The Use of an Exact Algorithm within a Tabu Search Maximum Clique Algorithm

Let G = (V, E) be an undirected graph with vertex set V and edge set E. A clique C of G is a subset of the vertices of V with every pair of vertices of C adjacent. A maximum clique is a clique with the maximum number of vertices. A tabu search algorithm for the maximum clique problem that uses an exact algorithm on subproblems is presented. The exact algorithm uses a graph coloring upper bound for pruning, and the best such algorithm to use in this context is considered. The final tabu search algorithm successfully finds the optimal or best known solution for all standard benchmarks considered. It is compared with a state-of-the-art algorithm that does not use exact search. It is slower to find the known optimal solution for most instances but is faster for five instances and finds a larger clique for two instances

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

Track-oriented multiple hypothesis tracking based on Tabu search and Gibbs sampling

In order to circumvent the curse of dimensionality in multiple hypothesis tracking data association, this paper proposes two efficient implementation algorithms using Tabu search and Gibbs sampling. As the first step, we formulate the problem of generating the best global hypothesis in multiple hypothesis tracking as the problem of finding a maximum weighted independent set of a weighted undirected graph. Then, the metaheuristic Tabu search with two basic movements is designed to find the global optimal solution of the problem formulated. To improve the computational efficiency, this paper also develops a sampling based algorithm based on Gibbs sampling. The problem formulated for the Tabu search-based algorithm is reformulated as a maximum product problem to enable the implementation of Gibbs sampling. The detailed algorithm is then designed and the convergence is also theoretically analyzed. The performance of the two algorithms proposed are verified through numerical simulations and compared with that of a mainstream multiple dimensional assignment implementation algorithm. The simulation results confirm that the proposed algorithms significantly improve the computational efficiency while maintaining or even enhancing the tracking performance

