Mixed Graph Research Articles

Mixed paths and cycles determined by their spectrum

A mixed graph is obtained from a graph by orienting some of its edges. The Hermitian adjacency matrix of a mixed graph with the vertex set {v1,…,vn}, is the matrix H=[hij]n×n, where hij=−hji=i if there is a directed edge from vi to vj, hij=1 if there exists an undirected edge between vi and vj, and hij=0 otherwise. The Hermitian spectrum of a mixed graph is defined to be the spectrum of its Hermitian adjacency matrix.In this paper we study mixed graphs which are determined by their Hermitian spectrum (DHS). First, we show that each mixed cycle is switching equivalent to either a mixed cycle with no directed edges (Cn), a mixed cycle with exactly one directed edge (Cn1), or a mixed cycle with exactly two consecutive directed edges with the same direction (Cn2) and we determine the spectrum of these three types of cycles. Next, we characterize all DHS mixed paths and mixed cycles. We show that all mixed paths of even order, except P8 and P14, are DHS. It is also shown that mixed paths of odd order, except P3, are not DHS. Also, all cospectral mates of P8, P14 and P4k+1 and two families of cospectral mates of P4k+3, where k≥1, are introduced. Finally, we show that the mixed cycles C2k and C2k2, where k≥3, are not DHS, but the mixed cycles C4, C42, C2k+1, C2k+12, C2k+11 and C2j1 except C71, C91, C121 and C151, are DHS, where k≥1 and j≥2.

On the characteristic polynomials and H-ranks of the weighted mixed graphs

Let DG˜ be an n-vertex weighted mixed graph with Hermitian-adjacency matrix H(DG˜). The characteristic polynomial of the weighted mixed graph DG˜ is defined asϕ(DG˜,λ)=det⁡(λIn−H(DG˜))=∑r=0nαrλn−r and the H-rank of DG˜, written as rk(DG˜), is the rank of H(DG˜).In this paper, we begin by interpreting all the coefficients of the characteristic polynomial ϕ(DG˜,λ). Then we establish recurrences for the characteristic polynomial ϕ(DG˜,λ). By these obtained results, as a unified approach, some main results obtained in Hou and Lei (2011) [13], Gong and Xu (2012) [10], Liu and Li (2015) [18] can be deduced consequently. Furthermore, for a weighted mixed bipartite cactus graph DG˜, we give a necessary and sufficient condition for rk(DG˜)=2t, where t is any integer no greater than the matching number of G. Finally, as an application of our results, we study the minimum H-rank problem, determining the minimum H-rank of bipartite cactus graphs.

Extended Linear Multicommodity Multicost Network And Maximal Concurent Flow Problems

Graph is a powerful mathematical tool applied in many fields as transportation, communication, informatics, economy, … In ordinary graph the weights of edges and vertexes are considered indepently where the length of a path is the sum of weights of the edges and the vertexes on this path. However, in many practical problems, weights at a vertex are not the same for all paths passing this vertex, but depend on coming and leaving edges. In the article, a kind of weights, called switch cost, is defined. The papers study multicomodity flow problems in ordinary networks. The papers study multicomodity flow problems in extended networks, where switch costs are defined for mixed graphs. The papers develops a model of extended linear multicommodity multicost network and studies the maximal linear multicomodity multicost flow problems. The papers study the maximal multicomodity multicost flow limited cost problems. Model of extended linear multicommodity multicost network can be applied to modelize many practical problems more exactly and effectively. The presented paper studies the maximal concurent linear multicomodity multicost flow problems, that are modelized as implicit linear programming problems. On the base of dual theory in linear programming an effective aproximate algorithms is developed.

