Maximum Cutting Research Articles

Large homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs

AbstractFor a bipartite graph , let be the largest such that either contains , a complete bipartite subgraph with parts of size , or the bipartite complement of contains as a subgraph. For a class of graphs , let . We say that a bipartite graph is strongly acyclic if neither nor its bipartite complement contains a cycle. By we denote the set of bipartite graphs with parts of size , which do not contain as an induced bipartite subgraph respecting the sides. One can easily show that for a positive if is not strongly acyclic. Here we ask whether is linear in for any strongly acyclic graph . We answer this question in the positive for all but four strongly acyclic graphs. We do not address this question for the remaining four graphs in this paper.

Accessing the high- ℓ frontier under the reduced shear approximation with k -cut cosmic shear

The precision of Stage IV cosmic shear surveys will enable us to probe smaller physical scales than ever before, however, model uncertainties from baryonic physics and non-linear structure formation will become a significant concern. The $k$-cut method -- applying a redshift-dependent $\ell$-cut after making the Bernardeau-Nishimichi-Taruya transform -- can reduce sensitivity to baryonic physics; allowing Stage IV surveys to include information from increasingly higher $\ell$-modes. Here we address the question of whether it can also mitigate the impact of making the reduced shear approximation; which is also important in the high-$\kappa$, small-scale regime. The standard procedure for relaxing this approximation requires the repeated evaluation of the convergence bispectrum, and consequently can be prohibitively computationally expensive when included in Monte Carlo analyses. We find that the $k$-cut cosmic shear procedure suppresses the $w_0w_a$CDM cosmological parameter biases expected from the reduced shear approximation for Stage IV experiments, when $\ell$-modes up to $5000$ are probed. The maximum cut required for biases from the reduced shear approximation to be below the threshold of significance is at $k = 5.37 \, h{\rm Mpc}^{-1}$. With this cut, the predicted $1\sigma$ constraints increase, relative to the case where the correction is directly computed, by less than $10\%$ for all parameters. This represents a significant improvement in constraints compared to the more conservative case where only $\ell$-modes up to 1500 are probed, and no $k$-cut is used. We also repeat this analysis for a hypothetical, comparable kinematic weak lensing survey. The key parts of code used for this analysis are made publicly available.

Multistart search for the Cyclic Cutwidth Minimization Problem

The Cyclic Cutwidth Minimization Problem (CCMP) is a Graph Layout Problem that consists of finding an embedding of the vertices of a candidate graph in a host graph, in order to minimize the maximum cut of a host edge. In this case, the host graph is restricted to be a cycle. In this paper, we identify a new lower bound for the problem, and also a property which allows search procedures to drastically reduce the neighborhood size during the search. Additionally, we propose the use of an alternative objective function for min–max optimization problems, never used before in the context of the CCMP. These strategies have been combined within a multistart search procedure for this problem. The obtained method is compared with the state of the art for the CCMP using sets of problem instances previously published. Statistical tests indicate the merit of our proposal.

To quantum or not to quantum: towards algorithm selection in near-term quantum optimization

The Quantum approximate optimization algorithm (QAOA) constitutes one of the often mentioned candidates expected to yield a quantum boost in the era of near-term quantum computing. In practice, quantum optimization will have to compete with cheaper classical heuristic methods, which have the advantage of decades of empirical domain-specific enhancements. Consequently, to achieve optimal performance we will face the issue of algorithm selection, well-studied in practical computing. Here we introduce this problem to the quantum optimization domain. Specifically, we study the problem of detecting those problem instances of where QAOA is most likely to yield an advantage over a conventional algorithm. As our case study, we compare QAOA against the well-understood approximation algorithm of Goemans and Williamson on the Max-Cut problem. As exactly predicting the performance of algorithms can be intractable, we utilize machine learning (ML) to identify when to resort to the quantum algorithm. We achieve cross-validated accuracy well over 96%, which would yield a substantial practical advantage. In the process, we highlight a number of features of instances rendering them better suited for QAOA. While we work with simulated idealised algorithms, the flexibility of ML methods we employed provides confidence that our methods will be equally applicable to broader classes of classical heuristics, and to QAOA running on real-world noisy devices.

