due to the effect of the internal swirling flow and the axial circulation flow in the spigot. The MS type has the lowest imperfection value and the cut size of 76.5 μm implying a sharpness separation sharpness and an appropriate separation result. CC type has the minimum cut size of 62.21 μm, while the maximum cut size of 120.68 μm for C type.

Phase-Binarized Spin Hall Nano-Oscillator Arrays: Towards Spin Hall Ising Machines

Ising machines (IMs) are physical systems designed to find solutions to combinatorial optimization (CO) problems mapped onto the IM via the coupling strengths between its binary spins. Using its intrinsic dynamics and different annealing schemes, the IM relaxes over time to its lowest-energy state, which is the solution to the CO problem. IMs have been implemented on different platforms, and interacting nonlinear oscillators are particularly promising candidates. Here we demonstrate a pathway towards an oscillator-based IM using arrays of nanoconstriction spin Hall nano-oscillators (SHNOs). We show how SHNOs can be readily phase binarized and how their resulting microwave power corresponds to well-defined global phase states. To distinguish between degenerate states, we use phase-resolved Brillouin-light-scattering microscopy and directly observe the individual phase of each nanoconstriction. Micromagnetic simulations corroborate our experiments and confirm that our proposed IM platform can solve CO problems, showcased by how the phase states of a $2\ifmmode\times\else\texttimes\fi{}2$ SHNO array are solutions to a modified max-cut problem. Compared with the commercially available D-Wave ${\mathrm{Advantage}}^{\mathrm{TM}}$, our architecture holds significant promise for faster sampling, substantially reduced power consumption, and a dramatically smaller footprint.

Efficient sampling of ground and low-energy Ising spin configurations with a coherent Ising machine

We show that the nonlinear stochastic dynamics of a measurement-feedback-based coherent Ising machine (MFB-CIM) in the presence of quantum noise can be exploited to sample degenerate ground and low-energy spin configurations of the Ising model. We formulate a general discrete-time Gaussian-state model of the MFB-CIM which faithfully captures the nonlinear dynamics present at and above system threshold. This model overcomes the limitations of both mean-field models, which neglect quantum noise, and continuous-time models, which assume long photon lifetimes. Numerical simulations of our model show that when the MFB-CIM is operated in a quantum-noise-dominated regime with short photon lifetimes (i.e., low cavity finesse), homodyne monitoring of the system can efficiently produce samples of low-energy Ising spin configurations, requiring many fewer roundtrips to sample than suggested by established high-finesse, continuous-time models. We find that sampling performance is robust to, or even improved by, turning off or altogether reversing the sign of the parametric drive, but performance is critically reduced in the absence of optical nonlinearity. For the class of MAX-CUT problems with binary-signed edge weights, the number of roundtrips sufficient to fully sample all spin configurations up to the first-excited Ising energy, including all degeneracies, scales as $1.08^N$. At a problem size of $N = 100$ with a few dozen (median of 20) such desired configurations per instance, we have found median sufficient sampling times of $6\times10^6$ roundtrips; in an experimental implementation of an MFB-CIM with a 10 GHz repetition rate, this corresponds to a wall-clock sampling time of 60 ms.

Searching Personalized $k$-wing in Bipartite Graphs

There are extensive studies focusing on the application scenario that all the bipartite cohesive subgraphs need to be discovered in a bipartite graph. However, we observe that, for some applications, one is interested in finding bipartite cohesive subgraphs containing a specific vertex. In this paper, we study a new query-dependent bipartite cohesive subgraph search problem based on <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-wing model, named as personalized <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-wing search problem. We study the <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-wing equivalence relationship to summarize the edges of a bipartite graph <inline-formula><tex-math notation="LaTeX">$G$</tex-math></inline-formula> into groups. Therefore, all the edges of <inline-formula><tex-math notation="LaTeX">$G$</tex-math></inline-formula> are segregated into different groups, i.e. <i><inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-wing equivalence class</i>, forming an efficient and wing number conserving index called <i>EquiWing-Graph</i>. Further, we propose a more compact index, <i>EquiWing-Tree</i>, which is achieved by using our proposed <i><inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-butterfly loose</i> approach and discovered hierarchy properties. These indices are used to expedite the personalized <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-wing search with a non-repetitive access to <inline-formula><tex-math notation="LaTeX">$G$</tex-math></inline-formula>, which leads to linear algorithms for searching the personalized <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-wing. Moreover, we conduct a thorough study on the maintenance of the proposed indices for evolving bipartite graphs. We discover novel properties that help us localize the scope of the maintenance at a low cost. By exploiting the discoveries, we propose novel algorithms for maintaining the two indices, which substantially reduces the cost of maintenance. We perform extensive experimental studies in real-world graphs to validate the efficiency and effectiveness of <i>EquiWing-Graph</i> and <i>EquiWing-Tree</i> compared to the baseline.

