Maximum Cut Problem Research Articles

Optimization of Cascading Processes in Arbitrary Networks with Stochastic Interactions

We consider the problem of optimal propagation of cascades in a network, where the propagation process depends on the values that a decision-maker assigns to the network and independent influence rates. Assuming that there are costs associated with changing the values of these rates, we investigate the question of resource allocation that minimizes the time by which the cascading process reaches all nodes. To this end, we propose a model that represents the cascading process in the network as a continuous time Markov chain (CTMC). We show how the propagation problem can be framed as an eigenvalue optimization problem on the generator of the CTMC. In turn, this problem is successively transformed into a maximum minimum cut problem and into a generalized maximum flow problem on a linearly-sized auxiliary graph. Therefore, the optimization problem can be solved in strongly polynomial time in terms of the size of the network. In addition, whenever there are no preexisting influence rates, the problem can also be transformed into the problem of finding a minimum spanning arborescence on a similar auxiliary graph. In this setting the optimal solution provides a hierarchy that classifies the nodes in terms of their importance to the cascade propagation.

Advanced Tabu Search Algorithms for Bipartite Boolean Quadratic Programs Guided by Strategic Oscillation and Path Relinking

The bipartite Boolean quadratic programming problem (BBQP) is a generalization of the well-studied NP-hard Boolean quadratic programming problem and can be regarded as a unified model for many graph theoretic optimization problems, including maximum weight-induced subgraph problems, maximum weight biclique problems, matrix factorization problems, and maximum cut problems on bipartite graphs. This paper introduces three main algorithms for solving the BBQP, based on three variants of tabu search, the first two consisting of strategic oscillation–tabu search (SO-TS) algorithms, which use destructive and constructive procedures to guide the search into unexplored and promising areas. The third algorithm, whichDoes also incorporates the SO-TS algorithms as solution improvement methods, uses a path relinking (PR) algorithm that is capable of further enhancing search performance. Experimental results demonstrate that all three algorithms perform very effectively compared with the best methods in the literature, and the PR algorithm joined with tabu search is able to discover new best solutions for two-thirds of the large problem instances and match the previous best known solutions for the other instances. Additional analysis discloses the contributions of the key ingredients of each of the proposed algorithms.

Improving Optimization Bounds Using Machine Learning: Decision Diagrams Meet Deep Reinforcement Learning

Finding tight bounds on the optimal solution is a critical element of practical solution methods for discrete optimization problems. In the last decade, decision diagrams (DDs) have brought a new perspective on obtaining upper and lower bounds that can be significantly better than classical bounding mechanisms, such as linear relaxations. It is well known that the quality of the bounds achieved through this flexible bounding method is highly reliant on the ordering of variables chosen for building the diagram, and finding an ordering that optimizes standard metrics is an NP-hard problem. In this paper, we propose an innovative and generic approach based on deep reinforcement learning for obtaining an ordering for tightening the bounds obtained with relaxed and restricted DDs. We apply the approach to both the Maximum Independent Set Problem and the Maximum Cut Problem. Experimental results on synthetic instances show that the deep reinforcement learning approach, by achieving tighter objective function bounds, generally outperforms ordering methods commonly used in the literature when the distribution of instances is known. To the best knowledge of the authors, this is the first paper to apply machine learning to directly improve relaxation bounds obtained by general-purpose bounding mechanisms for combinatorial optimization problems.

Subspace Selection via DR-Submodular Maximization on Lattices

The subspace selection problem seeks a subspace that maximizes an objective function under some constraint. This problem includes several important machine learning problems such as the principal component analysis and sparse dictionary selection problem. Often, these problems can be (exactly or approximately) solved using greedy algorithms. Here, we are interested in why these problems can be solved by greedy algorithms, and what classes of objective functions and constraints admit this property.In this study, we focus on the fact that the set of subspaces forms a lattice, then formulate the problems as optimization problems on lattices. Then, we introduce a new class of functions on lattices, directional DR-submodular functions, to characterize the approximability of problems. We prove that the principal component analysis, sparse dictionary selection problem, and these generalizations are monotone directional DRsubmodularity functions. We also prove the “quantum version” of the cut function is a non-monotone directional DR submodular function. Using these results, we propose new solvable feature selection problems (generalized principal component analysis and quantum maximum cut problem), and improve the approximation ratio of the sparse dictionary selection problem in certain instances.We show that, under several constraints, the directional DRsubmodular function maximization problem can be solved efficiently with provable approximation factors.

