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

Graph Isomorphism and the Lasserre Hierarchy

In this paper we show lower bounds for a certain large class of algorithms solving the Graph Isomorphism problem, even on expander graph instances. Spielman [25] shows an algorithm for isomorphism of strongly regular expander graphs that runs in time exp(O(n^(1/3)) (this bound was recently improved to expf O(n^(1/5) [5]). It has since been an open question to remove the requirement that the graph be strongly regular. Recent algorithmic results show that for many problems the Lasserre hierarchy works surprisingly well when the underlying graph has expansion properties. Moreover, recent work of Atserias and Maneva [3] shows that k rounds of the Lasserre hierarchy is a generalization of the k-dimensional Weisfeiler-Lehman algorithm for Graph Isomorphism. These two facts combined make the Lasserre hierarchy a good candidate for solving graph isomorphism on expander graphs. Our main result rules out this promising direction by showing that even Omega(n) rounds of the Lasserre semidefinite program hierarchy fail to solve the Graph Isomorphism problem even on expander graphs.Comment: 22 pages, 3 figures, submitted to CC

Novel techniques for automorphism group computation

Graph automorphism (GA) is a classical problem, in which the objective is to compute the automorphism group of an input graph. In this work we propose four novel techniques to speed up algorithms that solve the GA problem by exploring a search tree. They increase the performance of the algorithm by allowing to reduce the depth of the search tree, and by effectively pruning it. We formally prove that a GA algorithm that uses these techniques correctly computes the automorphism group of the input graph. We also describe how the techniques have been incorporated into the GA algorithm conauto, as conauto-2.03, with at most an additive polynomial increase in its asymptotic time complexity. We have experimentally evaluated the impact of each of the above techniques with several graph families. We have observed that each of the techniques by itself significantly reduces the number of processed nodes of the search tree in some subset of graphs, which justifies the use of each of them. Then, when they are applied together, their effect is combined, leading to reductions in the number of processed nodes in most graphs. This is also reflected in a reduction of the running time, which is substantial in some graph families

Symmetry in Finite Combinatorial Objects: Scalable Methods and Applications.

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 thesis, we describe highly scalable symmetry-detection algorithms for two widely-used combinatorial objects: graphs and Boolean functions. Our algorithms are based on a decision tree that combines elements of group-theoretic computation with branching and backtracking search. Moreover, we contrast the search for graph symmetries and a canonical labeling to dissect typical algorithms and identify their similarities and differences. We develop a novel approach to graph canonical labeling where symmetries are found first and then used to speed up the canonical-labeling routines. Empirical results are given for graphs with millions of vertices and Boolean functions with hundreds of I/Os, where our algorithms can often find all symmetry group generators or a canonical labeling in seconds.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/100003/1/hadik_1.pd