Tutte polynomials for directed graphs

The Tutte polynomial is a fundamental invariant of graphs. In this article, we define and study a generalization of the Tutte polynomial for directed graphs, that we name the B-polynomial. The B-polynomial has three variables, but when specialized to the case of graphs (that is, digraphs where arcs come in pairs with opposite directions), one of the variables becomes redundant and the B-polynomial is equivalent to the Tutte polynomial. We explore various properties, expansions, specializations, and generalizations of the B-polynomial, and try to answer the following questions:•what properties of the digraph can be detected from its B-polynomial (acyclicity, length of directed paths, number of strongly connected components, etc.)?•which of the marvelous properties of the Tutte polynomial carry over to the directed graph setting? The B-polynomial generalizes the strict chromatic polynomial of mixed graphs introduced by Beck, Bogart and Pham. We also consider a quasisymmetric function version of the B-polynomial which simultaneously generalizes the Tutte symmetric function of Stanley and the quasisymmetric chromatic function of Shareshian and Wachs.

The maximum likelihood threshold of a path diagram

Linear structural equation models postulate noisy linear relationships between variables of interest. Each model corresponds to a path diagram, which is a mixed graph with directed edges that encode the domains of the linear functions and bidirected edges that indicate possible correlations among noise terms. Using this graphical representation, we determine the maximum likelihood threshold, that is, the minimum sample size at which the likelihood function of a Gaussian structural equation model is almost surely bounded. Our result allows the model to have feedback loops and is based on decomposing the path diagram with respect to the connected components of its bidirected part. We also prove that if the sample size is below the threshold, then the likelihood function is almost surely unbounded. Our work clarifies, in particular, that standard likelihood inference is applicable to sparse high-dimensional models even if they feature feedback loops.

Multi‐layer distributed protocols for robust cooperative tracking in interconnected nonlinear multiagent systems

SummaryWe develop a mixed graph and optimal control theoretic formulation to design a robust cooperative control protocol for a large‐scale multiagent system with partially known interconnected first‐, second‐, or mixed first‐ and second‐order dynamics. In each case, we transform the control protocol design task to a robust communication graph design problem, which, from a cyber‐physical perspective, is interpreted as the control layer design problem for an interconnected system with unknown agent layer dynamics. According to this viewpoint, each state variable has its own control layer communication topology separate from the other state variable's communication topology and the unknown agent layer interconnection topologies. We prove that all cooperative, decentralized, and centralized tracking protocols can be treated as a single design problem and, by deriving closed‐form solutions for the robust control layer topologies, we further provide a simpler design procedure, which is only based on the matrix manipulations. Aside from the linear implementation of the protocol and the connection of the proposed formulation to the well known rules‐of‐thumb in optimal control theory, this creates a higher potential to transfer ideas to industry. Modeling uncertainties tolerable by a given control layer topology is analyzed, and a preliminary performance‐oriented analysis and design approach for large‐scale interconnected systems is discussed. We show that exactly the same steps can be followed to design appropriate control layers for both tracking and stabilization.

A Tight Lower Bound for Planar Steiner Orientation

In the Steiner Orientation problem, the input is a mixed graph G (it has both directed and undirected edges) and a set of k terminal pairs mathscr {T}. The question is whether we can orient the undirected edges in a way such that there is a directed sleadsto t path for each terminal pair (s,t)in mathscr {T}. Arkin and Hassin [DAM’02] showed that the Steiner Orientation problem is NP-complete. They also gave a polynomial time algorithm for the special case when k=2. From the viewpoint of exact algorithms, Cygan et al. [ESA’12, SIDMA’13] designed an XP algorithm running in n^{O(k)} time for all kge 1. Pilipczuk and Wahlström [SODA’16, TOCT’18] showed that the Steiner Orientation problem is W[1]-hard parameterized by k. As a byproduct of their reduction, they were able to show that under the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi and Zane [JCSS’01] the Steiner Orientation problem does not admit an f(k)cdot n^{o(k/log k)} algorithm for any computable function f. In this paper, we give a short and easy proof that the n^{O(k)} algorithm of Cygan et al. is asymptotically optimal, even if the input graph is planar. Formally, we show that the Planar Steiner Orientation problem is W[1]-hard parameterized by the number k of terminal pairs, and, under ETH, cannot be solved in f(k)cdot n^{o(k)} time for any computable function f. Moreover, under a stronger hypothesis called Gap-ETH of Dinur [ECCC’16] and Manurangsi and Raghavendra [ICALP’17], we are able to show that there is no constant vartheta >0 such that Planar Steiner Orientation admits an (frac{19}{20}+vartheta )-approximation in FPT time, i.e., no f(k)cdot n^{o(k)} time algorithm can distinguish between the case when all k pairs are satisfiable versus the case when less than k cdot (frac{19}{20}+vartheta ) pairs are satisfiable. To the best of our knowledge, this is the first FPT inapproximability result on planar graphs.