A heuristic method for bi-decomposition of partial Boolean functions

The problem of decomposition of a Boolean function is to represent a given Boolean function in the form of a superposition of some Boolean functions whose number of arguments are less than the number of given function. The bi-decomposition represents a given function as a logic algebra operation, which is also given, over two Boolean functions. The task is reduced to specification of those two functions. A method for bi-decomposition of incompletely specified (partial) Boolean function is suggested. The given Boolean function is specified by two sets, one of which is the part of the Boolean space of the arguments of the function where its value is 1, and the other set is the part of the space where the function has the value 0. The complete graph of orthogonality of Boolean vectors that constitute the definitional domain of the given function is considered. In the graph, the edges are picked out, any of which has its ends corresponding the elements of Boolean space where the given function has different values. The problem of bi-decomposition is reduced to the problem of a weighted two-block covering the set of picked out edges of considered graph by its complete bipartite subgraphs (bicliques). Every biclique is assigned with a disjunctive normal form (DNF) in definite way. The weight of a biclique is a pair of certain parameters of assigned DNF. According to each biclique of obtained cover, a Boolean function is constructed whose arguments are the variables from the term of minimal rank on the DNF. A technique for constructing the mentioned cover for two kinds of output function is described.

Analyzing Heuristic-based Randomized Search Strategies for the Quantum Circuit Compilation Problem

In this work we investigate the performance of greedy randomised search (GRS) techniques to the problem of compiling quantum circuits to emerging quantum hardware. Quantum computing (QC) represents the next big step towards power consumption minimisation and CPU speed boost in the future of computing machines. Quantum computing uses quantum gates that manipulate multi-valued bits (qubits). A quantum circuit is composed of a number of qubits and a series of quantum gates that operate on those qubits, and whose execution realises a specific quantum algorithm. Current quantum computing technologies limit the qubit interaction distance allowing the execution of gates between adjacent qubits only. This has opened the way to the exploration of possible techniques aimed at guaranteeing nearest-neighbor (NN) compliance in any quantum circuit through the addition of a number of so-called swap gates between adjacent qubits. In addition, technological limitations (decoherence effect) impose that the overall duration (makespan) of the quantum circuit realization be minimized. One core contribution of the paper is the definition of two lexicographic ranking functions for quantum gate selection, using two keys: one key acts as a global closure metric to minimise the solution makespan; the second one is a local metric, which favours the mutual approach of the closest qstates pairs. We present a GRS procedure that synthesises NN-compliant quantum circuits realizations, starting from a set of benchmark instances of different size belonging to the Quantum Approximate Optimization Algorithm (QAOA) class tailored for the MaxCut problem. We propose a comparison between the presented meta-heuristics and the approaches used in the recent literature against the same benchmarks, both from the CPU efficiency and from the solution quality standpoint. In particular, we compare our approach against a reference benchmark initially proposed and subsequently expanded in [1] by considering: (i) variable qubit state initialisation and (ii) crosstalk constraints that further restrict parallel gate execution.

Clustering Improves the Goemans–Williamson Approximation for the Max-Cut Problem

MAX-CUT is one of the well-studied NP-hard combinatorial optimization problems. It can be formulated as an Integer Quadratic Programming problem and admits a simple relaxation obtained by replacing the integer “spin” variables xi by unitary vectors v→i. The Goemans–Williamson rounding algorithm assigns the solution vectors of the relaxed quadratic program to a corresponding integer spin depending on the sign of the scalar product v→i·r→ with a random vector r→. Here, we investigate whether better graph cuts can be obtained by instead using a more sophisticated clustering algorithm. We answer this question affirmatively. Different initializations of k-means and k-medoids clustering produce better cuts for the graph instances of the most well known benchmark for MAX-CUT. In particular, we found a strong correlation of cluster quality and cut weights during the evolution of the clustering algorithms. Finally, since in general the maximal cut weight of a graph is not known beforehand, we derived instance-specific lower bounds for the approximation ratio, which give information of how close a solution is to the global optima for a particular instance. For the graphs in our benchmark, the instance specific lower bounds significantly exceed the Goemans–Williamson guarantee.

