Interesting statistics
At its core, graph theory models pair relationships between objects using mathematical structures. Nodes, vertices, edges, and arcs represent these objects.
Social networks are a simple example of graph theory. In it, nodes are people, and edges are their interactions.
Graph theory, which studies the interaction of nodes and edges in networks, began in 1736 when Leonhard Euler solved the problem of the seven bridges of Königsberg.
An edge connects each adjacent pair of vertices along the path of the graph. In graph theory, the shortest path is usually the most important one.
Graphs can be directed or undirected:
The edges from node A to node B are identical in undirected graphs.
In directed graphs (also known as digraphs), edges have directions, so the edge from node A to node B varies from node B to node A.
Graph theory has many technical terms and properties, such as loops, multiple edges, adjacent vertices, vertex degrees, subgraphs, and more. These components and principles provide real-world applications in computer science, operational research, bioinformatics, and project management.
Another important concept is a weighted graph, where each edge has a value or "weight". Weights can indicate distances or costs when planning a route.
Graph theory provides a powerful framework for studying and describing complex systems such as project management and planning.
Basic concepts
Graph theory is based on vertices (nodes) and edges (or arcs). Vertices and edges form a graph, an abstract description of connections and interactions.
Vertex and Edge: Vertices represent entities or objects. An edge connects two vertices. Cities are the vertices of the transport network, and the roads connecting them are the edges.
Directed and undirected graph: Arrows point to a directed graph. Undirected graphs are bidirectional. Twitter is a directed graph (user A follows user B, not the other way around). Telegram contacts are undirected graphs (reciprocal connection).
Weighted Graph. Weighted graphs assign values to edges based on distance, cost, bandwidth, etc. For example, distance or travel time between cities in a transportation network can weigh them.
Path and Cycle: A path is a sequence of vertices connected by edges. Cycles start and end at the same vertex. The warehouse delivery route is a cycle.
Degree: Degrees of vertices is the number of their edges. Directed graphs have input and output degrees (the number of outgoing edges). In a social network graph, the degree of a vertex can indicate a person's connections.
Adjacency and Incidence: Edges connect adjacent vertices. Incident edges have a common vertex. On the city network graph, passing roads meet in cities and directly connect nearby cities.
Subgraph: A subgraph is a graph segment with associated vertices and edges. It may be a subset of the network.
Connectivity: A graph is connected if any vertex can reach any other. Social networks are connected if everyone can reach a sequence of "each other" links.
Tree and Forest: A tree is a connected graph without cycles. A forest is a collection of trees. For example, a company's organizational structure can be represented as a tree with the CEO at the root.
Knowing these basic concepts helps to use graph theory in management and project planning.
The Role of graph theory in project planning
Project planning is a critical example where tasks are sequenced, and deadlines are set to ensure efficient completion.
Modeling project tasks as a graph: Project planning models tasks as vertices and dependencies as edges. This creates a directed graph or project network where tasks are ordered.
Dependency identification: Task dependencies are shown as edges in the project graph. The boundary from task A to task B indicates that task B cannot start until task A is completed. The laying of the foundation must precede the erection of the walls in the construction project.
Path and critical path. A path is a sequence of project tasks from start to finish. The longest path on the graph is the critical path, indicating the shortest time to complete the project. Any delay in the critical route task will delay the project.
Task Sequence Definition: A directed graph model can identify a task sequence. Vertices with no incoming edges can start immediately, but others must wait for their preliminary tasks to be completed.
Resource Allocation: Vertices can be weighted to show task resources. This helps identify resource conflicts and optimize resource allocation.
Risk assessment. Graph and path connectivity can reveal risks and bottlenecks. For example, a task with multiple dependent tasks can be risky.
Progress Monitoring: Project progress can be tracked on a graph. Marking completed tasks shows the status of the project and issues.
Project schedule update. The timeline can be modified to plan and track the project as tasks are completed dynamically.
Graph theory facilitates systematic project planning, resource management, and decision-making.
Project management and graph theory
Graph theory helps project managers make decisions, optimize and reduce risks.
Project network. The life cycles of projects are presented in the form of graphs. Each task is a vertex; the edges show interdependence, visualizing the workflow. Software development project graphs can contain requirements, design, coding, testing, and deployment.