Evaluation of Parameterized Quantum Circuits With Cross-Resonance Pulse-Driven Entanglers

Variational Quantum Algorithms (VQAs) have emerged as a powerful class of algorithms that is highly suitable for noisy quantum devices. Therefore, investigating their design has become key in quantum computing research. Previous works have shown that choosing an effective parameterized quantum circuit (PQC) or ansatz for VQAs is crucial to their overall performance, especially on near-term devices. In this paper, we utilize pulse-level access to quantum machines and our understanding of their two-qubit interactions to optimize the design of two-qubit entanglers in a manner suitable for VQAs. Our analysis results show that pulse-optimized ansatze reduce state preparation times by more than half, maintain expressibility relative to standard PQCs, and are more trainable through local cost function analysis. Our algorithm performance results show that in three cases, our PQC configuration outperforms the base implementation. Our algorithm performance results, executed on IBM Quantum hardware, demonstrate that our pulse-optimized PQC configurations are more capable of solving MaxCut and Chemistry problems compared to a standard configuration.

Энергосберегающее кодирование частичных состояний параллельного автомата

The problem of a low-power assignment of the partial states of a parallel automaton is considered. A method to solve that problem is suggested that provides minimizing the number of memory elements in the implementing circuit of the automaton and minimization of their switching activity. The problem is reduced to finding a minimal weighted cover of a graph with its complete bipartite sub-graphs (bi-cliques).

Effects of Dynamical Decoupling and Pulse-Level Optimizations on IBM Quantum Computers

Currently available quantum computers are prone to errors. Circuit optimization and error mitigation methods are needed to design quantum circuits to achieve better fidelity when executed on NISQ hardware. Dynamical decoupling (DD) is generally used to suppress the decoherence error and different DD strategies have been proposed. Moreover, the circuit fidelity can be improved by pulse-level optimization, such as creating hardware-native pulse-efficient gates. This paper implements all the popular DD sequences and evaluates their performances on IBM quantum chips with different characteristics for various well-known quantum applications. Also, we investigate combining DD with pulse-level optimization method and apply them to QAOA to solve Max-Cut problem. Based on the experimental results, we found that DD can be a benefit for only certain types of quantum algorithms, while the combination of DD and pulse-level optimization methods always has a positive impact. Finally, we provide several guidelines for users to learn how to use these noise mitigation methods to build circuits for quantum applications with high fidelity on IBM quantum computers.

Filtering variational quantum algorithms for combinatorial optimization

Current gate-based quantum computers have the potential to provide a computational advantage if algorithms use quantum hardware efficiently. To make combinatorial optimization more efficient, we introduce the filtering variational quantum eigensolver which utilizes filtering operators to achieve faster and more reliable convergence to the optimal solution. Additionally we explore the use of causal cones to reduce the number of qubits required on a quantum computer. Using random weighted MaxCut problems, we numerically analyze our methods and show that they perform better than the original VQE algorithm and the quantum approximate optimization algorithm. We also demonstrate the experimental feasibility of our algorithms on a Quantinuum trapped-ion quantum processor powered by Honeywell.

Improvement of Quantum Approximate Optimization Algorithm for Max-Cut Problems.

The objective of this short letter is to study the optimal partitioning of value stream networks into two classes so that the number of connections between them is maximized. Such kind of problems are frequently found in the design of different systems such as communication network configuration, and industrial applications in which certain topological characteristics enhance value–stream network resilience. The main interest is to improve the Max–Cut algorithm proposed in the quantum approximate optimization approach (QAOA), looking to promote a more efficient implementation than those already published. A discussion regarding linked problems as well as further research questions are also reviewed.

Regarding Two Conjectures on Clique and Biclique Partitions

For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures – one regarding the maximum possible value of $cp(G) + cp(\overline{G})$ (due to de Caen, Erdős, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $\max_G cp(G) + cp(\overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.

Freedom of the mixer rotation axis improves performance in the quantum approximate optimization algorithm

Variational quantum algorithms such as the quantum approximate optimization algorithm (QAOA) are particularly attractive candidates for implementation on near-term quantum processors. As hardware realities such as error and qubit connectivity will constrain achievable circuit depth in the near future, new ways to achieve high-performance at low depth are of great interest. In this work, we present a modification to QAOA that adds additional variational parameters in the form of freedom of the rotation-axis in the $XY$-plane of the mixer Hamiltonian. Via numerical simulation, we show that this leads to a drastic performance improvement over standard QAOA at finding solutions to the MAXCUT problem on graphs of up to 7 qubits. Furthermore, we explore the Z-phase error mitigation properties of our modified ansatz, its performance under a realistic error model for a neutral atom quantum processor, and the class of problems it can solve in a single round.