Impact of Symmetries in Graph Clustering

Diese Dissertation beschäftigt sich mit der durch die Automorphismusgruppe definierten Symmetrie von Graphen und wie sich diese auf eine Knotenpartition, als Ergebnis von Graphenclustering, auswirkt. Durch eine Analyse von nahezu 1700 Graphen aus verschiedenen Anwendungsbereichen kann gezeigt werden, dass mehr als 70 % dieser Graphen Symmetrien enthalten. Dies bildet einen Gegensatz zum kombinatorischen Beweis, der besagt, dass die Wahrscheinlichkeit eines zufälligen Graphen symmetrisch zu sein bei zunehmender Größe gegen Null geht. Das Ergebnis rechtfertigt damit die Wichtigkeit weiterer Untersuchungen, die auf mögliche Auswirkungen der Symmetrie eingehen. Bei der Analyse werden sowohl sehr kleine Graphen (10 000 000 Knoten/>25 000 000 Kanten) berücksichtigt. Weiterhin wird ein theoretisches Rahmenwerk geschaffen, das zum einen die detaillierte Quantifizierung von Graphensymmetrie erlaubt und zum anderen Stabilität von Knotenpartitionen hinsichtlich dieser Symmetrie formalisiert. Eine Partition der Knotenmenge, die durch die Aufteilung in disjunkte Teilmengen definiert ist, wird dann als stabil angesehen, wenn keine Knoten symmetriebedingt von der einen in die andere Teilmenge abgebildet werden und dadurch die Partition verändert wird. Zudem wird definiert, wie eine mögliche Zerlegbarkeit der Automorphismusgruppe in unabhängige Untergruppen als lokale Symmetrie interpretiert werden kann, die dann nur Auswirkungen auf einen bestimmten Bereich des Graphen hat. Um die Auswirkungen der Symmetrie auf den gesamten Graphen und auf Partitionen zu quantifizieren, wird außerdem eine Entropiedefinition präsentiert, die sich an der Analyse dynamischer Systeme orientiert. Alle Definitionen sind allgemein und können daher für beliebige Graphen angewandt werden. Teilweise ist sogar eine Anwendbarkeit für beliebige Clusteranalysen gegeben, solange deren Ergebnis in einer Partition resultiert und sich eine Symmetrierelation auf den Datenpunkten als Permutationsgruppe angeben lässt. Um nun die tatsächliche Auswirkung von Symmetrie auf Graphenclustering zu untersuchen wird eine zweite Analyse durchgeführt. Diese kommt zum Ergebnis, dass von 629 untersuchten symmetrischen Graphen 72 eine instabile Partition haben. Für die Analyse werden die Definitionen des theoretischen Rahmenwerks verwendet. Es wird außerdem festgestellt, dass die Lokalität der Symmetrie eines Graphen maßgeblich beeinflusst, ob dessen Partition stabil ist oder nicht. Eine hohe Lokalität resultiert meist in einer stabilen Partition und eine stabile Partition impliziert meist eine hohe Lokalität. Bevor die obigen Ergebnisse beschrieben und definiert werden, wird eine umfassende Einführung in die verschiedenen benötigten Grundlagen gegeben. Diese umfasst die formalen Definitionen von Graphen und statistischen Graphmodellen, Partitionen, endlichen Permutationsgruppen, Graphenclustering und Algorithmen dafür, sowie von Entropie. Ein separates Kapitel widmet sich ausführlich der Graphensymmetrie, die durch eine endliche Permutationsgruppe, der Automorphismusgruppe, beschrieben wird. Außerdem werden Algorithmen vorgestellt, die die Symmetrie von Graphen ermitteln können und, teilweise, auch das damit eng verwandte Graphisomorphie Problem lösen. Am Beispiel von Graphenclustering gibt die Dissertation damit Einblicke in mögliche Auswirkungen von Symmetrie in der Datenanalyse, die so in der Literatur bisher wenig bis keine Beachtung fanden

A SAT Approach to Clique-Width

Clique-width is a graph invariant that has been widely studied in combinatorics and computer science. However, computing the clique-width of a graph is an intricate problem, the exact clique-width is not known even for very small graphs. We present a new method for computing the clique-width of graphs based on an encoding to propositional satisfiability (SAT) which is then evaluated by a SAT solver. Our encoding is based on a reformulation of clique-width in terms of partitions that utilizes an efficient encoding of cardinality constraints. Our SAT-based method is the first to discover the exact clique-width of various small graphs, including famous graphs from the literature as well as random graphs of various density. With our method we determined the smallest graphs that require a small pre-described clique-width.Comment: proofs in section 3 updated, results remain unchange

The Impact of Graph Symmetry on Clustering

This article investigates the effect of graph symmetry on modularity optimal graph clustering partitions. The key finding is that there actually exists an impact of graph symmetry, as more than 22% of the analyzed graphs have an unstable partition. The results are based on an empirical analysis of 1254 symmetric graphs, which are a subset of the 1699 graphs that were analyzed by Ball and Geyer-Schulz (2018a). For each graph a modularity optimal partition is computed by a graph clustering algorithm. Additionally, the generating sets for the automorphism group of each graph are obtained. All computed partitions are tested for stability, which means that the symmetry that is captured by the automorphism group does not change this partition. Furthermore, definitions that allow to distinguish local and global symmetry of graphs are presented

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