Decision making: A well-designed graph improves the decision-making process. Managers can prioritize work, identify bottlenecks, and assign resources. The vertex degree helps to identify jobs with dependencies that require special attention.
Time Management: The duration of a task can be assigned to each of the edges. Optimizing the project graph with weighted edges improves time management.
Risk Reduction: Tasks with strong dependencies or on a critical path can be identified by examining the graph. Proactive measures can mitigate these impacts.
Resource optimization: manpower, equipment, and budget can also be weighted on the graph for resource optimization.
Project Monitoring: Graphs are not static. Completed tasks can be flagged or excluded as the project progresses, providing real-time visual status. This helps track progress and quickly detect deviations from the plan.
Change handling: The graph can reflect new tasks or dependencies as the scope changes, allowing for dynamic project management.
Graph theory helps project managers visualize and manage complex activities and dependencies.
Critical path method with graph theory
The Critical Path Method (CPM) plans project tasks in project management. He finds the "critical path" using graph theory.
The critical path is the longest path from the start node to the end node in a directed and weighted project graph. The length corresponds to the time of the project.
Critical Path Calculation: CPM is calculated forward and backward. The start-to-finish walk determines the start and end times of the project.
Implications of scheduling: Project completion time depends on the critical path. Delays in key road tasks will delay the project. Delays in non-critical path jobs may not affect the project schedule as long as they do not exceed their slack (delay tolerance).
Resource Management: Critical path tasks are often carefully funded due to their direct impact on project timelines. Project managers can allocate resources efficiently by recognizing these tasks.
Risk Management: The critical path method identifies high-risk areas. Critical path risks can delay a project. Thus, these tasks require careful risk reduction.
Project Control: The progress of a project can be tracked by monitoring critical path tasks. A project is more likely to be completed on time if key tasks are completed on time.
Network diagrams in graph theory
Network diagrams are mathematically based on graph theory. They help analyze and optimize complex systems.
Formation of network diagrams. Each node is a vertex of a network diagram, and its relationship or interaction is an edge. The vertices may represent a telecommunications network's cell towers and edge data cables.
Weighted networks: If there is a quantitative measure of the relationship between objects, such as the distance to a city or the project cost, these values can be assigned as weights to edges in a network diagram.
Flow Networks: A flow network diagram shows systems where items, data, or traffic moves through a network. In such diagrams, each edge has a capacity.
Pattern Identification: Network diagrams can identify system patterns and trends, such as highly interconnected regions or nodes with a particularly large number of connections. This is important in many areas, from social media analysis to infrastructure development.
Optimization: Graph theory algorithms such as shortest path and minimum spanning tree help solve optimization problems in network diagrams. The shortest path algorithm can find the fastest delivery route in the logistics network.
Troubleshooting. Network diagrams help you spot problems such as production bottlenecks or weaknesses in a computer network.
Resource allocation using graph theory
Graph theory helps project managers allocate resources. Resource allocation can improve productivity, save money, and complete projects.
Resource representation: Tasks or processes can be represented by vertices in a weighted graph, and weights can indicate resources. Time, money, equipment, and personnel are resources.
Resource Conflict Identification: Graph analysis can identify resource conflicts or resource overuse. A resource conflict occurs when two jobs that require the same resource are scheduled at the same time.
Optimal distribution. Resource allocation can be optimized using graph algorithms such as the maximum flow algorithm. This method can find the most efficient distribution of raw materials from warehouses (source nodes) to factories in a production scenario (destination nodes).
Task prioritization: The degree of a top can indicate its resource requirements. High-degree vertices can be resource intensive. Thus, prioritize them at the time of distribution.
Schedule Optimization: When time is a resource, task scheduling is resource allocation. Optimizing the schedule using the critical path method ensures no task runs out of resources.
Allocation Monitoring: Resource tracking becomes critical as a project grows. Resource usage graphs can display allocation status and over/under usage.
Resource Reallocation: The graph can be changed, and resources reallocated when new tasks or resources become unavailable. It maintains the project graph and helps dynamic resource management.
Graph-theoretic solutions for complex projects
Graph theory helps to plan and execute complex projects with many interconnected activities. The methods and principles of graph theory can help manage such projects.
Project structure modeling: tasks are vertices, and dependencies are edges in a complex project graph. This project graph shows task relationships for better understanding and planning.
Project scheduling: The task sequence determines the minimum time to complete a project. Project managers can optimize scheduling to reduce delays.
