Μετα-ευρεστικές Μέθοδοι Αποικίας Μυρμηγκιών και Σμήνους Σωματιδίων: Εφαρμογή στο Πρόβλημα της Εύρεσης Βαρύτερης Κλίκας σε Γράφο.

Στην παρούσα έρευνα γίνεται σύγκριση δύο μετα-ευρεστικών αλγορίθμων βελτιστοποίησης της συλλογικής νοημοσύνης, της Αποικίας μυρμηγκιών (ACO), που προτάθηκε από τον Marco Dorigo και της νέας εκδοχής του Δυαδικού Σμήνους σωματιδίων (NPSO), που εκδόθηκε από τους Khanesar, Teshnehlab και Shoorehdeli. Το πρόβλημα στο οποίο γίνεται η σύγκριση είναι αυτό της βαρύτερης κλίκας γράφου με βάρη στους κόμβους (MVWCP) το οποίο είναι NP-Hard. Σε αυτό το έγγραφο γίνεται ανάλυση των εκδοχών των προαναφερθέντων αλγορίθμων, οι οποίοι δοκιμάστηκαν σε ένα εύρος γράφων από εκατό μέχρι χίλιους κόμβους συμπεριλαμβανομένων κάποιων από των DIMACS benchmark instances.In this research, two meta-heuristic optimisation algorithms are compared. The Ant Colony Optimisation (ACO) introduced by Marco Dorigo and a newer version of the Binary Particle Swarm Optimization (NPSO), suggested by Khanesar, Teshnehlab, and Shoorehdeli (NPSO). The comparison is applied to the problem of the Maximum Vertex Weight Clique Problem (MVWCP), which is an NP-Hard problem. In this paper, there is an in-depth analysis of the above-mentioned algorithms, which were tested on graphs of one hundred to one thousand vertices, including some graphs of the well-known DIMACS benchmark instances

On Maximum Weight Clique Algorithms, and How They Are Evaluated

Maximum weight clique and maximum weight independent set solvers are often benchmarked using maximum clique problem instances, with weights allocated to vertices by taking the vertex number mod 200 plus 1. For constraint programming approaches, this rule has clear implications, favouring weight-based rather than degree-based heuristics. We show that similar implications hold for dedicated algorithms, and that additionally, weight distributions affect whether certain inference rules are cost-effective. We look at other families of benchmark instances for the maximum weight clique problem, coming from winner determination problems, graph colouring, and error-correcting codes, and introduce two new families of instances, based upon kidney exchange and the Research Excellence Framework. In each case the weights carry much more interesting structure, and do not in any way resemble the 200 rule. We make these instances available in the hopes of improving the quality of future experiments

Multi-neighborhood tabu search for the maximum weight clique problem

Given an undirected graph G=(V,E) with vertex set V={1,…,n} and edge set E⊆V×V. Let w:V→Z + be a weighting function that assigns to each vertex i∈V a positive integer. The maximum weight clique problem (MWCP) is to determine a clique of maximum weight. This paper introduces a tabu search heuristic whose key features include a combined neighborhood and a dedicated tabu mechanism using a randomized restart strategy for diversification. The proposed algorithm is evaluated on a total of 136 benchmark instances from different sources (DIMACS, BHOSLIB and set packing). Computational results disclose that our new tabu search algorithm outperforms the leading algorithm for the maximum weight clique problem, and in addition rivals the performance of the best methods for the unweighted version of the problem without being specialized to exploit this problem class

Low-Diameter Clusters in Network Analysis

In this dissertation, we introduce several novel tools for cluster-based analysis of complex systems and design solution approaches to solve the corresponding optimization problems. Cluster-based analysis is a subfield of network analysis which utilizes a graph representation of a system to yield meaningful insight into the system structure and functions. Clusters with low diameter are commonly used to characterize cohesive groups in applications for which easy reachability between group members is of high importance. Low-diameter clusters can be mathematically formalized using a clique and an s-club (with relatively small values of s), two concepts from graph theory. A clique is a subset of vertices adjacent to each other and an s-club is a subset of vertices inducing a subgraph with a diameter of at most s. A clique is actually a special case of an s-club with s = 1, hence, having the shortest possible diameter. Two topics of this dissertation focus on graphs prone to uncertainty and disruptions, and introduce several extensions of low-diameter models. First, we introduce a robust clique model in graphs where edges may fail with a certain probability and robustness is enforced using appropriate risk measures. With regard to its ability to capture underlying system uncertainties, finding the largest robust clique is a better alternative to the problem of finding the largest clique. Moreover, it is also a hard combinatorial optimization problem, requiring some effective solution techniques. To this aim, we design several heuristic approaches for detection of large robust cliques and compare their performance. Next, we consider graphs for which uncertainty is not explicitly defined, studying connectivity properties of 2-clubs. We notice that a 2-club can be very vulnerable to disruptions, so we enhance it by reinforcing additional requirements on connectivity and introduce a biconnected 2-club concept. Additionally, we look at the weak 2-club counterpart which we call a fragile 2-club (defined as a 2-club that is not biconnected). The size of the largest biconnected 2-club in a graph can help measure overall system reachability and connectivity, whereas the largest fragile 2-club can identify vulnerable parts of the graph. We show that the problem of finding the largest fragile 2-club is polynomially solvable whereas the problem of finding the largest biconnected 2-club is NP-hard. Furthermore, for the former, we design a polynomial time algorithm and for the latter - combinatorial branch-and-bound and branch-and-cut algorithms. Lastly, we once again consider the s-club concept but shift our focus from finding the largest s-club in a graph to the problem of partitioning the graph into the smallest number of non-overlapping s-clubs. This problem cannot only be applied to derive communities in the graph, but also to reduce the size of the graph and derive its hierarchical structure. The problem of finding the minimum s-club partitioning is a hard combinatorial optimization problem with proven complexity results and is also very hard to solve in practice. We design a branch-and-bound combinatorial optimization algorithm and test it on the problem of minimum 2-club partitioning

Variable Formulation and Neighborhood Search Methods for the Maximum Clique Problem in Graph

Doktorska disertacija se bavi temama rešavanja računarski teških problema kombinatorne optimizacije. Istaknut je problem maksimalne klike kao predstavnik određenih struktura u grafovima. Problem maksimalne klike i sa njim povezani problemi su formulisani kao nelinearne funkcije. Rešavani su sa ciljem otkrivanja novih metoda koje pronalaze dobre aproksimacije rešenja za neko razumno vreme. Predložene su varijante Metode promenljivih okolina na rešavanje maksimalne klike u grafu. Povezani problemi na grafovima se mogu primeniti na pretragu informacija, raspoređivanje, procesiranje signala, teoriju klasifikacije, teoriju kodiranja, itd. Svi algoritmi su implementirani i uspešno testirani na brojnim različitim primerima.This Ph.D. thesis addresses topics NP hard problem solving approaches in combinatorial optimization and according to that it is highlighted maximum clique problem as a representative of certain structures in graphs. Maximum clique problem and related problems with this have been formulated as non linear functions which have been solved to research for new methods and good solution approximations for some reasonable time. It has been proposed several different extensions of Variable Neighborhood Search method. Related problems on graphs could be applied on information retrieval, scheduling, signal processing, theory of classi_cation, theory of coding, etc. Algorithms are implemented and successfully tested on various different tasks