A multiple search operator heuristic for the max-k-cut problem

The max-k-cut problem is to partition the vertices of an edge-weighted graph G=(V,E) into k≥2 disjoint subsets such that the weight sum of the edges crossing the different subsets is maximized. The problem is referred as the max-cut problem when k=2. In this work, we present a multiple operator heuristic (MOH) for the general max-k-cut problem. MOH employs five distinct search operators organized into three search phases to effectively explore the search space. Experiments on two sets of 91 well-known benchmark instances show that the proposed algorithm is highly effective on the max-k-cut problem and improves the current best known results (lower bounds) of most of the tested instances for k∈[3,5]. For the popular special case k=2 (i.e., the max-cut problem), MOH also performs remarkably well by discovering 4 improved best known results. We provide additional studies to shed light on the key ingredients of the algorithm

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 v⃗ 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 v⃗ i⋅r⃗ with a random vector r⃗ . 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

A simple iterative algorithm for maxcut

We propose a simple iterative (SI) algorithm for the maxcut problem through fully using an equivalent continuous formulation. It does not need rounding at all and has advantages that all subproblems have explicit analytic solutions, the cut values are monotonically updated and the iteration points converge to a local optima in finite steps via an appropriate subgradient selection. Numerical experiments on G-set demonstrate the performance. In particular, the ratios between the best cut values achieved by SI and the best known ones are at least $0.986$ and can be further improved to at least $0.997$ by a preliminary attempt to break out of local optima.Comment: 30 pages, 1 figure. Subgradient selection, cost analysis and local breakout are adde

Order-of-magnitude differences in computational performance of analog Ising machines induced by the choice of nonlinearity

Ising machines based on nonlinear analog systems are a promising method to accelerate computation of NP-hard optimization problems. Yet, their analog nature is also causing amplitude inhomogeneity which can deteriorate the ability to find optimal solutions. Here, we investigate how the system's nonlinear transfer function can mitigate amplitude inhomogeneity and improve computational performance. By simulating Ising machines with polynomial, periodic, sigmoid and clipped transfer functions and benchmarking them with MaxCut optimization problems, we find the choice of transfer function to have a significant influence on the calculation time and solution quality. For periodic, sigmoid and clipped transfer functions, we report order-of-magnitude improvements in the time-to-solution compared to conventional polynomial models, which we link to the suppression of amplitude inhomogeneity induced by saturation of the transfer function. This provides insights into the suitability of systems for building Ising machines and presents an efficient way for overcoming performance limitations

Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem

According to the structural balance theory, a signed graph is considered structurally balanced when it can be partitioned into a number of modules such that positive and negative edges are respectively located inside and between the modules. In practice, real-world networks are rarely structurally balanced, though. In this case, one may want to measure the magnitude of their imbalance, and to identify the set of edges causing this imbalance. The correlation clustering (CC) problem precisely consists in looking for the signed graph partition having the least imbalance. Recently, it has been shown that the space of the optimal solutions of the CC problem can be constituted of numerous and diverse optimal solutions. Yet, this space is difficult to explore, as the CC problem is NP-hard, and exact approaches do not scale well even when looking for a single optimal solution. To alleviate this issue, in this work we propose an efficient enumeration method allowing to retrieve the complete space of optimal solutions of the CC problem. It combines an exhaustive enumeration strategy with neighborhoods of varying sizes, to achieve computational effectiveness. Results obtained for middle-sized networks confirm the usefulness of our method