Positive semidefinite penalty method for quadratically constrained quadratic programming

Abstract Quadratically constrained quadratic programming (QCQP) appears widely in engineering applications such as wireless communications and networking and multiuser detection with examples like the MAXCUT problem and boolean optimization. A general QCQP problem is NP-hard. We propose a penalty formulation for the QCQP problem based on semidefinite relaxation. Under suitable assumptions we show that the optimal solutions of the penalty problem are the same as those of the original QCQP problem if the penalty parameter is sufficiently large. Then, to solve the penalty problem, we present a proximal point algorithm and an update rule for the penalty parameter. Numerically, we test our algorithm on two well-studied QCQP problems. The results show that our proposed algorithm is very effective in finding high-quality solutions.

Connected max cut is polynomial for graphs without the excluded minor $$K_5\backslash e$$

Given a graph $$G=(V, E)$$ , a connected cut $$\delta (U)$$ is the set of edges of E linking all vertices of U to all vertices of $$V\backslash U$$ such that the induced subgraphs G[U] and $$G[V\backslash U]$$ are connected. Given a positive weight function w defined on E, the connected maximum cut problem (CMAX CUT) is to find a connected cut $$\varOmega $$ such that $$w(\varOmega )$$ is maximum among all connected cuts. CMAX CUT is NP-hard even for planar graphs, and thus for graph without the excluded minor $$K_5$$ . In this paper, we prove that CMAX CUT is polynomial for the class of graphs without the excluded minor $$K_5\backslash e$$ , denoted by $${\mathcal {G}}(K_5\backslash e)$$ . We deduce two quadratic time algorithms: one for the minimum cut problem in $${\mathcal {G}}(K_5\backslash e)$$ without computing the maximum flow, and another one for Hamilton cycle problem in the class of graphs without the two excluded minors the prism $$P_6$$ and $$K_{3, 3}$$ . This latter problem is NP-complete for maximal planar graphs.

Some Graphs Determined by their Signless Laplacian (Distance) Spectra

In literature, there are some results known about spectral determination of graphs with many edges. In [M.~C\'{a}mara and W.H.~Haemers. Spectral characterizations of almost complete graphs. {\em Discrete Appl. Math.}, 176:19--23, 2014.], C\'amara and Haemers studied complete graph with some edges deleted for spectral determination. In fact, they found that if the deleted edges form a matching, a complete graph $K_m$ provided $m \le n-2$, or a complete bipartite graph, then it is determined by its adjacency spectrum. In this paper, the graph $K_{n}\backslash K_{l,m}$ $(n>l+m)$ which is obtained from the complete graph $K_{n}$ by removing all the edges of a complete bipartite subgraph $K_{l,m}$ is studied. It is shown that the graph $K_{n}\backslash K_{1,m}$ with $m\ge4$ is determined by its signless Laplacian spectrum, and it is proved that the graph $K_{n}\backslash K_{l,m}$ is determined by its distance spectrum. The signless Laplacian spectral determination of the multicone graph $K_{n-2\alpha}\vee \alpha K_{2}$ was studied by Bu and Zhou in [C.~Bu and J.~Zhou. Signless Laplacian spectral characterization of the cones over some regular graphs. {\em Linear Algebra Appl.}, 436:3634--3641, 2012.] and Xu and He in [L. Xu and C. He. On the signless Laplacian spectral determination of the join of regular graphs. {\em Discrete Math. Algorithm. Appl.}, 6:1450050, 2014.] only for $n-2\alpha=1 ~\text{or}~ 2$. Here, this problem is completely solved for all positive integer $n-2\alpha$. The proposed approach is entirely different from those given by Bu and Zhou, and Xu and He.

