scholarly journals Improved upper bounds in clique partitioning problem

2019 ◽  
pp. 93-104
Author(s):  
Alexander B. Belyi ◽  
Stanislav L. Sobolevsky ◽  
Alexander N. Kurbatski ◽  
Carlo Ratti
Keyword(s):  
Exact Solution ◽  
Branch And Bound ◽  
Optimization Problems ◽  
Weighted Graph ◽  
Upper Bounds ◽  
Clique Partitioning ◽  
Clustering Problem ◽  
Combinatorial Optimization Problems ◽  
Clique Partitioning Problem ◽  
Partitioning Problem

In this work, a problem of partitioning a complete weighted graph into cliques in such a way that sum of edge weights between vertices belonging to the same clique is maximal is considered. This problem is known as a clique partitioning problem. It arises in many applications and is a varian of classical clustering problem. However, since the problem, as well as many other combinatorial optimization problems, is NP-hard, finding its exact solution often appears hard. In this work, a new method for constructing upper bounds of partition quality function values is proposed, and it is shown how to use these upper bounds in branch and bound technique for finding an exact solution. Proposed method is based on the usage of triangles constraining maximal possible quality of partition. Novelty of the method lies in possibility of using triangles overlapping by edges, which allows to find much tighter bounds than when using only non-overlapping subgraphs. Apart from constructing initial estimate, a method of its recalculation, when fixing edges on each step of branch and bound method, is described. Test results of proposed algorithm on generated sets of random graphs are provided. It is shown, that version that uses new bounds works several times faster than previously known methods.

Application of the “descent with mutations” metaheuristic to a clique partitioning problem

2019 ◽  
Vol 53 (3) ◽  
pp. 1083-1095 ◽  
Author(s):  
Olivier Hudry
Keyword(s):  
Cluster Analysis ◽  
Simulated Annealing ◽  
Weighted Graph ◽  
Local Transformation ◽  
Clique Partitioning ◽  
Clique Partitioning Problem ◽  
Partitioning Problem ◽  
And Cluster Analysis ◽  
Symmetric Relations ◽  
Annealing Method

We study here the application of the “descent with mutations” metaheuristic to a problem arising from the field of classification and cluster analysis (dealing more precisely with the aggregation of symmetric relations) and which can be represented as a clique partitioning of a weighted graph. In this problem, we deai with a complete undirected graphe G; the edges of G have weights which can be positive, negative or equal to 0; the aim is to partition the vertices of G into disjoint cliques (whose number depends on G in order to minimize the sum of the weights of the edges with their two extremities in a same clique; this problem is NP-hard. The “descent with mutations” is a local search metaheuristic, of which the design is very simple and is based on local transformation. It consists in randomly performing random elementary transformations, irrespective improvement or worsening with respect to the objective function. We compare it with another very efficient metaheuristic, which is a simulated annealing method improved by the addition of some ingredients coming from the noising methods. Experiments show that the descent with mutations is at least as efficient for the studied problem as this improved simulated annealing, usually a little better, while it is much easier to design and to tune.

Boosting branch-and-bound MaxSAT solvers with clause learning

2021 ◽  
pp. 1-21
Author(s):  
Chu-Min Li ◽  
Zhenxing Xu ◽  
Jordi Coll ◽  
Felip Manyà ◽  
Djamal Habet ◽  
...  
Keyword(s):  
Experimental Investigation ◽  
Combinatorial Optimization ◽  
Problem Solving ◽  
Branch And Bound ◽  
Real World ◽  
Optimization Problems ◽  
Satisfiability Problem ◽  
Maximum Satisfiability ◽  
Combinatorial Optimization Problems ◽  
Clause Learning

The Maximum Satisfiability Problem, or MaxSAT, offers a suitable problem solving formalism for combinatorial optimization problems. Nevertheless, MaxSAT solvers implementing the Branch-and-Bound (BnB) scheme have not succeeded in solving challenging real-world optimization problems. It is widely believed that BnB MaxSAT solvers are only superior on random and some specific crafted instances. At the same time, SAT-based MaxSAT solvers perform particularly well on real-world instances. To overcome this shortcoming of BnB MaxSAT solvers, this paper proposes a new BnB MaxSAT solver called MaxCDCL. The main feature of MaxCDCL is the combination of clause learning of soft conflicts and an efficient bounding procedure. Moreover, the paper reports on an experimental investigation showing that MaxCDCL is competitive when compared with the best performing solvers of the 2020 MaxSAT Evaluation. MaxCDCL performs very well on real-world instances, and solves a number of instances that other solvers cannot solve. Furthermore, MaxCDCL, when combined with the best performing MaxSAT solvers, solves the highest number of instances of a collection from all the MaxSAT evaluations held so far.

scholarly journals Antiferromagnetic spatial photonic Ising machine through optoelectronic correlation computing

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Junyi Huang ◽  
Yisheng Fang ◽  
Zhichao Ruan
Keyword(s):  
Spatial Light Modulator ◽  
Large Scale ◽  
Single Phase ◽  
Optimization Problems ◽  
Light Modulator ◽  
Practical Applications ◽  
Combinatorial Optimization Problems ◽  
Partitioning Problem ◽  
Statistical Systems ◽  
Antiferromagnetic Model

AbstractRecently, spatial photonic Ising machines (SPIM) have been demonstrated to compute the minima of Hamiltonians for large-scale spin systems. Here we propose to implement an antiferromagnetic model through optoelectronic correlation computing with SPIM. Also we exploit the gauge transformation which enables encoding the spins and the interaction strengths in a single phase-only spatial light modulator. With a simple setup, we experimentally show the ground-state-search acceleration of an antiferromagnetic model with 40000 spins in number-partitioning problem. Thus such an optoelectronic computing exhibits great programmability and scalability for the practical applications of studying statistical systems and combinatorial optimization problems.

On the minimum clique partitioning problem on weighted chordal graphs

2019 ◽  
Vol 791 ◽  
pp. 1-9
Author(s):  
Changseong Jo ◽  
Jihoon Choi ◽  
Suh-Ryung Kim ◽  
Yoshio Sano
Keyword(s):  
Chordal Graphs ◽  
Clique Partitioning ◽  
Clique Partitioning Problem ◽  
Partitioning Problem

Chapter 23. MaxSAT, Hard and Soft Constraints

2021 ◽  
Author(s):  
Chu Min Li ◽  
Felip Manyà
Keyword(s):  
Combinatorial Optimization ◽  
Lower Bounds ◽  
Branch And Bound ◽  
Data Structures ◽  
Optimization Problems ◽  
Soft Constraints ◽  
Inference Rules ◽  
Combinatorial Optimization Problems ◽  
Definition Of ◽  
Hard And Soft Constraints

MaxSAT solving is becoming a competitive generic approach for solving combinatorial optimization problems, partly due to the development of new solving techniques that have been recently incorporated into modern MaxSAT solvers, and to the challenge problems posed at the MaxSAT Evaluations. In this chapter we present the most relevant results on both approximate and exact MaxSAT solving, and survey in more detail the techniques that have proven to be useful in branch and bound MaxSAT and Weighted MaxSAT solvers. Among such techniques, we pay special attention to the definition of good quality lower bounds, powerful inference rules, clever variable selection heuristics and suitable data structures. Moreover, we discuss the advantages of dealing with hard and soft constraints in the Partial MaxSAT formalims, and present a summary of the MaxSAT Evaluations that have been organized so far as affiliated events of the International Conference on Theory and Applications of Satisfiability Testing.

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