On the Study of Fitness Landscapes and the Max-Cut Problem

The goal of this thesis is to study the complexity of NP-Hard problems, using the Max-Cut and the Max-k-Cut problems, and the study of fitness landscapes. The Max-Cut and Max-k-Cut problems are well studied NP-hard problems specially since the approximation algorithm of Goemans and Williamson (1995) which introduced the use of SDP to solve relaxed problems. In order to prove the existence of a performance guarantee, the rounding step from the SDP solution to a Max-Cut solution is simple and randomized. For the Max-k-Cut problem, there exist several approximation algorithms but many of them have been proved to be equivalent. Similarly as in Max-Cut, these approximation algorithms use a simple randomized rounding to be able to get a performance guarantee. Ignoring for now the performance guarantee, one could ask if there is a rounding process that takes into account the structure of the relaxed solution since it is the result of an optimization problem. In this thesis we answered this question positively by using clustering as a rounding method. In order to compare the performance of both algorithms, a series of experiments were performed using the so-called G-set benchmark for the Max-Cut problem and using the Random Graph Benchmark of Goemans1995 for the Max-k-Cut problem. With this new rounding, larger cut values are found both for the Max-Cut and the Max-k-Cut problems, and always above the value of the performance guarantee of the approximation algorithm. This suggests that taking into account the structure of the problem to design algorithms can lead to better results, possibly at the cost of a worse performance guarantee. An example for the vertex k-center problem can be seen in Garcia-Diaz et al. (2017), where a 3-approximation algorithm performs better than a 2-approximation algorithm despite having a worse performance guarantee. Landscapes over discrete configurations spaces are an important model in evolutionary and structural biology, as well as many other areas of science, from the physics of disordered systems to operations research. A landscape is a function defined on a very large discrete set V that carries an additional metric or at least topological structure into the real numbers R. We will consider landscapes defined on the vertex set of undirected graphs. Thus let G=G(V,E) be an undirected graph and f an arbitrary real-valued function taking values from V . We will refer to the triple (V,E,f) as a landscape over G. We say two configurations x,y in V are neutral if f(x)=f(y). We colloquially refer to a landscape as 'neutral'' if a substantial fraction of adjacent pairs of configurations are neutral. A flat landscape is one where f is constant. The opposite of flatness is ruggedness and it is defined as the number of local optima or by means of pair correlation functions. These two key features of a landscape, ruggedness and neutrality, appear to be two sides of the same coin. Ruggedness can be measured either by correlation properties, which are sensitive to monotonic transformation of the landscape, and by combinatorial properties such as the lengths of downhill paths and the number of local optima, which are invariant under monotonic transformations. The connection between the two views has remained largely unexplored and poorly understood. For this thesis, a survey on fitness landscapes is presented, together with the first steps in the direction to find this connection together with a relation between the covariance matrix of a random landscape model and its ruggedness