Greedy Maximization of Functions with Bounded Curvature under Partition Matroid Constraints

We investigate the performance of a deterministic GREEDY algorithm for the problem of maximizing functions under a partition matroid constraint. We consider non-monotone submodular functions and monotone subadditive functions. Even though constrained maximization problems of monotone submodular functions have been extensively studied, little is known about greedy maximization of non-monotone submodular functions or monotone subadditive functions.
 We give approximation guarantees for GREEDY on these problems, in terms of the curvature. We find that this simple heuristic yields a strong approximation guarantee on a broad class of functions.
 We discuss the applicability of our results to three real-world problems: Maximizing the determinant function of a positive semidefinite matrix, and related problems such as the maximum entropy sampling problem, the constrained maximum cut problem on directed graphs, and combinatorial auction games.
 We conclude that GREEDY is well-suited to approach these problems. Overall, we present evidence to support the idea that, when dealing with constrained maximization problems with bounded curvature, one needs not search for (approximate) monotonicity to get good approximate solutions.

Spectral bounds for graph partitioning with prescribed partition sizes

Given an undirected edge weighted graph, the graph partitioning problem consists in determining a partition of the node set of the graph into subsets of prescribed sizes, so as to maximize the sum of the weights of the edges having both endpoints in the same subset. We introduce a new class of bounds for this problem relying on the full spectral information of the weighted adjacency matrix A. The expression of these bounds involves the eigenvalues and particular geometrical parameters defined using the eigenvectors of A. A connection is established between these parameters and the maximum cut problem. We report computational results showing that the new bounds compare favorably with previous bounds in the literature.

On fractional cut covers

Given an undirected graph, a minimum cut cover is a collection of cuts covering the whole set of edges and having minimum cardinality. This paper is dedicated to the fractional version of this problem where a fractional weight is computed for each cut such that, for each edge, the sum of the weights of all cuts containing it is no less than 1, while the sum of all weights is minimized. The fractional cover is computed for different graph classes among which are the weakly bipartite graphs. Efficient algorithms are described to compute lower and upper bounds with worst-case performance guarantees. A general randomized approach is also presented giving new insights into Goemans and Williamson’s algorithm for the maximum cut problem. Some numerical experiments are included to assess the quality of bounds.

Maximum cuts in edge-colored graphs

The input of the Maximum Colored Cut problem consists of a graph G=(V,E) with an edge-coloring c:E→{1,2,3,…,p} and a positive integer k, and the question is whether G has a nontrivial edge cut using at least k colors. The Colorful Cut problem has the same input but asks for a nontrivial edge cut using all p colors. Unlike what happens for the classical Maximum Cut problem, we prove that both problems are NP-complete even on complete, planar, or bounded treewidth graphs. Furthermore, we prove that Colorful Cut is NP-complete even when each color class induces a clique of size at most three, but is trivially solvable when each color induces an edge. On the positive side, we prove that Maximum Colored Cut is fixed-parameter tractable when parameterized by either k or p, by constructing a cubic kernel in both cases.

Individuality for Quantum-Inspired Individuals Based on Nonuniform Convergence Speed in Maximum Cut Problem

Individuality for Quantum-Inspired Individuals Based on Nonuniform Convergence Speed in Maximum Cut Problem

A note on the sufficient initial condition ensuring the convergence of directly extended 3-block ADMM for special semidefinite programming

ABSTRACTIn this note, we consider three types of problems, H-weighted nearest correlation matrix problem and two types of important doubly non-negative semidefinite programming, derived from the binary integer quadratic programming and maximum cut problem. The dual of these three types of problems is a 3-block separable convex optimization problem with a coupling linear equation constraint. It is known that, the directly extended 3-block alternating direction method of multipliers (ADMM3d) is more efficient than many of its variants for solving these convex optimization, but its convergence is not guaranteed. By choosing initial points properly, we obtain the convergence of ADMM3d for solving the dual of these three types of problems. Furthermore, we simplify the iterative scheme of ADMM3d and show the equivalence of ADMM3d to the 2-block semi-proximal ADMM for solving the dual's reformulation, under these initial conditions.

