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In mathematics, particularly graph theory, and computer science, a directed acyclic graph is a finite directed graph with no directed cycles. That is, it consists of finitely many vertices and edges, with each edge directed from one vertex to another, such that there is no way to start at any vertex v and follow a consistently-directed sequence of edges that eventually loops back to v again. Equivalently, a DAG is a directed graph that has a topological ordering, a sequence of the vertices such that every edge is directed from earlier to later in the sequence.
This is a glossary of graph theory terms. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges.
In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices and each of the related pairs of vertices is called an edge. Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.
In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate.
In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs that maps adjacent vertices to adjacent vertices.
In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge uv from vertex u to vertex v, u comes before v in the ordering. For instance, the vertices of the graph may represent tasks to be performed, and the edges may represent constraints that one task must be performed before another; in this application, a topological ordering is just a valid sequence for the tasks. A topological ordering is possible if and only if the graph has no directed cycles, that is, if it is a directed acyclic graph (DAG). Any DAG has at least one topological ordering, and algorithms are known for constructing a topological ordering of any DAG in linear time.
In mathematics, and more specifically in graph theory, a polytree is a directed acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic.
In graph theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, containment graphs, and divisor graphs. An incomparability graph is an undirected graph that connects pairs of elements that are not comparable to each other in a partial order.
The Shifting Bottleneck Heuristic is a procedure intended to minimize the time it takes to do work, or specifically, the makespan in a job shop. The makespan is defined as the amount of time, from start to finish, to complete a set of multi-machine jobs where machine order is pre-set for each job. Assuming that the jobs are actually competing for the same resources (machines) then there will always be one or more resources that act as a 'bottleneck' in the processing. This heuristic, or 'rule of thumb' procedure minimises the effect of the bottleneck. The Shifting Bottleneck Heuristic is intended for job shops with a finite number of jobs and a finite number of machines.
Job shop scheduling or the job-shop problem (JSP) is an optimization problem in computer science and operations research in which jobs are assigned to resources at particular times. The most basic version is as follows: We are given n jobs J1, J2, ..., Jn of varying processing times, which need to be scheduled on m machines with varying processing power, while trying to minimize the makespan. The makespan is the total length of the schedule.
In graph theory, a strong orientation of an undirected graph is an assignment of a direction to each edge that makes it into a strongly connected graph.
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or by the sum of the weights of its edges. In contrast to the shortest path problem, which can be solved in polynomial time in graphs without negative-weight cycles, the longest path problem is NP-hard and the decision version of the problem, which asks whether a path exists of at least some given length, is NP-complete. This means that the decision problem cannot be solved in polynomial time for arbitrary graphs unless P = NP. Stronger hardness results are also known showing that it is difficult to approximate. However, it has a linear time solution for directed acyclic graphs, which has important applications in finding the critical path in scheduling problems.
In mathematics, and more specifically in graph theory, a directed graph is a graph that is made up of a set of vertices connected by edges, where the edges have a direction associated with them.
In graph theory and combinatorial optimization, a closure of a directed graph is a set of vertices with no outgoing edges. That is, the graph should have no edges that start within the closure and end outside the closure. The closure problem is the task of finding the maximum-weight or minimum-weight closure in a vertex-weighted directed graph. It may be solved in polynomial time using a reduction to the maximum flow problem. It may be used to model various application problems of choosing an optimal subset of tasks to perform, with dependencies between pairs of tasks, one example being in open pit mining.
In order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two simple composition operations.
In job shop scheduling and graph drawing, the Coffman–Graham algorithm is an algorithm, named after Edward G. Coffman, Jr. and Ronald Graham, for arranging the elements of a partially ordered set into a sequence of levels. The algorithm chooses an arrangement such that an element that comes after another in the order is assigned to a lower level, and such that each level has a number of elements that does not exceed a fixed width bound W. When W = 2, it uses the minimum possible number of distinct levels, and in general it uses at most 2 − 2/W times as many levels as necessary.
In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph.
In graph theory, an acyclic orientation of an undirected graph is an assignment of a direction to each edge that does not form any directed cycle and therefore makes it into a directed acyclic graph. Every graph has an acyclic orientation.
In graph theory, the Gallai–Hasse–Roy–Vitaver theorem is a form of duality between the colorings of the vertices of a given undirected graph and the orientations of its edges. It states that the minimum number of colors needed to properly color any graph G equals one plus the length of a longest path in an orientation of G chosen to minimize this path's length. The orientations for which the longest path has minimum length always include at least one acyclic orientation.
A mixed graphG = is a mathematical object consisting of a set of vertices V, a set of (undirected) edges E, and a set of directed edges A.
References
- ↑ Gröflin, H.; Klinkert, A. (2002), "Scheduling with generalized disjunctive graphs: Feasibility issues", XV Conference of the European Chapter on Combinatorial Optimization (ECCO XV), May 30 - June 1, 2002, Lugano, Switzerland.
- ↑ Olson, Lars E. (August 2003), Querying Disjunctive Databases in Polynomial Time (PDF), Master's thesis, Brigham Young University, Department of Computer Science.
- ↑ Roy, S.; Sussman, B. (December 1964), Les problemes d'ordonnancement avec contraintes disjonctives, Note D.S. No. 9 bis, SEMA.