Standardizing Monte Carlo simulation parameters for a reproducible dose-averaged linear energy transfer.

Dose-averaged linear energy transfer (LETD) is one of the factors which determines relative biological effectiveness (RBE) for treatment planning in proton therapy. It is usually determined from Monte Carlo (MC) simulation. However, no standard simulation protocols were established for sampling of LETD. Simulation parameters like maximum step length and range cut will affect secondary electrons production and have an impact on the accuracy of dose distribution and LETD. We aim to show how different combinations of step length and range cut in GEANT4 will affect the result in sampling of LETD using different MC scoring methods. In this work, different step length and range cut value in a clinically relevant voxel geometry were used for comparison. Different LETD scoring methods were established and the concept of covariance between energy deposition per step and step length is used to explain the differences between them. We recommend a maximum step length of 0.05 mm and a range cut of 0.01 mm in MC simulation as this yields the most consistent LETD value across different scoring methods. Different LETD scoring methods are also compared and variation up to 200% can be observed at the plateau of 80 MeV proton beam. Scoring Method one has one of the lowest percentage differences compared across all simulation parameters. We have determined a set of maximum step length and range cut parameters to be used for LETD scoring in a 1 mm voxelized geometry. LETD scoring method should also be clearly defined and standardized to facilitate cross-institutional studies. Establishing a standard simulation protocol for sampling LETD would reduce the discrepancy when comparing data across different centres, and this can improve the calculation for RBE.

Intersection graph of maximal stars

A biclique of a graph G is an induced complete bipartite subgraph of G such that neither part is empty. A star is a biclique of G such that one part has exactly one vertex. The star graph of G is the intersection graph of the maximal stars of G. A graph H is star-critical if its star graph is different from the star graph of any of its proper induced subgraphs. We begin by showing that a star-critical pre-image of an n-vertex star graph has at most On2vertices. We then describe a Krausz-type characterization for star graphs. We combine these results to show that the problem of recognizing star graphs is in NP. We also present some properties of the class. In particular, we show that they are biconnected, that every edge belongs to at least one triangle, characterize the structures the pre-image must have in order to generate degree two vertices in its star graph, and bound the diameter of the star graph with respect to the diameter of its pre-image. Finally, we prove a monotonicity theorem, which we apply in order to list every star graph on at most eight vertices.

Solving Max-Cut Optimization Problem

The goal of this paper is to find a better method that converges faster of Max-Cut problem. One strategy is to the comparison between Bundle Method and the Augmented Lagrangian method. We have also developed the theoretical convergence properties of these methods.

Power-efficient combinatorial optimization using intrinsic noise in memristor Hopfield neural networks

To tackle important combinatorial optimization problems, a variety of annealing-inspired computing accelerators, based on several different technology platforms, have been proposed, including quantum-, optical- and electronics-based approaches. However, to be of use in industrial applications, further improvements in speed and energy efficiency are necessary. Here, we report a memristor-based annealing system that uses an energy-efficient neuromorphic architecture based on a Hopfield neural network. Our analogue–digital computing approach creates an optimization solver in which massively parallel operations are performed in a dense crossbar array that can inject the needed computational noise through the analogue array and device errors, amplified or dampened by using a novel feedback algorithm. We experimentally show that the approach can solve non-deterministic polynomial-time (NP)-hard max-cut problems by harnessing the intrinsic hardware noise. We also use experimentally grounded simulations to explore scalability with problem size, which suggest that our memristor-based approach can offer a solution throughput over four orders of magnitude higher per power consumption relative to current quantum, optical and fully digital approaches. A memristor-based annealing system that uses an analogue neuromorphic architecture based on a Hopfield neural network can solve non-deterministic polynomial (NP)-hard max-cut problems in an approach that is potentially more efficient than current quantum, optical and digital approaches.

Assessing visible aerosol generation during vitrectomy in the era of Covid-19.