Computation of maximum likelihood estimates in cyclic structural equation models

Software for computation of maximum likelihood estimates in linear structural equation models typically employs general techniques from nonlinear optimization, such as quasi-Newton methods. In practice, careful tuning of initial values is often required to avoid convergence issues. As an alternative approach, we propose a block-coordinate descent method that cycles through the considered variables, updating only the parameters related to a given variable in each step. We show that the resulting block update problems can be solved in closed form even when the structural equation model comprises feedback cycles. Furthermore, we give a characterization of the models for which the block-coordinate descent algorithm is well defined, meaning that for generic data and starting values all block optimization problems admit a unique solution. For the characterization, we represent each model by its mixed graph (also known as path diagram), which leads to criteria that can be checked in time that is polynomial in the number of considered variables.

Mixed Energy of a Mixed Hourglass Graph

In this paper we discuss a complete mixed graph called mixed hourglass graph. The direct representation of hourglass matrix in graph gives a weighted mixed hourglass graph. Then, we obtain a mixed hourglass graph from the weighted mixed hourglass graph by assigning its edge-labelled a numerical value of weight 1. Next, we derive the determinant, spectrum and mixed energy of the graph to conclude that the energy of a mixed hourglass graph coincides with twice the number of edges in the graph and the sum of the square of its eigenvalues.

Parameterized Mixed Graph Coloring

Coloring of mixed graphs that contain both directed arcs and undirected edges is relevant for scheduling of unit-length jobs with precedence constraints and conflicts. The classic GHRV theorem (attributed to Gallai, Hasse, Roy, and Vitaver) relates graph coloring to longest paths. It can be extended to mixed graphs. In the present paper we further extend the GHRV theorem to weighted mixed graphs. As a byproduct this yields a kernel and a parameterized algorithm (with the number of undirected edges as parameter) that is slightly faster than the brute-force algorithm. The parameter is natural since the directed version is polynomial whereas the undirected version is NP-complete. Furthermore we point out a new polynomial case where the edges form a clique.

Minors of Hermitian (quasi-) Laplacian matrix of a mixed graph

ABSTRACT A mixed graph is obtained from an unoriented graph by orienting a subset of its edges. Yu, Liu and Qu in 2017 have established the expression for the determinant of the Hermitian (quasi-) Laplacian matrix of a mixed graph. Here we find general expressions for all minors of Hermitian (quasi-) Laplacian matrix of a mixed graph.

Relation between the [formula omitted]-rank of a mixed graph and the rank of its underlying graph

Given a simple graph G=(VG,EG) with vertex set VG and edge set EG, the mixed graph G˜ is obtained from G by orienting some of its edges. Let H(G˜) denote the Hermitian adjacency matrix of G˜ and A(G) be the adjacency matrix of G. The H-rank (resp. rank) of G˜ (resp. G), written as rk(G˜) (resp. r(G)), is the rank of H(G˜) (resp. A(G)). Denote by d(G) the dimension of cycle space of G, that is d(G)=|EG|−|VG|+ω(G), where ω(G) denotes the number of connected components of G. In this paper, we concentrate on the relation between the H-rank of G˜ and the rank of G. We first show that −2d(G)⩽rk(G˜)−r(G)⩽2d(G) for every mixed graph G˜. Then we characterize all the mixed graphs that attain the above lower (resp. upper) bound. By these obtained results in the current paper, all the main results obtained in Luo et al. (2018); Wong et al. (2016) may be deduced consequently.

