Minimum Cycle Basis Research Articles

IMCB-PGO: Incremental Minimum Cycle Basis Construction and Application to Online Pose Graph Optimization

iMCB-PGO: Incremental Minimum Cycle Basis Construction and Application to Online Pose Graph Optimization

Time-Resolved Graphs of Polymorphic Cycles for H-Bonded Network Identification in Flexible Biomolecules.

A novel approach based on a coarse-grained representation of topological graphs is proposed for the automatic analysis of molecular dynamics (MD) trajectories of hydrogen-bonded (H-Bonded) flexible biomolecules. Herein, our approach models an H-Bonded biomolecule by its H-Bonded cycles and its graph of cycles in which the vertices and links represent the intersections between these cycles. We propose a methodology in which each identified conformer/isomer from the MD is represented by a well-chosen set of H-Bonded cycles called a minimum cycle basis. The key component is the "polycycles" that distinguish the cycles that play the same polymorphic role in the molecule from the ones that lead to an actual conformational change of the molecule. The relevance of our proposed method is evaluated on MD trajectories of gas-phase biomolecules, for which the covalent bonds are unchanged over time and only the hydrogen bonds change over time. The polygraphs and their time evolution are shown to reveal the dynamicity of the metastructure(s) of the H-Bonded biomolecules while providing polymorphic information on the cycles. Such information on the dynamics and changes in the H-bond network, as some cycles change identity while retaining the same role in the overall structure, is not easily captured at the atomic level of representation. Such information can instead be captured by polymorphic cycles.

Incremental Cycle Bases for Cycle-Based Pose Graph Optimization

Pose graph optimization is a special case of the simultaneous localization and mapping problem where the only variables to be estimated are pose variables and the only measurements are inter-pose constraints. The vast majority of pose graph optimization techniques are vertex based (variables are robot poses), but recent work has parameterized the pose graph optimization problem in a relative fashion (variables are the transformations between poses) that utilizes a minimum cycle basis to maximize the sparsity of the problem. We explore the construction of a cycle basis in an incremental manner while maximizing the sparsity. We validate an algorithm that constructs a sparse cycle basis incrementally and compare its performance with a minimum cycle basis. Additionally, we present an algorithm to approximate the minimum cycle basis of two graphs that are sparsely connected as is common in multi-agent scenarios. Lastly, the relative parameterization of pose graph optimization has been limited to using rigid body transforms on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SE(2)$</tex-math></inline-formula> or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SE(3)$</tex-math></inline-formula> as the constraints between poses. We introduce a methodology to allow for the use of lower-degree-of-freedom measurements in the relative pose graph optimization problem. We provide extensive validation of our algorithms on standard benchmarks, simulated datasets, and custom hardware.

Cycle analysis of Directed Acyclic Graphs

In this paper, we employ the decomposition of a directed network as an undirected graph plus its associated node meta-data to characterise the cyclic structure found in directed networks by finding a Minimal Cycle Basis of the undirected graph and augmenting its components with direction information. We show that only four classes of directed cycles exist, and that they can be fully distinguished by the organisation and number of source–sink node pairs and their antichain structure. We are particularly interested in Directed Acyclic Graphs and introduce a set of metrics that characterise the Minimal Cycle Basis using the Directed Acyclic Graphs meta-data information. In particular, we numerically show that transitive reduction stabilises the properties of Minimal Cycle Bases measured by the metrics we introduced while retaining key properties of the Directed Acyclic Graph. This makes the metrics a consistent characterisation of Directed Acyclic Graphs and the systems they represent. We measure the characteristics of the Minimal Cycle Bases of four models of transitively reduced Directed Acyclic Graphs and show that the metrics introduced are able to distinguish the models and are sensitive to their generating mechanisms.

Timetable merging for the Periodic Event Scheduling Problem

We propose a new mixed integer programming based heuristic for computing new benchmark primal solutions for instances of the PESPlib. The PESPlib is a collection of instances for the Periodic Event Scheduling Problem (PESP), comprising periodic timetabling problems inspired by real-world railway timetabling settings, and attracting several international research teams during the last years. We describe two strategies to merge a set of good periodic timetables. These make use of the instance structure and minimum weight cycle bases, finally leading to restricted mixed integer programming formulations with tighter variable bounds. Implementing this timetable merging approach in a concurrent solver, we improve the objective values of the best known solutions for the smallest and largest PESPlib instances by 1.7 and 4.3 percent, respectively.