An inexact dual logarithmic barrier method for solving sparse semidefinite programs

A dual logarithmic barrier method for solving large, sparse semidefinite programs is proposed in this paper. The method avoids any explicit use of the primal variable X and therefore is well-suited to problems with a sparse dual matrix S. It relies on inexact Newton steps in dual space which are computed by the conjugate gradient method applied to the Schur complement of the reduced KKT system. The method may take advantage of low-rank representations of matrices $$A_i$$ to perform implicit matrix-vector products with the Schur complement matrix and to compute only specific parts of this matrix. This allows the construction of the partial Cholesky factorization of the Schur complement matrix which serves as a good preconditioner for it and permits the method to be run in a matrix-free scheme. Convergence properties of the method are studied and a polynomial complexity result is extended to the case when inexact Newton steps are employed. A Matlab-based implementation is developed and preliminary computational results of applying the method to maximum cut and matrix completion problems are reported.

Search performance analysis of qubit convergence measure for quantum-inspired evolutionary algorithm introducing on maximum cut problem

The quantum-inspired evolutionary algorithm (QEA) and QEA with a pair-swap strategy (QEAPS), where each gene is represented by a quantum bit (qubit) and the qubit is updated by a unitary transformation in both algorithms. QEA and QEAPS can automatically shift the evolution from a global search to a local search and have shown superior search performance to the classical genetic algorithm. However, the population gets into a locally optimal solution and the solution search stagnates when the probability amplitudes of qubit excessively converge to |0› or |1›. In this study, we have proposed a measure that can confirm convergence state of qubits. From the results of the computational experiment in the maximum cut problem, we have clarified that the proposed measure can estimate the state of the qubit, and the quality of the obtained solution is improved by applying the method for maintenance of diversity.

Search performance analysis of qubit convergence measure for quantum-inspired evolutionary algorithm introducing on maximum cut problem

Search performance analysis of qubit convergence measure for quantum-inspired evolutionary algorithm introducing on maximum cut problem

Maximum Cuts in Edge-colored Graphs

The input of the Maximum Colored Cut problem consists of a graph G=(V,E) with an edge-coloring c:E→{1,2,3,…,p} and a positive integer k>0, and the question is whether G has a nontrivial edge cut using at least k colors. The Colorful Cut problem has the same input but asks for a nontrivial edge cut using all colors. Unlike what happens for the classical Maximum Cut problem, we prove that both problems are NP-complete even on complete, planar, or bounded treewidth graphs. Furthermore, we prove that Colorful Cut is NP-complete even when each color class induces a clique of size at most 3, but is trivially solvable when each color induces a K2. On the positive side, we prove that Maximum Colored Cut is fixed-parameter tractable when parameterized by either k or p, and that it admits a cubic kernel in both cases.

Linear Kernels and Linear-Time Algorithms for Finding Large Cuts

The maximum cut problem in graphs and its generalizations are fundamental combinatorial problems. Several of these cut problems were recently shown to be fixed-parameter tractable and admit polynomial kernels when parameterized above the tight lower bound measured by the size and order of the graph. In this paper we continue this line of research and considerably improve several of those results:We show that an algorithm by Crowston et al. (Algorithmica 72(3):734–757, 2015) for (Signed) Max-Cut Above Edwards−Erdős Bound can be implemented so as to run in linear time8^kcdot O(m); this significantly improves the previous analysis with run time 8^kcdot O(n^4).We give an asymptotically optimal kernel for (Signed) Max-Cut Above Edwards−Erdős Bound with O(k) vertices, improving a kernel with O(k^3) vertices by Crowston et al. (Theor Comput Sci 513:53–64, 2013).We improve all known kernels for parameterizations above strongly lambda -extendible properties (a generalization of the Max-Cut results) by Crowston et al. (Proceedings of FSTTCS 2013, Leibniz international proceedings in informatics, Guwahati, 2013) from O(k^3) vertices to O(k) vertices.Therefore, Max Acyclic Subdigraph parameterized above Poljak–Turzík bound admits a kernel with O(k) vertices and can be solved in 2^{O(k)}cdot n^{O(1)} time; this answers an open question by Crowston et al. (Proceedings of FSTTCS 2012, Leibniz international proceedings in informatics, Hyderabad, 2012). All presented kernels can be computed in time O(km).

