Some optimization problems for timed marked graphs

Discrete event dynamic systems are frequently encountered in complex man-made systems. Such a system is often described in terms of a timed Petri net. For periodic behaviour of such a system, a timed marked graph provides a convenient and powerful tool of modelling and analysis. In this paper we define three optimization problems for such a system. All of these problems are formulated as mixed integer linear programming (MIP) problems. To improve computational efficiency, we introduce ‘cuts’ to these MIP problems, and the effect of such cuts is examined through computational experiments on some test problems.     

Approximate mean value analysis for stochastic marked graphs

An iterative technique for the computation of approximate performance indices of a class of stochastic Petri net models is presented. The proposed technique is derived from the mean value analysis algorithm for product-form solution stochastic Petri nets. In this paper, we apply the approximation technique to stochastic marked graphs. In principle, the proposed technique can be used for other stochastic Petri net subclasses. In this paper, some of these possible applications are presented. Several examples are presented in order to validate the approximate results     

Linear control of live marked graphs

Given a linear constraint on the firing vectors of a live marked graph with uncontrollable/unobservable transitions, bounded or unbounded, we apply linear programming techniques to compute the most liberal controller enforcing this constraint.     

Approximate throughput computation of stochastic marked graphs

A general iterative technique for approximate throughput computation of stochastic strongly connected marked graphs is presented. It generalizes a previous technique based on net decomposition through a single input-single output cut, allowing the split of the model through any cut. The approach has two basic foundations. First, a deep understanding of the qualitative behavior of marked graphs leads to a general decomposition technique. Second, after the decomposition phase, an iterative response time approximation method is applied for the computation of the throughput. Experimental results on several examples generally have an error of less than 3%. The state space is usually reduced by more than one order of magnitude; therefore, the analysis of otherwise intractable systems is possible     

Supervisory control of partially observable marked graphs

This note presents a control synthesis approach for discrete event systems modeled by marked graphs with unobservable transitions. It solves forbidden state problems characterized by a set of general mutual exclusion constraints. We prove that for any sequence of observable transitions, there exist a unique marking from which all other possible current markings can be reached unobservably. This salient feature allows us to design efficient control policies based on proper separation of observation and control.     

Overlapping community detection in labeled graphs

We present a new approach for the problem of finding overlapping communities in graphs and social networks. Our approach consists of a novel problem definition and three accompanying algorithms. We are particularly interested in graphs that have labels on their vertices, although our methods are also applicable to graphs with no labels. Our goal is to find k communities so that the total edge density over all k communities is maximized. In the case of labeled graphs, we require that each community is succinctly described by a set of labels. This requirement provides a better understanding for the discovered communities. The proposed problem formulation leads to the discovery of vertex-overlapping and dense communities that cover as many graph edges as possible. We capture these properties with a simple objective function, which we solve by adapting efficient approximation algorithms for the generalized maximum-coverage problem and the densest-subgraph problem. Our proposed algorithm is a generic greedy scheme. We experiment with three variants of the scheme, obtained by varying the greedy step of finding a dense subgraph. We validate our algorithms by comparing with other state-of-the-art community-detection methods on a variety of performance measures. Our experiments confirm that our algorithms achieve results of high quality in terms of the reported measures, and are practical in terms of performance.     

Performance evaluation of marked graphs by linear programming

Performance evaluation is an important issue for discrete event dynamic systems, which in many cases can be described in terms of the language of Petri nets. In particular, for marked graphs, a subclass of Petri nets, a formula for performance evaluation is widely known. That is, the ratio of the total execution time to the number of tokens in a cycle gives a lower bound for the cycle time of the system, and the maximum of these ratios over all cycles determines the overall system performance. However, we need to enumerate all directed cycles in a graph to apply this formula of performance evaluation. Carrying out this enumeration for an actual system is often impractical, since the number of cycles in a graph usually grows exponentially with the size of a system. We give a linear algebraic characterization for directed cycles, and based on this result, transform the problem of performance evaluation into a simple linear programming problem. A few explanatory examples are also given.     

A note of Pearlman's state-space equivalence theorem

Using Carleman linearization and known results on state-affine systems, a simplified proof of Pearlman's [3] state-space equivalence theorem is given.     

A structure theorem for maximum internal matchings in graphs

A counterpart of the Gallai-Edmonds Structure Theorem is proved for matchings that are not required to cover the external vertices of graphs. The appropriate counterpart of Tutte's Theorem is derived from this result for the existence of perfect internal matchings in graphs.     