Sparse Pose Graph Optimization in Cycle Space

The state-of-the-art modern pose-graph optimization (PGO) systems are vertex based. In this context the number of variables might be high, albeit the number of cycles in the graph (loop closures) is relatively low. For sparse problems particularly, the cycle space has a significantly smaller dimension than the number of vertices. By exploiting this observation, in this paper we propose an alternative solution to PGO, that directly exploits the cycle space. We characterize the topology of the graph as a cycle matrix, and re-parameterize the problem using relative poses, which are further constrained by a cycle basis of the graph. We show that by using a minimum cycle basis, the cycle-based approach has superior convergence properties against its vertex-based counterpart, in terms of convergence speed and convergence to the global minimum. For sparse graphs, our cycle-based approach is also more time efficient than the vertex-based. As an additional contribution of this work we present an effective algorithm to compute the minimum cycle basis. Albeit known in computer science, we believe that this algorithm is not familiar to the robotics community. All the claims are validated by experiments on both standard benchmarks and simulated datasets. To foster the reproduction of the results, we provide a complete open-source C++ implementation (Code: \url{https://bitbucket.org/FangBai/cycleBasedPGO) of our approach.

The isothermal evolution of nanoporous gold from the ring perspective - an application of graph theory

The ring structures of five isothermally annealed nanoporous gold (npg) samples were analyzed explicitly by applying results and algorithms from graph theory to skeletonized 3D reconstructions from focused ion beam (FIB) tomography data. Simplified skeletons of the reconstructions were utilized, in which the real ligaments are reduced to straight edges between the branching points of the npg microstructure. So-called minimum weight cycle bases of each skeleton graph’s cycle vector space were calculated, assigning different weight functions to these straight edges: equal weights, Euclidean lengths, and the real ligament lengths from backmapping the Euclidean skeleton edges to the skeletonized real ligament sections. These cycle bases contain the maximum number of linearly independent rings that cannot be generated by smaller rings via the ring sum specified in the cycle vector space. Such a decomposition of the npg network structures into the fundamental ring building blocks served to provide a new perspective of the isothermal evolution of npg, since the coarsening of the npg network structure could be examined from analyzing the local ring topologies and the classification of the ring topological classes. Our results suggest an increasing relative dominance of ligament pinch-off events over ring collapse events, manifesting in a broadening of the distribution of topological classes, and leading to a small but steady increase of the average number of ring edges. Furthermore, self-similar evolution of the investigated sample series cannot be stated. The implications on the topological evolution of npg as a function of the solid volume fraction are discussed.

Thermal state entanglement entropy on a quantum graph.

A particle jumps between the nodes of a graph interacting with local spins. We show that the entanglement entropy of the particle with the spin network is related to the length of the minimum cycle basis. The structure of the thermal state is reminiscent of the string net of spin liquids.

Minimum fundamental cycle basis of some bipartite graphs

We determine the number of cycles of different lengths in the minimum fundamental cycle basis of some bipartite graphs with centralized spanning trees. As applications, we characterize the minimum fundamental cycle basis of some (n−k)-regular balanced graphs with order 2n.

Detecting Determinism in Time Series with Complex Networks Constructed Using a Compression Algorithm

The properties of complex networks derived from applying a compression algorithm to time series subject to symbolic ordinal-based encoding is explored. The information content of compression codewords can be used to detect forbidden symbolic patterns indicative of nonlinear determinism. The connectivity structure of ordinal-based compression networks summarized by their minimal cycle basis structure can also be used in tests for nonlinear determinism, in particular, detection of time irreversibility in a signal.

Length bounds for cycle bases of graphs

The cycle space of a graph G is a vector space over GF(2) which is formed by all Eulerian subgraphs of G with vector addition X⊕Y:=(X∪Y)∖(X∩Y) and scalar multiplication 1⋅X=X, 0⋅X=∅. A base of this vector space is called a cycle base. A cycle base is used to examine the cyclic structure of a graph. The length of a cycle base is the number of its edges in the base. A minimum cycle base is that having the least number of edges. In this paper, we study the length bounds for cycle bases and the minimum cycle bases. A complete characterization is given for a 2-connected graph G with a cycle base of length 2|E(G)|−|V(G)| (it is a lower bound obtained by Leydold and Stadler (1998) [11]). In addition, we derive a sharp lower length bound for minimum cycle bases. As for upper bounds, Horton (1987) showed that the length of a minimum cycle base of a graph with n vertices is at most 3(n−1)(n−2)/2. We improve the bound substantially for graphs on the projective plane where it is at most ⌊13n/2⌋−9.

