Exploiting single-cycle symmetries in branch-and-prune algorithms

The final publication is available at link.springer.comAs a first attempt to exploit symmetries in continuous constraint problems, we focus on permutations of the variables consisting of one single cycle. We propose a procedure that takes advantage of these symmetries by interacting with a Branch-and-Prune algorithm without interfering with it. A key concept in this procedure are the classes of symmetric boxes formed by bisecting a n-dimensional cube at the same point in all dimensions at the same time. We quantify these classes as a function of n. Moreover, we propose a simple algorithm to generate the representatives of all these classes for any number of variables at very high rates. A problem example from the chemical field and a kinematics solver are used to show the performance of the approach in practice.Peer ReviewedPostprint (author's final draft

Exploiting single-cycle symmetries in branch-and-prune algorithms

As a first attempt to exploit symmetries in continuous con- straint problems, we focus on permutations of the variables consisting of one single cycle. We propose a procedure that takes advantage of these symmetries by interacting with a Branch-and-Prune algorithm without interfering with it. A key concept in this procedure are the classes of symmetric boxes formed by bisecting a n-dimensional cube at the same point in all dimensions at the same time. We quantify these classes as a function of n. Moreover, we propose a simple algorithm to generate the representatives of all these classes for any number of variables at very high rates. A problem example from the chemical field and a kinematics solver are used to show the performance of the approach in practice.Peer Reviewe

Graph Symmetry Detection and Canonical Labeling: Differences and Synergies

Symmetries of combinatorial objects are known to complicate search algorithms, but such obstacles can often be removed by detecting symmetries early and discarding symmetric subproblems. Canonical labeling of combinatorial objects facilitates easy equivalence checking through quick matching. All existing canonical labeling software also finds symmetries, but the fastest symmetry-finding software does not perform canonical labeling. In this work, we contrast the two problems and dissect typical algorithms to identify their similarities and differences. We then develop a novel approach to canonical labeling where symmetries are found first and then used to speed up the canonical labeling algorithms. Empirical results show that this approach outperforms state-of-the-art canonical labelers.Comment: 15 pages, 10 figures, 1 table, Turing-10

Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

Symmetry Breaking for Answer Set Programming

In the context of answer set programming, this work investigates symmetry detection and symmetry breaking to eliminate symmetric parts of the search space and, thereby, simplify the solution process. We contribute a reduction of symmetry detection to a graph automorphism problem which allows to extract symmetries of a logic program from the symmetries of the constructed coloured graph. We also propose an encoding of symmetry-breaking constraints in terms of permutation cycles and use only generators in this process which implicitly represent symmetries and always with exponential compression. These ideas are formulated as preprocessing and implemented in a completely automated flow that first detects symmetries from a given answer set program, adds symmetry-breaking constraints, and can be applied to any existing answer set solver. We demonstrate computational impact on benchmarks versus direct application of the solver. Furthermore, we explore symmetry breaking for answer set programming in two domains: first, constraint answer set programming as a novel approach to represent and solve constraint satisfaction problems, and second, distributed nonmonotonic multi-context systems. In particular, we formulate a translation-based approach to constraint answer set solving which allows for the application of our symmetry detection and symmetry breaking methods. To compare their performance with a-priori symmetry breaking techniques, we also contribute a decomposition of the global value precedence constraint that enforces domain consistency on the original constraint via the unit-propagation of an answer set solver. We evaluate both options in an empirical analysis. In the context of distributed nonmonotonic multi-context system, we develop an algorithm for distributed symmetry detection and also carry over symmetry-breaking constraints for distributed answer set programming.Comment: Diploma thesis. Vienna University of Technology, August 201

Fastest mixing Markov chain on graphs with symmetries

We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of large-scale instances that are otherwise computationally infeasible. We obtain analytic or semi-analytic results for particular classes of graphs, such as edge-transitive and distance-transitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and block-diagonalization, respectively. We also establish the connection between these two approaches.Comment: 39 pages, 15 figure

Detection of permutation symmetries in numerical constraint satisfaction problems

Technical reportThis technical report summarizes the work done by Ieva Dauzickaite during her traineeship at IRI from 2016-10-03 to 2017-03-31. The main objectives of the stay at IRI were to study detection of variable symmetries in numerical constraint satisfaction problems and to implement a method that detects such symmetries.Peer ReviewedPostprint (published version

Exploiting Symmetry in Linear and Integer Linear Programming

This thesis explores two algorithmic approaches for exploiting symmetries in linear and integer linear programs. The first is orbital crossover, a novel method of crossover designed to exploit symmetry in linear programs. Symmetry has long been considered a curse in combinatorial optimization problems, but significant progress has been made. Up until recently, symmetry exploitation in linear programs was not worth the upfront cost of symmetry detection. However, recent results involving a generalization of symmetries, equitable partitions, has made the upfront cost much more manageable. The motivation for orbital crossover is that many highly symmetric integer linear programs exist, and thus, solving symmetric linear programs is of major interest in order to efficiently solve symmetric integer linear programs. The results of this work indicate that a specialized linear programming algorithm that exploits symmetry is likely to be useful in the toolbox of linear programming solvers. The second algorithm is orbital cut generation. The main issue brought forward by symmetric integer linear programs is multiple symmetric solutions having an equivalent objective value. This massively increases the search space for algorithms such as branch and bound or branch and cut. Orbital cut generation aims to tackle the issues of multiple equivalent symmetric solutions using symmetrically valid inequalities. Chapter 2 shows how to effectively exploit symmetry in integer linear programs by generating symmetric cutting planes that remove multiple symmetric solutions in one go. Further, the method is strengthened using symmetry to aggregate integer linear programs and generate cutting planes in aggregate spaces before lifting them to the original problem

Almost Symmetries and the Unit Commitment Problem

This thesis explores two main topics. The first is almost symmetry detection on graphs. The presence of symmetry in combinatorial optimization problems has long been considered an anathema, but in the past decade considerable progress has been made. Modern integer and constraint programming solvers have automatic symmetry detection built-in to either exploit or avoid symmetric regions of the search space. Automatic symmetry detection generally works by converting the input problem to a graph which is in exact correspondence with the problem formulation. Symmetry can then be detected on this graph using one of the excellent existing algorithms; these are also the symmetries of the problem formulation.The motivation for detecting almost symmetries on graphs is that almost symmetries in an integer program can force the solver to explore nearly symmetric regions of the search space. Because of the known correspondence between integer programming formulations and graphs, this is a first step toward detecting almost symmetries in integer programming formulations. Though we are only able to compute almost symmetries for graphs of modest size, the results indicate that almost symmetry is definitely present in some real-world combinatorial structures, and likely warrants further investigation.The second topic explored in this thesis is integer programming formulations for the unit commitment problem. The unit commitment problem involves scheduling power generators to meet anticipated energy demand while minimizing total system operation cost. Today, practitioners usually formulate and solve unit commitment as a large-scale mixed integer linear program.The original intent of this project was to bring the analysis of almost symmetries to the unit commitment problem. Two power generators are almost symmetric in the unit commitment problem if they have almost identical parameters. Along the way, however, new formulations for power generators were discovered that warranted a thorough investigation of their own. Chapters 4 and 5 are a result of this research.Thus this work makes three contributions to the unit commitment problem: a convex hull description for a power generator accommodating many types of constraints, an improved formulation for time-dependent start-up costs, and an exact symmetry reduction technique via reformulation