최대 가중 클릭 문제의 동적 생성법을 이용한 온라인 다중 카메라 다중 물체 추적 기법

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2016. 8. 최진영.In this dissertation, we propose an online and real-time algorithm for tracking of multiple targets with multiple cameras that have overlapping field of views. Because of its applicability, multiple target tracking with a visual sensor has been studied intensively during the recent decades. Especially, algorithms using multiple overlapping cameras have been proposed to overcome the occlusion and missing problem of target that cannot be resolved by a single camera. Since the multiple camera multiple target tracking (MCMTT) problem is more complicated than the single camera multiple target tracking (SCMTT) problem, most of MCMTT algorithms are based on a batch process which considers a whole sequence at a time. Although the batch-based algorithms have been achieved the robust performance, their usability is limited because many practical applications need an instantaneous result. The objective of this dissertation is to develop an online MCMTT algorithm that has compatible tracking performance compared to the batch-based algorithms, but requires a small amount of computations. The proposed algorithm generates track hypotheses (or simply called `track') with all possible data associations between object detections from multiple cameras through frames. Then, it picks a set of tracks that best describes the tracking of targets. To identify a good track, the quality of each track is measured by our score function. The tracking solution is, then, a set of tracks that has the maximum total score. To get the solution, we formulate the problem of finding those track set as the maximum weighted clique problem (MWCP), which is one of the widely adopted formulations for a combinatorial problem that has the pairwise compatibility relationship among the variables. MWCP is well-known NP-complete problem and its worst-case computation time is proportional to the exponent of the number of tracks. Thus, solving MWCP is intractable because the number of candidate tracks exponentially increases when the tracking progresses. To alleviate the huge computational load, we propose an online scheme that dynamically formates multiple MWCPs with small-sized subsets of candidate tracks in every frame. The scheme is motivated by that the tracking solutions from consecutive frames are very similar because the status of each target is not abruptly changed between one frame. When we assume that a specific track set is an actual solution of the previous frame, only a small number of tracks have a possibility to become a solution track of the current frame. Thus, we can narrow down the size of candidate track set with the previous solution. However, propagating only the best solution of each frame can cause irreducible error when a wrong track set is chosen as the solution because of the tracking ambiguity. To hedge the risk of this error, we find multiple good solutions at each frame and propagate the K-best solutions among them to the next frame instead of a single solution. When the candidate tracks are updated and generated with newly obtained detections at the next frame, we generate multiple subsets of the entire candidate tracks with the K-best previous solutions. Each subset consists of candidate solution tracks with respect to each of the previous solutions, and a small-sized MWCP is formated with the subset. Then, our algorithm finds multiple solutions from each MWCP and repeats above procedures until the tracking is terminated. Even the proposed algorithm solves multiple MWCPs, it has lower computational complexity than solving a single MWCP with the entire candidate tracks because the overall computational load is mainly affected by the size of the largest MWCP. Moreover, when an instantaneous result is demanded, our algorithm finds better solution than solving a single large-sized MWCP because it finds more diverse solutions under a limited solving time. Although our dynamic formulation remarkably moderates the overall computational complexity, it is still challenging to satisfy the real-time capability of the tracking system. Thus, we apply three more strategies to reduce the computation time. First, we generate tracklets, robust fragments of a target's trajectory, at each camera and generate candidate tracks with those tracklets instead of detections. This prevents a generation of many absurd tracks. Second, we adopt a heuristic algorithm called a breakout local search (BLS) to solve each MWCP. With BLS, multiple suboptimal solutions can be found efficiently within a short time. Last, we prune the candidate tracks with a probability that is calculated with the K-best solutions. The probability represents the quality of each track with respect to the overall tracking situation instead of an individual track. Thus, utilizing this probability ensures a proper pruning of candidate tracks. In the experiments with a public benchmark dataset, our algorithm shows the compatible performance compared to the state-of-the-art batch-based MCMTT algorithms. Moreover, our algorithm shows a real-time capability by achieving a satisfactory performance within a reasonable computation time. We also conduct a self-comparison to verify our dynamic MWCP formation with respect to the tracking performance and solving time. When a sufficient number of solutions are propagated, our algorithm performs better and takes shorter time than solving a single MWCP considering the entire candidate tracks.Chapter 1 Introduction 1 1.1 Background 1 1.2 Related Works 3 1.2.1 Reconstruction-and-tracking methods 4 1.2.2 Tracking-and-reconstruction methods 6 1.2.3 Unified frameworks 7 1.3 Contents of the Research 8 1.4 Thesis Organization 11 Chapter 2 Preliminaries 13 2.1 Bayesian Tracking 14 2.1.1 Recursive Bayesian Tracking 16 2.1.2 Bayesian Tracking for Multiple Targets 17 2.1.3 Multiple Hypothesis Tracking (MHT) 19 2.2 Maximum Weighted Clique Problem (MWCP) 24 2.2.1 Clique Problems 24 2.2.2 Solving MWCP 26 2.3 Breakout Local Search (BLS) 27 2.3.1 Solution exploration 28 2.3.2 Perturbation Strategies 30 2.3.3 Initial Solution and Termination Condition 32 Chapter 3 Proposed Approach 35 3.1 Problem Statements 35 3.2 Tracklet Generation 40 3.2.1 Detection-to-tracklet Matching 43 3.2.2 Matching Score with Motion Estimation 46 3.2.3 Matching Validation 49 3.3 Track Hypothesis 51 3.3.1 Tracklet Association 51 3.3.2 Online Generation of Association Sets 55 3.3.3 Track Generation 57 3.3.4 Track Score 59 3.4 Global Hypothesis 64 3.4.1 MWCP for MCMTT 65 3.4.2 BLS for MCMTT 69 3.5 Pruning 70 3.5.1 Approximated Global Track Probability 71 3.5.2 Track Pruning Scheme 72 Chapter 4 Experiments 75 4.1 Comparison with the State-of-the-art Methods 81 4.2 Influence of Parameters 84 4.3 Score Function Analysis 87 4.4 Solving Scheme Analysis 88 4.5 Qualitative Results 90 Chapter 5 Concluding Remarks 97 5.1 Conclusions 97 5.2 Future Works 98 초록 117Docto

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

Literature Review on Big Data Analytics Methods

Companies and industries are faced with a huge amount of raw data, which have information and knowledge in their hidden layer. Also, the format, size, variety, and velocity of generated data bring complexity for industries to apply them in an efficient and effective way. So, complexity in data analysis and interpretation incline organizations to deploy advanced tools and techniques to overcome the difficulties of managing raw data. Big data analytics is the advanced method that has the capability for managing data. It deploys machine learning techniques and deep learning methods to benefit from gathered data. In this research, the methods of both ML and DL have been discussed, and an ML/DL deployment model for IOT data has been proposed

