Combinatorial optimization and metaheuristics

Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods

Hybrid algorithms for distributed constraint satisfaction.

A Distributed Constraint Satisfaction Problem (DisCSP) is a CSP which is divided into several inter-related complex local problems, each assigned to a different agent. Thus, each agent has knowledge of the variables and corresponding domains of its local problem together with the constraints relating its own variables (intra-agent constraints) and the constraints linking its local problem to other local problems (inter-agent constraints). DisCSPs have a variety of practical applications including, for example, meeting scheduling and sensor networks. Existing approaches to Distributed Constraint Satisfaction can be mainly classified into two families of algorithms: systematic search and local search. Systematic search algorithms are complete but may take exponential time. Local search algorithms often converge quicker to a solution for large problems but are incomplete. Problem solving could be improved through using hybrid algorithms combining the completeness of systematic search with the speed of local search. This thesis explores hybrid (systematic + local search) algorithms which cooperate to solve DisCSPs. Three new hybrid approaches which combine both systematic and local search for Distributed Constraint Satisfaction are presented: (i) DisHyb; (ii) Multi-Hyb and; (iii) Multi-HDCS. These approaches use distributed local search to gather information about difficult variables and best values in the problem. Distributed systematic search is run with a variable and value ordering determined by the knowledge learnt through local search. Two implementations of each of the three approaches are presented: (i) using penalties as the distributed local search strategy and; (ii) using breakout as the distributed local search strategy. The three approaches are evaluated on several problem classes. The empirical evaluation shows these distributed hybrid approaches to significantly outperform both systematic and local search DisCSP algorithms. DisHyb, Multi-Hyb and Multi-HDCS are shown to substantially speed-up distributed problem solving with distributed systematic search taking less time to run by using the information learnt by distributed local search. As a consequence, larger problems can now be solved in a more practical timeframe

Reinforcement learning based local search for grouping problems: A case study on graph coloring

Grouping problems aim to partition a set of items into multiple mutually disjoint subsets according to some specific criterion and constraints. Grouping problems cover a large class of important combinatorial optimization problems that are generally computationally difficult. In this paper, we propose a general solution approach for grouping problems, i.e., reinforcement learning based local search (RLS), which combines reinforcement learning techniques with descent-based local search. The viability of the proposed approach is verified on a well-known representative grouping problem (graph coloring) where a very simple descent-based coloring algorithm is applied. Experimental studies on popular DIMACS and COLOR02 benchmark graphs indicate that RLS achieves competitive performances compared to a number of well-known coloring algorithms

A Parameterisation of Algorithms for Distributed Constraint Optimisation via Potential Games

This paper introduces a parameterisation of learning algorithms for distributed constraint optimisation problems (DCOPs). This parameterisation encompasses many algorithms developed in both the computer science and game theory literatures. It is built on our insight that when formulated as noncooperative games, DCOPs form a subset of the class of potential games. This result allows us to prove convergence properties of algorithms developed in the computer science literature using game theoretic methods. Furthermore, our parameterisation can assist system designers by making the pros and cons of, and the synergies between, the various DCOP algorithm components clear

An Order-based Algorithm for Minimum Dominating Set with Application in Graph Mining

Dominating set is a set of vertices of a graph such that all other vertices have a neighbour in the dominating set. We propose a new order-based randomised local search (RLS$_o$) algorithm to solve minimum dominating set problem in large graphs. Experimental evaluation is presented for multiple types of problem instances. These instances include unit disk graphs, which represent a model of wireless networks, random scale-free networks, as well as samples from two social networks and real-world graphs studied in network science. Our experiments indicate that RLS$_o$ performs better than both a classical greedy approximation algorithm and two metaheuristic algorithms based on ant colony optimisation and local search. The order-based algorithm is able to find small dominating sets for graphs with tens of thousands of vertices. In addition, we propose a multi-start variant of RLS$_o$ that is suitable for solving the minimum weight dominating set problem. The application of RLS$_o$ in graph mining is also briefly demonstrated