The periodic rural postman problem with irregular services on mixed graphs

In this paper, we deal with an extension of the rural postman problem in which some links of a mixed graph must be traversed a given number of times over a time horizon. These links represent entities that must be serviced a specified number of times in some subsets of days (or periods) of the time horizon. The aim is to design a set of minimum-cost tours, one for each day/period of the time horizon, that satisfy the service requirements. We refer to this problem as the periodic rural postman problem with irregular services (PRPP–IS). Some practical applications of the problem can be found in road maintenance operations and road network surveillance, for example. In order to solve the PRPP–IS, we propose a mathematical model and a branch-and-cut algorithm. As far as we know, this is the first exact method devised for a periodic arc routing problem. In the solution framework, constraints ensuring connectivity and other valid inequalities are identified by using specific separation procedures. Most valid inequalities consider the particular nature of the PRPP–IS. We show the effectiveness of the solution approach through an extensive experimental phase.

A new general family of mixed graphs

A new general family of mixed graphs is presented, which generalizes both the pancake graphs and the cycle prefix digraphs. The obtained graphs are vertex transitive and, for some values of the parameters, they constitute the best infinite families with asymptotically optimal (or quasi-optimal) number of vertices for their diameter.

A method for parallel scheduling of multi-rate co-simulation on multi-core platforms

The design of cyber-physical systems is a complex process and relies on the simulation of the system behavior before its deployment. Such is the case, for instance, of joint simulation of the different subsystems that constitute a hybrid automotive powertrain. Co-simulation allows system designers to simulate a whole system composed of a number of interconnected subsystem simulators. Traditionally, these subsystems are modeled by experts of different fields using different tools, and then integrated into one tool to perform simulation at the system-level. This results in complex and compute-intensive co-simulations and requires the parallelization of these co-simulations in order to accelerate their execution. The simulators composing a co-simulation are characterized by the rates of data exchange between the simulators, defined by the engineers who set the communication steps. The RCOSIM approach allows the parallelization on multi-core processors of co-simulations using the FMI standard. This approach is based on the exploitation of the co-simulation parallelism where dependent functions perform different computation tasks. In this paper, we extend RCOSIM to handle additional co-simulation requirements. First, we extend the co-simulation to multi-rate, i.e. where simulators are assigned different communication steps. We present graph transformation rules and an algorithm that allow the execution of each simulator at its respective rate while ensuring correct and efficient data exchange between simulators. Second, we present an approach based on acyclic orientation of mixed graphs for handling mutual exclusion constraints between functions that belong to the same simulator due to the non-thread-safe implementation of FMI. We propose an exact algorithm and a heuristic for performing the acyclic orientation. The final stage of the proposed approach consists in scheduling the co-simulation on a multi-core architecture. We propose an algorithm and a heuristic for computing a schedule which minimizes the total execution time of the co-simulation. We evaluate the performance of our proposed approach in terms of the obtained execution speed. By applying our approach on an industrial use case, we obtained a maximum speedup of 2.91 on four cores.

Relation between the Hermitian energy of a mixed graph and the matching number of its underlying graph

ABSTRACT Given a graph G, the mixed graph is obtained from G by orienting some of its edges (G is also called the underlying graph of ). Let be the Hermitian energy of and let be the matching number of the underlying graph G. In this paper, we first establish the inequality and characterize all the mixed graphs which make hold. Furthermore, we obtain , where and are, respectively, the vertex cover number, the number of odd cycles and the maximum degree of graph G. The lower bound is attained if and only if is switching equivalent to its underlying graph G, where G is the disjoint union of some complete bipartite graphs each of which contains a perfect matching, together with some isolated vertices. The upper bound is best possible. By our results in this paper, some main results in Tian FL, Wong D. [Relation between the skew energy of an oriented graph and its matching number. Discrete Appl Math. 2017;222:179–184]; Wang L, Ma XB. [Bounds of graph energy in terms of vertex cover number. Linear Algebra Appl. 2017;517:207–216]; Wong D, Wang XL, Chu R. [Lower bounds of graph energy in terms of matching number. Linear Algebra Appl. 2018;549:276–286] can be deduced in a unified approach.

