Graph Coloring for class scheduling

Abstract

Graphs are discrete structures composed of vertices and edges connecting these vertices. Graphs are used in almost all disciplines as abstract models for the representation and study of a wide range of relations and processes in physical, biological, social and information systems. Many practical problems in a variety of areas like computer and communication networks,-imply the existence of edges between corresponding vertices. Thus, focusing on the abstract graph model instead of studying each particular instance as a different real-world problem reveals common underlying properties, deficiencies and principles. In this way, efficient approaches to real-world problems emerge from the theoretical study of their abstractions. In this work, we use graph coloring to propose efficient solutions to scheduling problems arising in higher education. The objective of the graph coloring problem is to assign colors to graph vertices so that adjacent vertices, i.e., vertices connected by an edge, receive different colors. We consider as the objective of scheduling problems in higher education, like lecture and exam scheduling, to assign time/day slots to teaching or examination activities so that the maximum number of students can attend them with the fewest possible conflicts. Our main motivation has been the crucial issue of efficient course and exam schedules often arising in departments of the University of Patras, Greece. Students usually have to attend lectures or exams scheduled in overlapping or simultaneous time slots. However, course and exam schedules are created based on heuristic approaches which may work well on average but certainly leave several room for improvement. What if a graph-theoretic approach were used? Courses correspond to vertices of a graph and there is an edge between two vertices if and only if an appropriately selected minimum population of students attends corresponding courses (lectures/exams). Then, a coloring of such an underlying graph suggests an appropriate schedule for teaching/examination activities. Using a simple coloring algorithm and the MATLAB programming environment, we have designed and developed a scheduling application which receives as input courses and constraints and outputs an efficient lecture/examination schedule. Experimental evaluation suggests that our application works well in practice. Ongoing work focuses on the use of a more involved coloring algorithm for addressing more complex course scheduling instances while minimizing required time resources.

A Classroom Scheduling Service for Smart Classes

“…For example, [12] and [13] both conduct a graph coloring scheme in scheduling. Of the two approaches, [12] propose a heuristic approach to promote uniform distribution of courses over the colors and also to balance course load for each time slot over the available class rooms. In contrast, [13] present an alternative graph coloring method for university timetabling that incorporates room assignment during the coloring process.…”

A literature review for course scheduling problem

“…Bir eniyileme tekniği olan matematiksel programlamanın kullanıldığı çalışmalarda, çoğunlukla matematiksel programlamanın uzantıları olan tamsayılı programlama ve doğrusal programlama (DP) tekniklerinden yararlanılmıştır. [44]- [51]. Matematiksel programlama tekniğine benzer olarak, bu yöntem de büyük boyutlu problemlerin çözümünde yetersiz kalmaktadır.…”

Global Optimization Using Local Search Approach for Course Scheduling Problem

“…The most traditional technique for solving the course scheduling problem is the graph coloring technique [14,15]. Graph coloring algorithm comes from a classical problem in graph theory which implies the problem to color the nodes of an undirected graph such that no two adjacent nodes share the same color.…”

“…The course scheduling problem is modeled by letting the nodes and edges represent the courses and the common students, respectively. Dandashi and Al-Mouhamed [15] have given a good historical review on the graph coloring technique since 1967 up to recently.…”