Question

A) Explain (step by step) how the Bellman-Ford algorithm can be used to solve the following difference constraints:

x
1
​
−x
2
​
≤7
x
2
​
−x
3
​
≤−3
x
1
​
−x
3
​
≤4
​
b) The longest path in a graph can be computed by negating the costs of all edges in the graph and then running the Floyd-Warshall algorithm. True or False? Explain: If it is True then give a proof. If it is False, then give a counterexample and explain what the algorithm returns. (c) Suppose that there are 3 students, {A,B,C}, and 4 graduation projects, {p,q,r,y}. Each student specifies a set of projects they would like to work on, and each supervisor of the project specifies a set of students whom they would like to work with:
A:{p,q,r}
B:{r,y}
C:{q,r}
​

p:{A,B,C}
q:{C}
r:{C,D}
y:{A,B}
​
Explain (step by step) how the Ford-Fulkerson algorithm can be used to assign projects to the maximum number of students, such that the following two conditions hold: there is no student matched to two different projects, and there is no project assigned to two different students.

Answer 1

The Bellman-Ford algorithm can be used to solve difference constraints. The statement that the longest path in a graph can be computed by negating edge costs and using the Floyd-Warshall algorithm is False. The Ford-Fulkerson algorithm can be used to assign projects to the maximum number of students while satisfying given conditions.

Step-by-step explanation:

Part a: Bellman-Ford Algorithm for Difference Constraints

The Bellman-Ford algorithm can be used to solve the given difference constraints step by step. Here's how:

Create a graph with vertices for each variable and an additional vertex for the source (S) and the sink (T).

Add directed edges from S to each variable vertex with the corresponding constraint as the weight.

Add directed edges from each variable vertex to the sink T with weight 0.

Run the Bellman-Ford algorithm starting from the source S.

If there is a negative cycle, the constraints are not feasible. Otherwise, the algorithm will provide the shortest path distances from S to each variable vertex, which represent the maximum values for each variable.

Part b: Longest Path and Floyd-Warshall Algorithm

The statement is False. The Floyd-Warshall algorithm is used to find the shortest paths between all pairs of vertices in a graph with positive or negative edge weights. Negating the costs of all edges and running the algorithm would not yield the longest path in the graph.

Part c: Ford-Fulkerson Algorithm for Project Assignment

To use the Ford-Fulkerson algorithm for assigning projects to the maximum number of students while satisfying the conditions, follow these steps:

Create a bipartite graph with a vertex for each student and each project.

Add a source S and connect it to each student vertex with an edge of capacity 1.

Add a sink T and connect each project vertex to it with an edge of capacity 1.

Add edges between the student and project vertices based on their preferences.

Run the Ford-Fulkerson algorithm starting from the source S and find the maximum flow.

The maximum flow will represent the maximum number of students assigned to projects without violating the conditions.