Applications of Ear Decomposition to Efficient Heterogeneous Algorithms for Shortest Path/Cycle Problems

Graph algorithms play an important role in several fields of sciences and engineering. Prominent among them are the All-Pairs-Shortest-Paths ( APSP ) and related problems. Indeed there are several efficient implementations for such problems on a variety of modern multi- and many-core architectures. It can be noticed that for several graph problems, parallelism offers only a limited success as current parallel architectures have severe short-comings when deployed for most graph algorithms. At the same time, some of these graphs exhibit clear structural properties due to their sparsity. This calls for particular solution strategies aimed at scalable processing of large, sparse graphs on modern parallel architectures. In this paper, we study the applicability of an ear decomposition of graphs to problems such as all-pairs-shortest-paths and minimum cost cycle basis. Through experimentation, we show that the resulting solutions are scalable in terms of both memory usage and also their speedup over best known current implementations. We believe that our techniques have the potential to be relevant for designing scalable solutions for other computations on large sparse graphs.

Cycle bases to the rescue

Cycle bases of graph theory are introduced for the analysis of transition data deposited in line-by-line rovibronic spectroscopic databases. The principal advantage of using cycle bases is that outlier transitions –almost always present in spectroscopic databases built from experimental data originating from many different sources– can be detected and identified straightforwardly and automatically. The data available for six water isotopologues, H216O, H217O, H218O, HD16O, HD17O, and HD18O, in the HITRAN2012 and GEISA2015 databases are used to demonstrate the utility of cycle-basis-based outlier-detection approaches. The spectroscopic databases appear to be sufficiently complete so that the great majority of the entries of the minimum cycle basis have the minimum possible length of four. More than 2000 transition conflicts have been identified for the isotopologue H216Oin the HITRAN2012 database, the seven common conflict types are discussed. It is recommended to employ cycle bases, and especially a minimum cycle basis, for the analysis of transitions deposited in high-resolution spectroscopic databases.

Minimum cycle and homology bases of surface-embedded graphs

We study the problems of finding a minimum cycle basis (a minimum-weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum-weight set of cycles that generates the 1-dimensional $(\mathbb{ Z}_ 2 )$-homology classes) of an undirected graph cellularly embedded on a surface. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the 1-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic $ O ( n^ ω + 2^{ 2 g} n^ 2 + m )$-time algorithm for graphs cellularly embedded on an orientable surface of genus $ g$ . Prior to this work, the best known algorithms for surface-embedded graphs were those for general graphs: an $ O ( m^ ω )$-time Monte Carlo algorithm and a deterministic $ O ( nm^ 2 / log + n^ 2 m )$-time algorithm . For the minimum homology basis problem, we give a deterministic $ O (( g + b )^ 3 n log + m )$-time algorithm for graphs cellularly embedded on an orientable or non-orientable surface of genus $ g$ with $ b$ boundary components, improving on existing algorithms for many values of $ g$ and $ n$ . The algorithm assumes that shortest paths are unique; this assumption can be avoided by either using random perturbations of the edge weights guaranteeing a high probability of success or by deterministic means at a cost of an $ O (\log )$ factor increase in running time.

Cycle-Based Singleton Local Consistencies

We propose to exploit cycles in the constraint network of a Constraint Satisfaction Problem (CSP) to vehicle constraint propagation and improve the effectiveness of local consistency algorithms. We focus our attention on the consistency property Partition-One Arc-Consistency (POAC), which is a stronger variant of Singleton Arc-Consistency (SAC). We modify the algorithm for enforcing POAC to operate on a minimum cycle basis (MCB) of the incidence graph of the CSP. We empirically show that our approach improves the performance of problem solving and constitutes a novel and effective localization of consistency algorithms. Although this paper focuses on POAC, we believe that exploiting cycles, such as MCBs, is applicable to other consistency algorithms and that our study opens a new direction in the design of consistency algorithms. This research is documented in a technical report (Woordward, Choueiry, and Bessiere 2016). http://consystlab.unl.edu/our_work/StudentReports/TR-UNL-CSE-2016-0004.pdf