Global Optimization of the Maximum K-Cut Problem

RÉSUMÉ: Le problème de la k-coupe maximale (max-k-cut) est un problème de partitionnement de graphes qui est un des représentatifs de la classe des problèmes combinatoires NP-difficiles. Le max-kcut peut être utilisé dans de nombreuses applications industrielles. L’objectif de ce problème est de partitionner l’ensemble des sommets en k parties de telle façon que le poids total des arrêtes coupées soit maximisé. Les méthodes proposées dans la littérature pour résoudre le max-k-cut emploient, généralement, la programmation semidéfinie positive (SDP) associée. En comparaison avec les relaxations de la programmation linéaire (LP), les relaxations SDP sont plus fortes mais les temps de calcul sont plus élevés. Par conséquent, les méthodes basées sur la SDP ne peuvent pas résoudre de gros problèmes. Cette thèse introduit une méthode efficace de branchement et de résolution du problème max-k-cut en utilisant des relaxations SDP et LP renforcées. Cette thèse présente trois approches pour améliorer les solutions du max-k-cut. La première approche se concentre sur l’identification des classes d’inégalités les plus pertinentes des relaxations de max-k-cut. Cette approche consiste en une étude expérimentale de quatre classes d’inégalités de la littérature : clique, general clique, wheel et bicycle wheel. Afin d’inclure ces inégalités dans les formulations, nous utilisons un algorithme de plan coupant (CPA) pour ajouter seulement les inégalités les plus importantes . Ainsi, nous avons conçu plusieurs procédures de séparation pour trouver les violations. Les résultats suggèrent que les inégalités de wheel sont les plus fortes. De plus, l’inclusion de ces inégalités dans le max-k-cut peut améliorer la borne de la SDP de plus de 2%. La deuxième approche introduit les contraintes basées sur formulation SDP pour renforcer la relaxation LP. De plus, le CPA est amélioré en exploitant la technique de terminaison précoce d’une méthode de points intérieurs. Les résultats montrent que la relaxation LP avec les inégalités basées sur la SDP surpasse la relaxation SDP pour de nombreux cas, en particulier pour les instances avec un grand nombre de partitions (k � 7). La troisième approche étudie la méthode d’énumération implicite en se basant sur les résultats des dernières approches. On étudie quatre composantes de la méthode. Tout d’abord, nous présentons quatre méthodes heuristiques pour trouver des solutions réalisables : l’heuristique itérative d’agrégation, l’heuristique d’opérateur multiple, la recherche à voisinages variables, et la procédure de recherche aléatoire adaptative gloutonne. La deuxième procédure analyse les stratégies dichotomiques et polytomiques pour diviser un sous-problème. La troisième composante étudie cinq règles de branchement. Enfin, pour la sélection des noeuds de l’arbre de branchement, nous considérons les stratégies suivantes : meilleur d’abord, profondeur d’abord, et largeur d’abord. Pour chaque stratégie, nous fournissons des tests pour différentes valeurs de k. Les résultats montrent que la méthode exacte proposée est capable de trouver de nombreuses solutions. Chacune de ces trois approches a contribué à la conception d’une méthode efficace pour résoudre le problème du max-k-cut. De plus, les approches proposées peuvent être étendues pour résoudre des problèmes génériques d’optimisation en variables mixtes.----------ABSTRACT: In graph theory, the maximum k-cut (max-k-cut) problem is a representative problem of the class of NP-hard combinatorial optimization problems. It arises in many industrial applications and the objective of this problem is to partition vertices of a given graph into at most k partitions such that the total weight of the cut is maximized. The methods proposed in the literature to optimally solve the max-k-cut employ, usually, the associated semidefinite programming (SDP) relaxation in a branch-and-bound framework. In comparison with the linear programming (LP) relaxation, the SDP relaxation is stronger but it suffers from high CPU times. Therefore, methods based on SDP cannot solve large problems. This thesis introduces an efficient branch-and-bound method to solve the max-k-cut problem by using tightened SDP and LP relaxations. This thesis presents three approaches to improve the solutions of the problem. The first approach focuses on identifying relevant classes of inequalities to tighten the relaxations of the max-k-cut. This approach carries out an experimental study of four classes of inequalities from the literature: clique, general clique, wheel and bicycle wheel. In order to include these inequalities, we employ a cutting plane algorithm (CPA) to add only the most important inequalities in practice and we design several separation routines to find violations in a relaxed solution. Computational results suggest that the wheel inequalities are the strongest by far. Moreover, the inclusion of these inequalities in the max-k-cut improves the bound of the SDP formulation by more than 2%. The second approach introduces the SDP-based constraints to strengthen the LP relaxation. Moreover, the CPA is improved by exploiting the early-termination technique of an interior-point method. Computational results show that the LP relaxation with the SDP-based inequalities outperforms the SDP relaxations for many instances, especially for a large number of partitions (k � 7). The third approach investigates the branch-and-bound method using both previous approaches. Four components of the branch-and-bound are considered. First, four heuristic methods are presented to find a feasible solution: the iterative clustering heuristic, the multiple operator heuristic, the variable neighborhood search, and the greedy randomized adaptive search procedure. The second procedure analyzes the dichotomic and polytomic strategies to split a subproblem. The third feature studies five branching rules. Finally, for the node selection, we consider the following strategies: best-first search, depth-first search, and breadth-first search. For each component, we provide computational tests for different values of k. Computational results show that the proposed exact method is able to uncover many solutions. Each one of these three approaches contributed to the design of an efficient method to solve the max-k-cut problem. Moreover, the proposed approaches can be extended to solve generic mixinteger SDP problems