ObjectiveTo assess visible aerosol generation during simulated vitrectomy surgery.MethodsA model comprising a human cadaveric corneoscleral rim mounted on an artificial anterior chamber was used. Three-port 25 gauge vitrectomy simulated surgery was performed with any visible aerosol production recorded using high-speed 4K camera. The following were assessed: (1) vitrector at maximum cut rate in static and dynamic conditions inside the model, (2) vitrector at air–fluid interface in a physical model, (3) passive fluid–air exchange with a backflush hand piece, (4) valved cannulas under air, and (5) a defective valved cannula under air.ResultsNo visible aerosol or droplets were identified when the vitrector was used within the model. In the physical model, no visible aerosol or droplets were seen when the vitrector was engaged at the air–fluid interface. Droplets were produced from the opening of backflush hand piece during passive fluid–air exchange. No visible aerosol was produced from the intact valved cannulas under air pressure, but droplets were seen at the beginning of fluid–air exchange when the valved cannula was defective.ConclusionsWe found no evidence of visible aerosol generation during simulated vitrectomy surgery with competent valved cannulas. In the physical model, no visible aerosol was generated by the high-speed vitrector despite cutting at the air–fluid interface.

Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices

The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems. Despite its promise for near-term quantum applications, not much is currently understood about QAOA's performance beyond its lowest-depth variant. An essential but missing ingredient for understanding and deploying QAOA is a constructive approach to carry out the outer-loop classical optimization. We provide an in-depth study of the performance of QAOA on MaxCut problems by developing an efficient parameter-optimization procedure and revealing its ability to exploit non-adiabatic operations. Building on observed patterns in optimal parameters, we propose heuristic strategies for initializing optimizations to find quasi-optimal $p$-level QAOA parameters in $O(\text{poly}(p))$ time, whereas the standard strategy of random initialization requires $2^{O(p)}$ optimization runs to achieve similar performance. We then benchmark QAOA and compare it with quantum annealing, especially on difficult instances where adiabatic quantum annealing fails due to small spectral gaps. The comparison reveals that QAOA can learn via optimization to utilize non-adiabatic mechanisms to circumvent the challenges associated with vanishing spectral gaps. Finally, we provide a realistic resource analysis on the experimental implementation of QAOA. When quantum fluctuations in measurements are accounted for, we illustrate that optimization will be important only for problem sizes beyond numerical simulations, but accessible on near-term devices. We propose a feasible implementation of large MaxCut problems with a few hundred vertices in a system of 2D neutral atoms, reaching the regime to challenge the best classical algorithms.

The subformula property of natural deduction derivations and analytic cuts

Abstract In derivations of a sequent system, $\mathcal{L}\mathcal{J}$, and a natural deduction system, $\mathcal{N}\mathcal{J}$, the trails of formulae and the subformula property based on these trails will be defined. The derivations of $\mathcal{N}\mathcal{J}$ and $\mathcal{L}\mathcal{J}$ will be connected by the map $g$, and it will be proved the following: an $\mathcal{N}\mathcal{J}$-derivation is normal $\Longleftrightarrow $ it has the subformula property based on trails $\Longleftrightarrow $ its $g$-image in $\mathcal{L}\mathcal{J}$ is without maximum cuts $\Longrightarrow $ that $g$-image has the subformula property based on trails. In $\mathcal{L}\mathcal{J}$-derivations, another type of cuts, sub-cuts, will be introduced, and it will be proved the following: all cuts of an $\mathcal{L}\mathcal{J}$-derivation are sub-cuts $\Longleftrightarrow $ it has the subformula property based on trails.