Generating labeled planar graphs uniformly at random

We present a deterministic polynomial time algorithm to sample a labeled planar graph uniformly at random. Our approach uses recursive formulae for the exact number of labeled planar graphs with n$n$ vertices and m$m$ edges, based on a decomposition into 1-, 2-, and 3-connected components. We can then use known sampling algorithms and counting formulae for 3-connected planar graphs.     

Efficient coding of labeled directed acyclic graphs

 We use an adaptation of the Prüfer code for trees to encode labeled directed acyclic graphs, which are often abbreviated to DAGs (or ADGs). In this paper, each DAG is assigned a unique DAG code, which allows an easy handling for several purposes. The set of all possible DAG codes (and therefore the set of all DAGs) for a fixed number of n vertices can be generated efficiently. Furthermore, we are able to rank DAGs, i.e., we provide an algorithm that assigns every DAG a unique number in the set {0,…,a _n −1}, where a _n is the cardinality of the set of labeled DAGs with n≥1 vertices, and we are able to unrank DAGs, which is the inverse operation. We also gain recurrence relations, which can be used to calculate a _n and a _n,q , i.e., the number of DAGs with n vertices and q edges. Finally, it is possible to generate, enumerate, rank and unrank DAGs with given number of edges and also DAGs with bounded indegree. RID="*" ID="*" This research was supported by the Austrian Science Fund (FWF), P13261-INF. I want to thank the reviewers, specially the one who suggested to add the algorithm for unranking DAG codes, for reading the paper very carefully and for the helpful comments.     

The labeled perfect matching in bipartite graphs

In this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs. Given a simple graph G=(V,E) with |V|=2n vertices such that E contains a perfect matching (of size n), together with a color (or label) function , the labeled perfect matching problem consists in finding a perfect matching on G that uses a minimum or a maximum number of colors.     

A new and simple proof of the equivalence theorem for behaviors

Two full row rank representations of the same behavior are related through a left unimodular transformation. We present a new and extremely simple and insightful proof for this well-established fact.     

Graphillion: software library for very large sets of labeled graphs

Several graph libraries have been developed in the past few decades, and they were basically designed to work with a few graphs. However, there are many problems in which we have to consider all subgraphs satisfying certain constraints on a given graph. Since the number of subgraphs can increase exponentially with the graph size, explicitly representing these sets is infeasible. Hence, libraries concerned with efficiently representing a single graph instance are not suitable for such problems. In this paper, we develop Graphillion, a software library for very large sets of (vertex-)labeled graphs, based on zero-suppressed binary decision diagrams. Graphillion is not based on a traditional representation of graphs. Instead, a graph set is simply regarded as a “set of edge sets” ignoring vertices, which allows us to employ powerful tools of a “family of sets” (a set of sets) and permits large graph sets to be handled efficiently. We also utilize advanced graph enumeration algorithms, which enable the simple family tools to understand the graph structure. Graphillion is implemented as a Python library to encourage easy development of its applications, without introducing significant performance overheads. In experiments, we consider two case studies, a puzzle solver and a power network optimizer, in which several operations and heavy optimization have to be performed over very large sets of constrained graphs (i.e., cycles or forests with complicated conditions). The results show that Graphillion allows us to manage a huge number of graphs with very low development effort.     

A multilevel approach for learning from labeled and unlabeled data on graphs

The recent years have witnessed a surge of interest in graph-based semi-supervised learning methods. The common denominator of these methods is that the data are represented by the nodes of a graph, the edges of which encode the pairwise similarities of the data. Despite the theoretical and empirical success, these methods have one major bottleneck which is the high computational complexity (since they usually need to solve a large-scale linear system of equations). In this paper, we propose a multilevel scheme for speeding up the traditional graph based semi-supervised learning methods. Unlike other accelerating approaches based on pure mathematical derivations (like conjugate gradient descent and Lanczos iteration) or intuitions, our method (1) has explicit physical meanings with some graph intuitions; (2) has guaranteed performance since it is closely related to the algebraic multigrid methods. Finally experimental results are presented to show the effectiveness of our method.     

A theorem of equivalence on the methods of least-squares estimation

A theorem on the methods of least-squares estimation is stated and proved. This theorem enables the replacement of a system of correlated measurements by an equivalent system of uncorrelated measurements without a whitening process, thus simplifying the analysis while resulting in the same minimum-variance estimate.