A Computational Comparison of Optimization Methods for the Golomb Ruler Problem

The Golomb ruler problem is defined as follows: Given a positive integer n, locate n marks on a ruler such that the distance between any two distinct pair of marks are different from each other and the total length of the ruler is minimized. The Golomb ruler problem has applications in information theory, astronomy and communications, and it can be seen as a challenge for combinatorial optimization algorithms. Although constructing high quality rulers is well-studied, proving optimality is a far more challenging task. In this paper, we provide a computational comparison of different optimization paradigms, each using a different model (linear integer, constraint programming and quadratic integer) to certify that a given Golomb ruler is optimal. We propose several enhancements to improve the computational performance of each method by exploring bound tightening, valid inequalities, cutting planes and branching strategies. We conclude that a certain quadratic integer programming model solved through a Benders decomposition and strengthened by two types of valid inequalities performs the best in terms of solution time for small-sized Golomb ruler problem instances. On the other hand, a constraint programming model improved by range reduction and a particular branching strategy could have more potential to solve larger size instances due to its promising parallelization features

Towards hybrid methods for solving hard combinatorial optimization problems

Tesis doctoral leída en la Escuela Politécnica Superior de la Universidad Autónoma de Madrid el 4 de septiembre de 200

Dynamically weakened constraints in bounded search for constraint optimisation problems

Combinatorial optimisation problems, where the goal is to an optimal solution from the set of solutions of a problem involving resources, constraints on how these resources can be used, and a ranking of solutions are of both theoretical and practical interest. Many real world problems (such as routing vehicles or planning timetables) can be modelled as constraint optimisation problems, and solved via a variety of solver technologies which rely on differing algorithms for search and inference. The starting point for the work presented in this thesis is two existing approaches to solving constraint optimisation problems: constraint programming and decision diagram branch and bound search. Constraint programming models problems using variables which have domains of values and valid value assignments to variables are restricted by constraints. Constraint programming is a mature approach to solving optimisation problems, and typically relies on backtracking search algorithms combined with constraint propagators (which infer from incomplete solutions which values can be removed from the domains of variables which are yet to be assigned a value). Decision diagram branch and bound search is a less mature approach which solves problems modelled as dynamic programming models using width restricted decision diagrams to provide bounds during search. The main contribution of this thesis is adapting decision diagram branch and bound to be the search scheme in a general purpose constraint solver. To achieve this we propose a method in which we introduce a new algorithm for each constraint that we wish to include in our solver and these new algorithms weaken individual constraints, so that they respect the problem relaxations introduced while using decision diagram branch and bound as the search algorithm in our solver. Constraints are weakened during search based on the problem relaxations imposed by the search algorithm: before search begins there is no way of telling which relaxations will be introduced. We attempt to provide weakening algorithms which require little to no changes to existing propagation algorithms. We provide weakening algorithms for a number of built-in constraints in the Flatzinc specifi- cation, as well as for global constraints and symmetry reduction constraints. We implement a solver in Go and empirically verify the competitiveness of our approach. We show that our solver can be parallelised using Goroutines and channels and that our approach scales well. Finally, we also provide an implementation of our approach in a solver which is tailored towards solving extremal graph problems. We use the forbidden subgraph problem to show that our approach of using decision diagram branch and bound as a search scheme in a constraint solver can be paired with canonical search. Canonical search is a technique for graph search which ensures that no two isomorphic graphs are returned during search. We pair our solver with the Nauty graph isomorphism algorithm to achieve this, and explore the relationship between branch and bound and canonical search

Global Constraint Catalog, 2nd Edition

This report presents a catalogue of global constraints where each constraint is explicitly described in terms of graph properties and/or automata and/or first order logical formulae with arithmetic. When available, it also presents some typical usage as well as some pointers to existing filtering algorithms