On maximum leaf trees and connections to connected maximum cut problems

In an instance of the (directed) Max Leaf Tree (MLT) problem we are given a vertex-weighted (di)graph G(V,E,w) and the goal is to compute a subtree with maximum weight on the leaves. The weighted Connected Max Cut (CMC) problem takes in an undirected edge-weighted graph G(V,E,w) and seeks a subset S⊆V such that the induced graph G[S] is connected and ∑e∈δ(S)w(e) is maximized.We obtain a constant approximation algorithm for MLT when the weights are chosen from {0,1}, which in turn implies a Ω(1/log⁡n) approximation for the general case. We show that the MLT and CMC problems are related and use the algorithm for MLT to improve the factor for CMC from Ω(1/log2⁡n) (Hajiaghayi et al., ESA 2015) to Ω(1/log⁡n).

Smoothed Analysis of Local Search for the Maximum-Cut Problem

Even though local search heuristics are the method of choice in practice for many well-studied optimization problems, most of them behave poorly in the worst case. This is, in particular, the case for the Maximum-Cut Problem, for which local search can take an exponential number of steps to terminate and the problem of computing a local optimum is PLS-complete. To narrow the gap between theory and practice, we study local search for the Maximum-Cut Problem in the framework of smoothed analysis in which inputs are subject to a small amount of random noise. We show that the smoothed number of iterations is quasi-polynomial, that is, it is bounded from above by a polynomial in n log n and ϕ, where n denotes the number of nodes and ϕ denotes the perturbation parameter. This shows that worst-case instances are fragile, and it is a first step in explaining why they are rarely observed in practice.

A polynomial-time algorithm for the maximum cardinality cut problem in proper interval graphs

It is known that the maximum cardinality cut problem is NP-hard even in chordal graphs. On the positive side, the problem is known to be polynomial time solvable in some subclasses of proper interval graphs which is in turn a subclass of chordal graphs. In this paper, we consider the time complexity of the problem in proper interval graphs, and propose a polynomial-time dynamic programming algorithm.

A coherent Ising machine for 2000-node optimization problems.

The analysis and optimization of complex systems can be reduced to mathematical problems collectively known as combinatorial optimization. Many such problems can be mapped onto ground-state search problems of the Ising model, and various artificial spin systems are now emerging as promising approaches. However, physical Ising machines have suffered from limited numbers of spin-spin couplings because of implementations based on localized spins, resulting in severe scalability problems. We report a 2000-spin network with all-to-all spin-spin couplings. Using a measurement and feedback scheme, we coupled time-multiplexed degenerate optical parametric oscillators to implement maximum cut problems on arbitrary graph topologies with up to 2000 nodes. Our coherent Ising machine outperformed simulated annealing in terms of accuracy and computation time for a 2000-node complete graph.

A nonmonotone GRASP

A greedy randomized adaptive search procedure (GRASP) is an iterative multistart metaheuristic for difficult combinatorial optimization problems. Each GRASP iteration consists of two phases: a construction phase, in which a feasible solution is produced, and a local search phase, in which a local optimum in the neighborhood of the constructed solution is sought. Repeated applications of the construction procedure yields different starting solutions for the local search and the best overall solution is kept as the result. The GRASP local search applies iterative improvement until a locally optimal solution is found. During this phase, starting from the current solution an improving neighbor solution is accepted and considered as the new current solution. In this paper, we propose a variant of the GRASP framework that uses a new “nonmonotone” strategy to explore the neighborhood of the current solution. We formally state the convergence of the nonmonotone local search to a locally optimal solution and illustrate the effectiveness of the resulting Nonmonotone GRASP on three classical hard combinatorial optimization problems: the maximum cut problem (MAX-CUT), the weighted maximum satisfiability problem (MAX-SAT), and the quadratic assignment problem (QAP).