Solving the Problems of Routing Interconnects with a Resynthesis for Reconfigurable Systems on a Chip

The existing means of design automation are focused mainly on the technology of foreign manufacturers, which makes it necessary to adapt the existing methods and means for designing reconfigurable systems-on-a-chip and to develop domestic specialized CAD tools to solve urgent problems in this field. Methods are proposed for solving interconnection routing problems in conjunction with logical resynthesis, applied to the architecture of a reconfigurable system-on-a-chip based on the domestic field-programmable gate array (FPGA) of the Almaz-14 family. Developers from JSC “NIIME” and PJSC “Micron” laid broad configurational solutions, which do not have foreign analogues, into this crystal. A wide range of additional elements for the configuration, as well as the potentialities for the logical resynthesis of the FPGA Almaz-14 chip, have led to the need to develop new methods for routing interconnections that would allow us to take into account and use these architectural features. An efficient algorithm for the automatic routing of interconnections for a reconfigurable system on a chip based on FPGAs belonging to the Almaz-14 family based on algorithm A* is developed. This algorithm represents a modification of the classical algorithm searching for the shortest path on a graph, the Dijkstra algorithm, including a mixed switching graph model. To describe the variety of additional switching elements, a special generalized mathematical model, as well as a special interface in the command language Tcl are developed, the latter includes a list of elements for configuration, as well as their description and functional purpose. This work has increased the efficiency of computer-aided design using programmed mechanisms developed and implemented in the C programming language for the optimal use of the configuration and routing elements of FPGAs, as well as mechanisms for the complete and entire routing of interconnections.

On reachability mixed arborescence packing

As a generalization of Edmonds’ arborescence packing theorem, Kamiyama–Katoh–Takizawa (2009) provided a good characterization of directed graphs that contain arc-disjoint arborescences spanning the set of vertices reachable from each root. Fortier–Király–Léonard–Szigeti–Talon (2018) asked whether the result can be extended to mixed graphs by allowing both directed arcs and undirected edges. In this paper, we solve this question by developing a polynomial-time algorithm for finding a collection of edge and arc-disjoint arborescences spanning the set of vertices reachable from each root in a given mixed graph.

A Genetic Algorithm for Hybrid Job-Shop Scheduling Problems with Minimizing the Makespan or Mean Flow Time

We address a generalization of the classical job-shop problem which is called a hybrid job-shop problem. The criteria under consideration are the minimization of the makespan and mean flow time. In the hybrid job-shop, machines of type [Formula: see text] are available for processing the specific subset [Formula: see text] of the given operations. Each set [Formula: see text] may be partitioned into subsets for their processing on the machines of type [Formula: see text]. Solving the hybrid job-shop problem implies the solution of two subproblems: an assignment of all operations from the set [Formula: see text] to the machines of type [Formula: see text] and finding optimal sequences of the operations for their processing on each machine. In this paper, a genetic algorithm is developed to solve these two subproblems simultaneously. For solving the subproblems, a special chromosome is used in the genetic algorithm based on a mixed graph model. We compare our genetic algorithms with a branch-and-bound algorithm and three other recent heuristic algorithms from the literature. Computational results for benchmark instances with 10 jobs and up to 50 machines show that the proposed genetic algorithm is rather efficient for both criteria. Compared with the other heuristics, the new algorithm gives most often an optimal solution and the average percentage deviation from the optimal function value is about 4%.

Unifying Markov properties for graphical models

Several types of graphs with different conditional independence interpretations—also known as Markov properties—have been proposed and used in graphical models. In this paper, we unify these Markov properties by introducing a class of graphs with four types of edges—lines, arrows, arcs and dotted lines—and a single separation criterion. We show that independence structures defined by this class specialize to each of the previously defined cases, when suitable subclasses of graphs are considered. In addition, we define a pairwise Markov property for the subclass of chain mixed graphs, which includes chain graphs with the LWF interpretation, as well as summary graphs (and consequently ancestral graphs). We prove the equivalence of this pairwise Markov property to the global Markov property for compositional graphoid independence models.