A robust contingency-constrained unit commitment with an [formula omitted] security criterion

We propose a generalized N-k security criterion, which is called N-αk, and applies it to a robust contingency-constrained unit commitment (CCUC) model. In the previous works, the N-k security criterion takes into account the simultaneous loss of up to k components. However, a robust solution of N-k CCUC model may be over-conservative in most cases. To mitigate this over-conservativeness, we propose a robust N-αk CCUC model which guarantees N-2 security criterion throughout the service hours and adopts N-k security criterion only for a few hours. The operational information is used to define an uncertainty set of N-αk security criterion. In addition, we introduce a constant, which we denote as Δ, to control the conservativeness of the N-αk security criterion. We propose a mixed-integer linear programming formulation for the robust N-αk CCUC problem. The proposed N-αk CCUC model can provide cost savings as well as the robustness of the solutions with a proper criterion setting. Benders decomposition (BD) type cutting plane algorithm is used to solve the proposed model, and maximum feasible subsystem cuts are additionally generated at each BD iteration to improve the performance of the BD algorithm. Numerical case studies on a modified IEEE 118-bus system illustrate the effectiveness of the proposed robust N-αk CCUC model and provide some guidelines for α criterion setting.

Benchmarking the quantum approximate optimization algorithm

The performance of the quantum approximate optimization algorithm is evaluated by using three different measures: the probability of finding the ground state, the energy expectation value, and a ratio closely related to the approximation ratio. The set of problem instances studied consists of weighted MaxCut problems and 2-satisfiability problems. The Ising model representations of the latter possess unique ground states and highly degenerate first excited states. The quantum approximate optimization algorithm is executed on quantum computer simulators and on the IBM Q Experience. Additionally, data obtained from the D-Wave 2000Q quantum annealer are used for comparison, and it is found that the D-Wave machine outperforms the quantum approximate optimization algorithm executed on a simulator. The overall performance of the quantum approximate optimization algorithm is found to strongly depend on the problem instance.

ЭВОЛЮЦИОННЫЙ АЛГОРИТМ РАЗБИЕНИЯ МЕТОДОМ КРИСТАЛЛИЗАЦИИ РОССЫПИ АЛЬТЕРНАТИВ

The operation of the partitioning algorithm is based on the use of collective evolutionarymemory, which means information that reflects the history of the search for a solution and isstored independently of individuals. The algorithm associated with evolutionary memory seeks tomemorize and reuse ways to achieve better results. The collective evolutionary memory of thepartitioning algorithm is a set of statistical indicators that reflect, for each implemented alternative,the number θ of its occurrences in the best solutions at previous iterations of the algorithmand the number δ indicating the usefulness of the implemented alternative when constructing solutionsat previous iterations of the algorithm. The team does not have centralized management, andits features are the presence of indirect exchange of information. Indirect exchange consists inperforming certain actions, at different times, during which some parts of evolutionary memorychange by one agent. In the future, this changed information is used by other agents in these parts.First, at each iteration, a constructive algorithm generates nk solutions Qk. Each solution Qk is amapping Fk=V→X, is represented as a bipartite subgraph Dk and is formed by sequentially assigningelements to nodes. The formation of each solution Qk is performed by the set of agents A,by means of the probabilistic choice by each agent ai of the node vj. The process of assigning anelement to a node involves two stages. In the first stage, agent ai is selected, and in the secondstage, the node. In this case, the restriction must be fulfilled: each agent of the set A correspondsto one unique node of the set V. The estimate ξk of the solution Qk and the utility estimate δk of theset of alternatives implemented by the agents in the solution Qk are calculated. At the second stage,the agents increase the integral utility of the set of alternatives in the integral placer of alternativesR* by the value δk. At the third stage, the utility estimates δk of the integral placer of alternatives arereduced by μ. The paper uses the cyclic method of forming decisions. In this case, the building up ofestimates of the integral utility δk of the set of positions P is performed after the complete formationof the set of solutions Q at iteration l. Experimental studies were carried out on the basis of formedtest cases with the optimal solution obtained earlier. The results obtained were compared with theresults obtained by other well-known algorithms for dividing circuits into parts. For comparison, aset of standard benchmarks was formed. After analyzing the results, we can conclude that the proposedmethod allows you to get 4–5 % better solutions than its analogues.