On the landscape of combinatorial optimization problems

This paper carries out a comparison of the fitness landscape for four classic optimization problems: Max-Sat, graph-coloring, traveling salesman, and quadratic assignment. We have focused on two types of properties, local average properties of the landscape, and properties of the local optima. For the local optima we give a fairly comprehensive description of the properties, including the expected time to reach a local optimum, the number of local optima at different cost levels, the distance between optima, and the expected probability of reaching the optima. Principle component analysis is used to understand the correlations between the local optima. Most of the properties that we examine have not been studied previously, particularly those concerned with properties of the local optima. We compare and contrast the behavior of the four different problems. Although the problems are very different at the low level, many of the long-range properties exhibit a remarkable degree of similarity

Analysis of the fitness landscape for the class of combinatorial optimisation problems

Anatomy of the fitness landscape for a group of well known combinatorial optimisation problems is studied in this research and the similarities and the differences between their landscapes are pointed out. In this research we target the analysis of the fitness landscape for MAX-SAT, Graph-Colouring, Travelling Salesman and Quadratic Assignment problems. Belonging to the class of NP-Hard problems, all these problems become exponentially harder as the problem size grows. We study a group of properties of the fitness landscape for these problems and show what properties are shared by different problems and what properties are different. The properties we investigate here include the time it takes for a local search algorithm to find a local optimum, the number of local and global optima, distance between local and global optima, expected cost of found optima, probability of reaching a global optimum and the cost of the best configuration in the search space. The relationship between these properties and the system size and other parameters of the problems are studied, and it is shown how these properties are shared or differ in different problems. We also study the long-range correlation within the search space, including the expected cost in the Hamming sphere around the local and global optima, the basin of attraction of the local and global optima and the probability of finding a local optimum as a function of its cost. We believe these information provide good insight for algorithm designers

Distributed guided local search for solving binary DisCSPs.

We introduce the Distributed Guided Local Search (Dist- GLS) algorithm for solving Distributed Constraint Satisfaction Problems. Our algorithm is based on the centralised Guided Local Search algorithm, which is extended with additional heuristics in order to enhance its efficiency in distributed scenarios. We discuss the strategies we use for dealing with local optima in the search for solutions and compare performance of Dist-GLS with that of Distributed Breakout (DBA). In addition, we provide the results of our experiments with distributed versions of random binary constraint satisfaction and graph colouring problems

Distributed constraint satisfaction for coordinating and integrating a large-scale, heterogeneous enterprise

Market forces are continuously driving public and private organisations towards higher productivity, shorter process and production times, and fewer labour hours. To cope with these changes, organisations are adopting new organisational models of coordination and cooperation that increase their flexibility, consistency, efficiency, productivity and profit margins. In this thesis an organisational model of coordination and cooperation is examined using a real life example; the technical integration of a distributed large-scale project of an international physics collaboration. The distributed resource constraint project scheduling problem is modelled and solved with the methods of distributed constraint satisfaction. A distributed local search method, the distributed breakout algorithm (DisBO), is used as the basis for the coordination scheme. The efficiency of the local search method is improved by extending it with an incremental problem solving scheme with variable ordering. The scheme is implemented as central algorithm, incremental breakout algorithm (IncBO), and as distributed algorithm, distributed incremental breakout algorithm (DisIncBO). In both cases, strong performance gains are observed for solving underconstrained problems. Distributed local search algorithms are incomplete and lack a termination guarantee. When problems contain hard or unsolvable subproblems and are tightly or overconstrained, local search falls into infinite cycles without explanation. A scheme is developed that identifies hard or unsolvable subproblems and orders these to size. This scheme is based on the constraint weight information generated by the breakout algorithm during search. This information, combined with the graph structure, is used to derive a fail first variable order. Empirical results show that the derived variable order is 'perfect'. When it guides simple backtracking, exceptionally hard problems do not occur, and, when problems are unsolvable, the fail depth is always the shortest. Two hybrid algorithms, BOBT and BOBT-SUSP are developed. When the problem is unsolvable, BOBT returns the minimal subproblem within the search scope and BOBT-SUSP returns the smallest unsolvable subproblem using a powerful weight sum constraint. A distributed hybrid algorithm (DisBOBT) is developed that combines DisBO with DisBT. The distributed hybrid algorithm first attempts to solve the problem with DisBO. If no solution is available after a bounded number of breakouts, DisBO is terminated, and DisBT solves the problem. DisBT is guided by a distributed variable order that is derived from the constraint weight information and the graph structure. The variable order is incrementally established, every time the partial solution needs to be extended, the next variable within the order is identified. Empirical results show strong performance gains, especially when problems are overconstrained and contain small unsolvable subproblems