Cycle bases of reduced powers of graphs

We define what appears to be a new construction. Given a graph G and a positive integer k , the reduced k th power of G , denoted G ( k ) , is the configuration space in which k indistinguishable tokens are placed on the vertices of G , so that any vertex can hold up to k tokens. Two configurations are adjacent if one can be transformed to the other by moving a single token along an edge to an adjacent vertex. We present propositions related to the structural properties of reduced graph powers and, most significantly, provide a construction of minimum cycle bases of G ( k ) . The minimum cycle basis construction is an interesting combinatorial problem that is also useful in applications involving configuration spaces. For example, if G is the state-transition graph of a Markov chain model of a stochastic automaton, the reduced power G ( k ) is the state-transition graph for k identical (but not necessarily independent) automata. We show how the minimum cycle basis construction of G ( k ) may be used to confirm that state-dependent coupling of automata does not violate the principle of microscopic reversibility, as required in physical and chemical applications.

Directed Graphs and Induced Magnetic Multipoles in Polycyclic Hydrocarbons

Abstract Time-odd interactions in molecules may give rise to a current along the bonds connecting the atoms. In a graph representation of a molecule the analogue of these currents corresponds to directed edges. We consider the distribution of currents in polycyclic molecules, using a minimal cycle basis. The irreducible representations of this basis in the automorphism group of the molecular graph delineate specific magnetic multipoles. This is illustrated for the case of corazulene with fourfold symmetry which may sustain a magnetic octopole. It is further shown how breaking of the fourfold symmetry can give rise to the induction of an octopolar magnetic moment by a homogeneous magnetic field. Two conjugated polycyclic hydrocarbon systems showing an induced octopolar moment are investigated by ab initio current density calculations.

Finding Shorter Cycles in a Weighted Graph

In this paper we investigate the structures of short cycles in a weighted graph. By Thomassen's $$3$$3-path-condition theory (Thomassen in J Comb Theory Ser 48:155---177, 1990), it is easy to find a shortest cycle in a collection of cycles beyond a subspace of the cycle space of a graph. What is more difficult for one to do is to find a shortest cycle within a subspace of the cycle space in polynomial time. By using the Dijkstra's algorithm (Dijkstra in Numer Math 1:55---61, 1959) we find a collection $$\mathcal {C}$$C of cycles containing many types of short cycles within a given subspace of the cycle space of a graph and this implies a polynomial time algorithm (called extended fundamental cycle algorithm) to locate all the possible shortest cycles in a weighted graph. In the case of unweighted graphs, the algorithm may also find every shortest even cycle in a graph, this greatly improved a result of Grotschel and Pulleyblank (Oper Res Lett 1:23---27, 1981/82), Monien (Computing 31:355---369, 1983), Yuster and Zwick (SIAM J Discrete Math 10:209---222, 1997). In fact, our algorithm shows that there are at most $$O(n^4)$$O(n4) many such short cycles in an unweighted graph of order $$n$$n. Further more, our fundamental cycle method may find a minimum cycle base (or simply MCB as some scholars named) in the cycle space of a graph. Since the structure of MCB's is unique (Ren and Deng in Discrete Math 307:2654---2660, 2007), this shows that, in the sense, cycles in a MCB are nearly-fundamental (i.e., each element in a MCB is a sum of at most two fundamental cycles). This provides a new way to study MCB.

Min st -Cut Oracle for Planar Graphs with Near-Linear Preprocessing Time

For an undirected n -vertex planar graph G with nonnegative edge weights, we consider the following type of query: given two vertices s and t in G , what is the weight of a min st -cut in G ? We show how to answer such queries in constant time with O ( n log 4 n ) preprocessing time and O ( n log n ) space. We use a Gomory-Hu tree to represent all the pairwise min cuts implicitly. Previously, no subquadratic time algorithm was known for this problem. Since all-pairs min cut and the minimum-cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum-cycle basis in O ( n log 4 n ) time and O ( n log n ) space. Additionally, an explicit representation can be obtained in O ( C ) time and space where C is the size of the basis. These results require that shortest paths are unique. This can be guaranteed either by using randomization without overhead or deterministically with an additional log 2 n factor in the preprocessing times.

On minimum average stretch spanning trees in polygonal 2-trees

A spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges. For a polygonal 2-tree on n vertices, we present an algorithm to compute a minimum average stretch spanning tree in O(nlog⁡n) time. This algorithm also finds a minimum fundamental cycle basis in polygonal 2-trees. We show that there is a unique minimum cycle basis in a polygonal 2-tree and it can be computed in linear time.