IJAEDU- International E-Journal of Advances in Education

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, social networks, transportation networks, cellular networks,
linguistics, chemistry, physics, biology can be represented and studied by graphs.

Real-world entities - like molecules, persons, groups, roles,
species, computing and communication devices, terms - correspond to vertices.
Relations among such entities - like preference, domination, independence,
interference, proximity, constraints - 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.