Sulfonitrocarburizing of High-Speed Steel Cutting Tools: Kinetics and Performances.

The scholarly literature records information related to the performance increase of the cutting tools covered by the superficial layers formed “in situ” when applying thermochemical processing. In this context, information is frequently reported on the carbamide role in processes aiming carbon and nitrogen surface saturation. Sulfur, together with these elements adsorbed and diffused in the cutting tools superficial layers, undoubtedly ensures an increase of their operating sustainability. The present paper discusses the process of sulfonitrocarburizing in pulverulent solid media of high-speed tools steel (AISI T1, HS18-0-1) and its consequences. The peculiarity of the considered process is that the source of nitrogen and carbon is mainly carbamide (CON2H4), which is found in solid powdery mixtures together with components that do not lead to cyan complex formation (non-toxic media), and the sulfur source is native sulfur. The kinetics of the sulfonitrocarburizing process, depending on the carbamide proportion in the powdered solid mixture and the processing temperature, was studied. The consequences of the achieved sulfonitrocarburized layers on the cutting tools’ performance are expressed by the maximum permissible cutting speed and the maximum cut length. An interesting aspect is highlighted, namely the possibility of using chemically active mixtures. Their components, by initiation of the metallothermic reduction reaction, become able to provide both elements of interest and the amount of heat needed for the ultrafast saturation of the targeted metal surfaces.

Annealing Processing Architecture of 28-nm CMOS Chip for Ising Model With 512 Fully Connected Spins

With the development of the Internet of things (IoT), sensors are being mounted on various objects. This trend has prompted demand for low-power, high-performance information processing on the edge side. Here, an Ising model architecture that can efficiently solve optimization problems would be an efficient processing solution for edges. In this study, we implemented a 512- spin fully connected Ising model on an LSI chip fabricated in a 28-nm CMOS process. The fully connected Ising model was implemented in the chip by using pseudo-annealing (PA), which is easier to implement than simulated annealing (SA). In addition, we devised a multi-spin-thread structure, concurrent update structure, and a folded interaction placement for accuracy, speed, and compactness. Because eight spin threads are implemented, the calculation throughput could be increased by a factor of eight in comparison with a single spin-thread implementation. Moreover, as a measure of solution accuracy, the average route length of a 22-city traveling salesman problem was reduced by 19% and the standard deviation (SD) was reduced by 46%. Likewise, the average cut value of a 512- node max-cut problem was increased by 1.6% and SD was decreased by 60%. The concurrent update almost doubled the calculation speed in comparison with the case of no concurrent update. In addition, the circuit area was reduced by about 38% as a result of the folded interaction placement. The time required to obtain a solution was 128 ms. The chip at annealing processing (main processing) had a power consumption of 12 mW at 1 MHz.

Duals of Feynman integrals. Part I. Differential equations

We elucidate the vector space (twisted relative cohomology) that is Poincaré dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces — an algebraic invariant called the intersection number — extracts integral coefficients for a minimal basis, bypassing the generation of integration-by-parts identities. Dual forms turn out to be much simpler than their Feynman counterparts: they are supported on maximal cuts of various sub-topologies (boundaries). Thus, they provide a systematic approach to generalized unitarity, the reconstruction of amplitudes from on-shell data. In this paper, we introduce the idea of dual forms and study their mathematical structures. As an application, we derive compact differential equations satisfied by arbitrary one-loop integrals in non-integer spacetime dimension. A second paper of this series will detail intersection pairings and their use to extract integral coefficients.

On the edge‐biclique graph and the iterated edge‐biclique operator

AbstractA biclique of a graph is a maximal induced complete bipartite subgraph of . The edge‐biclique graph of , , is the edge‐intersection graph of the bicliques of . A graph diverges (resp. converges or is periodic) under an operator whenever (resp. for some or for some and ). The iterated edge‐biclique graph of , , is the graph obtained by applying the edge‐biclique operator successive times to . In this article, we first study the connectivity relation between and . Next, we study the iterated edge‐biclique operator . In particular, we give sufficient conditions for a graph to be convergent or divergent under the operator , we characterize the behavior of burgeon graphs and we propose some general conjectures on the subject.