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Chapter 10: Problem 17

Schedule the final exams for Math 115, Math 116, Math 185, Math 195, CS 101, CS 102, CS 273, and CS 473, using the fewest number of different time slots, if there are no students taking both Math 115 and CS 473, both Math 116 and CS 473, both Math 195 and CS 101, both Math 195 and CS 102, both Math 115 and Math 116, both Math 115 and Math 185, and both Math 185 and Math 195, but there are students in every other pair of courses.

Short Answer

Expert verified
Use 4 times slots: 1: Math 115, Math 185, CS 101, CS 273 2: Math 116, Math 195, CS 102, CS 473.

Step by step solution

01

Understand the Problem

We are given eight courses and constraints on which pairs of courses have common students. The goal is to schedule the courses in the minimum number of time slots such that no students have overlapping exams.
02

Represent the Problem as a Graph

Create a graph where each course is a node and an edge exists between two nodes if there are students enrolled in both courses. Use the given constraints to draw edges.
03

Draw the Graph

Based on the given data, draw the graph: - Edges: (Math 115, Math 195), (Math 115, CS 101), (Math 115, CS 102), (Math 115, CS 273), (Math 115, CS 473), (Math 116, Math 185), (Math 116, Math 195), (Math 116, CS 101), (Math 116, CS 102), (Math 116, CS 273), (Math 116, CS 473), (Math 185, CS 101), (Math 185, CS 102), (Math 185, CS 273), (Math 185, CS 473), (Math 195, CS 273), (Math 195, CS 473), (CS 101, CS 102), (CS 101, CS 273), (CS 101, CS 473), (CS 102, CS 273), (CS 102, CS 473), (CS 273, CS 473).
04

Find a Minimum Vertex Coloring

The problem requires finding the minimum number of colors (time slots) needed to color the graph such that no two adjacent nodes share the same color. Use a graph coloring algorithm or heuristic.
05

Identify Clicks and Apply Colors

Identify maximal cliques in the graph to guide coloring. One possible configuration: - Time Slot 1: Math 115 - Time Slot 2: Math 116 - Time Slot 3: Math 185 - Time Slot 4: Math 195 - Time Slot 5: CS 101 - Time Slot 6: CS 102 - Time Slot 7: CS 273 - Time Slot 8: CS 473
06

Check for Overlap

Verify that each pair of courses sharing a student do not share a time slot, ensuring that no student has overlapping exams.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Coloring
Graph coloring is a method of assigning colors to the elements of a graph subject to certain constraints. In the context of exam scheduling, each course represents a vertex, and an edge between two vertices signifies that there are students taking both courses, creating a conflict. The aim is to color the graph so that no two adjacent vertices share the same color. This translates to scheduling exams such that no student has two exams at the same time.
Vertex Coloring
Vertex coloring specifically refers to the assignment of colors to the vertices of a graph. More practically, this involves determining the minimum number of time slots needed for the exams, ensuring that no two connected vertices (i.e., courses with common students) have the same color. In our exercise, each course is colored such that conflicting courses (those sharing a student) don鈥檛 share the same color, which translates into separate time slots.
Graph Theory
Graph theory is a branch of mathematics focusing on the properties of graphs. A graph in this context is made up of vertices (nodes) and edges (connections). Exam scheduling can be modeled using graph theory where each course is a vertex and edges represent overlapping student enrollments. By applying the principles of graph theory, specifically graph coloring, we can derive the optimal way to avoid conflicts in exam scheduling.
Minimal Coloring
Minimal coloring is the challenge of finding the smallest number of colors needed to color a graph. This is important in scheduling because fewer colors (time slots) mean a more efficient timetable. In the given exercise, the goal is to minimize time slots while ensuring no conflicting courses are scheduled simultaneously. Applying graph coloring algorithms or heuristics helps achieve the minimal coloring, ensuring the timetable is resource-efficient and manageable for students.

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Most popular questions from this chapter

Show that the problem of finding a circuit of minimum total weight that visits every vertex of a weighted graph at least once can be reduced to the problem of finding a circuit of minimum total weight that visits each vertex of a weighted graph exactly once. Do so by constructing a new weighted graph with the same vertices and edges as the original graph but whose weight of the edge connecting the vertices u and v is equal to the minimum total weight of a path from u to v in the original graph.

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Show that every pair of processors in a mesh network of \(n=m^{2}\) processors can communicate using \(O(\sqrt{n})=\) \(O(m)\) hops between directly connected processors.

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A knight is a chess piece that can move either two spaces horizontally and one space vertically or one space horizontally and two spaces vertically. That is, a knight on square \((x, y)\) can move to any of the eight squares \((x \pm 2, y \pm 1),(x \pm 1, y \pm 2),\) if these squares are on the chessboard, as illustrated here. A knight's tour is a sequence of legal moves by a knight starting at some square and visiting each square exactly once. A knight's tour is called reentrant if there is a legal move that takes the knight from the last square of the tour back to where the tour began. We can model knight's tours using the graph that has a vertex for each square on the board, with an edge connecting two vertices if a knight can legally move between the squares represented by these vertices. Show that there is no knight's tour on a \(4 \times 4\) chessboard.
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