Resource Allocation: A graph theory approach to maximum flow optimizes resource allocation. This ensures that no task is resource-constrained and increases the efficiency of the project.
How to deal with complexity: Graph theory simplifies complex tasks with many tasks and dependencies. Tasks can be organized into subgraphs or modules based on their relationships for better administration.
Problem-Solving: Graphs can solve several problems in a project. For example, logistical routes can be optimized using the shortest path problem and the distribution of tasks over resources - using the maximum matching problem.
Decision Making: Visualizing a project's interconnected structure helps make strategic decisions.
Change Management: Dynamic projects often change. Graph theory makes it easy to incorporate and visualize these changes. This helps to assess the impact of changes and manage them effectively.
Monitoring and control: Graph theory helps to make strategic decisions by visually showing the interconnected structure of the project.
The problem of seven bridges of Koenigsberg
This is a historical problem that Leonhard Euler solved in the 18th century, starting the field of graph theory.
In Prussia, Königsberg was a metropolis built around the Pregel River, with two islands connected to the mainland by seven bridges. The question was whether moving around the city on foot was possible, crossing each bridge only once.
Euler approached this question abstractly and reduced it to its main components:
He presented each land mass (two islands and two parts of the mainland) as nodes (tops).
Each of the seven bridges was represented by an edge connecting two nodes.
This formed the first graph, and the challenge was to find a path that went through each edge exactly once. Such a path is now known as the Euler path.
As a result, the graph looked something like this:
A
/ | \
B | C
|\ | /|
| D |
|_/ \_|A, B, C, and D represent land masses.
The lines connecting these letters represent bridges.
Euler noticed that after each entry into the node, it is also necessary to leave it, which requires an even number of edges. The knots in Koenigsberg had an odd number of edges, which did not allow one to go around all the bridges simultaneously without repetition.
This breakthrough marked the beginning of graph theory. DNA sequencing, routing, network architecture, and other fields use Euler routes and schemas.
Critical path example 1
Consider a simplified construction project:
Task A: Making plans (2 days)
Task B: Obtain permits (7 days)
Task C: Purchase of materials (3 days)
Task D: Laying the foundation (5 days)
Task E: Building walls (6 days)
Task F: Installing the roof (4 days)
Task G: Installing windows (2 days)
Task J: Painting (3 days)
Now we also have some dependencies:
Task A must be completed before B, C, and D.
Tasks B, C, and D must be completed before E.
Task E must be completed before F and G.
Tasks F and G must be completed before J.
Graphically, the project looks like this:
A(2) ---> B(7)
| |
| V
v E(6) ---> F(4)
C(3) ---> D(5) |
| | |
V V V
G(2) <---.J(3) <----|To find the shortest project duration, we need to find the longest path in this graph, also known as the critical path. Here's how:
Start with task A. This will take 2 days, so write "2" as the earliest start for tasks B, C, and D.
Look at each task associated with A. B takes 7 days, so its earliest finish is 2 (start time) + 7 = 9. C and D have the earliest finish 2 (start time) + 3 = 5 and 2 (time start). ) + 5 = 7, respectively.
Task E's earliest start is the latest end time of its dependencies B, C, and D, which is 9. Thus, task E's earliest finish is 9 (start time) + 6 = 15.
Tasks F and G depend on E, so their earliest start is 15, and their finish time is 15 + 4 = 19 and 15 + 2 = 17, respectively.
Task J depends on F and G, so its earliest start is between 19 and 17, which is 19. Its end is 19 (start time) + 3 = 22.
So, the shortest project duration is 22 days, and the critical path is A - B - E - F - J. This means that any delay in these tasks will delay the entire project.
Critical path example 2
Let's take an example from e-commerce, particularly the order fulfillment process. Tasks may include:
Task A: Get the order (1 day)
Task B: Inventory Check (2 days)
Task C: Reorder elements (if necessary) (7 days)
Task D: Prepare the order (1 day)
Task E: Quality Control Review (1 day)
Task F: Shipment of the order (1 day)
Task G: Delivery of the order (3 days)
Here are the dependencies:
Task A must be completed before B and D.
Task B must be completed before C.
Task C must be completed before D.
Tasks B and D must be completed before E.
Task E must be completed before F.
Task F must be completed before G.