High-Quality Hypergraph Partitioning

This dissertation focuses on computing high-quality solutions for the NP-hard balanced hypergraph partitioning problem: Given a hypergraph and an integer $k$, partition its vertex set into $k$ disjoint blocks of bounded size, while minimizing an objective function over the hyperedges. Here, we consider the two most commonly used objectives: the cut-net metric and the connectivity metric. Since the problem is computationally intractable, heuristics are used in practice - the most prominent being the three-phase multi-level paradigm: During coarsening, the hypergraph is successively contracted to obtain a hierarchy of smaller instances. After applying an initial partitioning algorithm to the smallest hypergraph, contraction is undone and, at each level, refinement algorithms try to improve the current solution. With this work, we give a brief overview of the field and present several algorithmic improvements to the multi-level paradigm. Instead of using a logarithmic number of levels like traditional algorithms, we present two coarsening algorithms that create a hierarchy of (nearly) $n$ levels, where $n$ is the number of vertices. This makes consecutive levels as similar as possible and provides many opportunities for refinement algorithms to improve the partition. This approach is made feasible in practice by tailoring all algorithms and data structures to the $n$-level paradigm, and developing lazy-evaluation techniques, caching mechanisms and early stopping criteria to speed up the partitioning process. Furthermore, we propose a sparsification algorithm based on locality-sensitive hashing that improves the running time for hypergraphs with large hyperedges, and show that incorporating global information about the community structure into the coarsening process improves quality. Moreover, we present a portfolio-based initial partitioning approach, and propose three refinement algorithms. Two are based on the Fiduccia-Mattheyses (FM) heuristic, but perform a highly localized search at each level. While one is designed for two-way partitioning, the other is the first FM-style algorithm that can be efficiently employed in the multi-level setting to directly improve $k$-way partitions. The third algorithm uses max-flow computations on pairs of blocks to refine $k$-way partitions. Finally, we present the first memetic multi-level hypergraph partitioning algorithm for an extensive exploration of the global solution space. All contributions are made available through our open-source framework KaHyPar. In a comprehensive experimental study, we compare KaHyPar with hMETIS, PaToH, Mondriaan, Zoltan-AlgD, and HYPE on a wide range of hypergraphs from several application areas. Our results indicate that KaHyPar, already without the memetic component, computes better solutions than all competing algorithms for both the cut-net and the connectivity metric, while being faster than Zoltan-AlgD and equally fast as hMETIS. Moreover, KaHyPar compares favorably with the current best graph partitioning system KaFFPa - both in terms of solution quality and running time

A Polyhedral Study of Mixed 0-1 Set

We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set

Cooperation of Combinatorial Solvers for Air Traffic Management and Control

In the context of the SESAR project, Air Traffic Control (ATC) and Management (ATM) in Europe is undergoing a paradigm shift to be able to accommodate the current traffic growth forecast: many expert-based systems will be enhanced by optimization software to improve the decisionmaking process and regulation planning. Current state-of-the-art combinatorial optimization techniques that are applied to ATC and ATM include approximation algorithms like metaheuristics (e.g. Genetic Algorithm, Tabu Search, Simulated Annealing, etc.) and complete algorithms like Constraint Programming (CP) and Mixed Integer Programming. However, the large scale of the considered instances and the handling of their inherent uncertainties result in very hard problems, which can hinder or even defeat either of the previously mentioned optimization methods alone. To overcome these difficulties and improve the resolution efficiency of standard algorithms, we propose to study the generic cooperation of any set of combinatorial solvers by sharing solutions, optimization bounds and possibly other information in order to speed up the overall process. In this thesis, we have specified and implemented a distributed system which is able to integrate any combinatorial solver with the suitable interface, adapt existing solvers to take into account and provide information on the state of the search from and to other solvers, and applied this framework to two ATC and ATM problems: the en-route conflict resolution problem and the Gate Allocation Problem (GAP). For the first one, we have presented a new generic framework for the modeling and resolution of en-route conflicts in three dimensions as well as a large set of realistic instances, which have been solved with the cooperation of a Memetic Algorithm and Integer Linear Programming (ILP) solver. For the GAP, we have presented a new CP model, as well as new optimization constraints to maximize the robustness of the schedule, and search strategies together with their parallel cooperation. The solver, implemented with the FaCiLe CP library, outperforms a state-of-the-art ILP solver on real instances