Graphically, the project looks like this:
A(1) ---> B(2) ---> C(7) ---> D(1)
| |
| V
v E(1) ---> F(1) ---> G(3)
D(1)
|
V
E(1)
|
V
F(1)
|
V
G(3)
To calculate the critical path:
Start with task A, which takes 1 day. Thus, the earliest start for tasks B and D is day 1.
Task B has the earliest finish of 1 (start time) + 2 = 3 days.
Task C starts after B and ends at 3 (start time) + 7 = 10 days.
Task D starts after the last task, A and C, so it starts on day 10 and ends 10 + 1 = 11 days later.
Task E starts after B and D, starting on day 11 and ending on day 11 + 1 = 12 days.
Task F starts after E and ends at 12 (start time) + 1 = 13 days.
Finally, task G starts after F and ends at 13 (start time) + 3 = 16 days.
So, the shortest project duration is 16 days, and the critical path is A - B - C - D - E - F - G. Any delay in these tasks will delay the entire project. This analysis can help the e-commerce company manage the order fulfillment process and minimize delivery time.
Example of the traveling salesman problem
We can consider another problem from graph theory, the traveling salesman problem (TSP), which often occurs in logistics and delivery scenarios in e-commerce companies.
Consider a scenario in which a courier has to deliver parcels to four locations. The distances between these points are known, and the goal is to find the shortest possible route that visits each point once and returns to the starting point.
Suppose:
A -> B: 10 miles
A -> C: 15 miles
A -> D: 20 miles
B -> C: 35 miles
B -> D: 25 miles
C -> D: 30 milesLet's represent these addresses and distances in the form of a graph:
A--10---B
|\ |
| \ |
20 15 35
| \ |
| \ |
D--30---C
|
|
25
|
|
BOn this graph, each location is a node, and each edge represents the distance between two locations.
To solve TSP, we must find the shortest route that visits each node exactly once and returns to the starting node. In this case, an example of a valid route would be A-B-C-D-A with a total distance of 10 (A to B) + 35 (B to C) + 30 (C to D) + 20 (D to A) = 95 miles.
Try different combinations or use a more complex algorithm to determine the shortest path. If the number of nodes is small, test all combinations. For more locations, heuristic or approximation algorithms may be required.
Solving this problem helps in efficient route planning.
Graph coloring example
Another production management scenario is assembly line planning using graph theory. This is especially important in industries such as automotive, electronics, etc.
Imagine a simple assembly line scenario where we must assemble a product (like a smartphone). This process includes several tasks, such as:
Task A: Assembling the motherboard
Task B: Installing the Camera
Task C: Screen attachment
Task D: Installing Components
Task E: Final quality check
Dependencies:
Task A must be completed before B, C, and D.
Tasks B, C, and D must be completed before E.
If we assign tasks to two workers, the goal will be to distribute the tasks between them in such a way as to minimize the total time spent. This is where graph theory comes into play, specifically the "graph coloring" concept.
Graph coloring is a special case of graph labeling. It is the assignment of labels (called colors) to the vertices of a graph so that no two neighboring vertices have the same color. In our case, the colors are workers, and we will try to color the graph (assign tasks) so that no two dependents (linked in the graph) are assigned to the same worker.
Here's what the build process graph might look like:
B
/
A - C
\
D
\
EOne way to color this graph: Worker 1: A, C, E
Working 2: B, D
This distribution provides a conflict-free workflow by separating activities for each worker.
Graph theory optimizes assembly line planning, saving time, resources, and production.
Vertex Cover Example
Here's an example from networking: setting up data centers to efficiently manage internet traffic.
Internet Service Providers (ISPs) must establish regional data centers to provide users with the best possible service. This service should minimize delays and delays in data transmission.
Each important city in the region is a node. The goal is to determine the smallest number of data centers and their placement to make each city a data center.
A vertex cover is a set of vertices that has at least one vertex incident to each edge of the graph.
There should be a data center on each edge between the two cities.
Suppose we have 5 major cities (A, B, C, D, E) with the following connections:
A -- B -- C
| |
D -- E -- In graph theory, B, D, and E are minimal vertex covers. Therefore, B, D, and E can host data centers. This ensures that each city is either a data center or near one, which reduces user latency.
An ISP can strategically plan the location of their data centers to optimize service for all consumers and reduce infrastructure costs by implementing the idea of vertex coverage.
