Operations Research: Assignment Problem

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Questions and Answers

What is the first step in finding the maximum flow in a network?

Find an open path from source to sink. (correct)

Subtract the flow amount from all paths.

Calculate the edge weights.

Draw a minimum cut through the edges.

In the context of the Hungarian Algorithm, when should you return to Step 3?

When the number of lines is less than the number of rows. (correct)

When at least one entry has been circled.

After completing Step 6.

When all entries have been paired.

What is a minimum cut in a flow network?

The total flow in all paths chosen.

The maximum flow available from source to sink.

A line that separates the source from the sink minimizing flow. (correct)

The greatest edge weight within the network.

Which of the following describes the role of the source in a flow network?

It is the start point where flow comes from. Signup and view all the answers

During Step 5 of the Hungarian algorithm, what action is taken with the smallest uncovered entry?

It is added to all uncovered entries and subtracted from covered entries. Signup and view all the answers

Which statement is true regarding the outcome of the Hungarian Algorithm?

It can yield multiple optimal solutions. Signup and view all the answers

What does maximum flow refer to in the context of a flow network?

The greatest flow achievable given the edge weight constraints. Signup and view all the answers

What happens when all possible open paths are chosen in the maximum flow algorithm?

The algorithm stops when no further flow can be added. Signup and view all the answers

What is the correct definition of a circuit in the context of Eulerian graphs?

A closed trail that visits all edges once and may repeat vertices Signup and view all the answers

What condition must be met for a graph to be classified as Eulerian?

All vertices must have an even degree Signup and view all the answers

Which of the following statements is true regarding minimum cut and maximum flow?

Maximum flow equals minimum cut Signup and view all the answers

In a semi-Eulerian graph, how many vertices can have an odd degree?

Two Signup and view all the answers

What is the capacity of the cut that includes edges with flows measuring 5, 0, 1, and 6?

12 Signup and view all the answers

What kind of path characterizes Hamiltonian graphs?

A closed path that visits all vertices once except for start and end Signup and view all the answers

The maximum flow through a network can be verified through what process?

Measuring the capacity of the minimum cut Signup and view all the answers

In a closed trail within Eulerian graphs, what condition must hold for the trail's vertices?

All vertices must have even degrees Signup and view all the answers

Study Notes

Hungarian Algorithm

The algorithm aims to find minimum cost assignments in a cost matrix.
Steps include covering lines, finding the smallest uncovered entry, and manipulating the matrix.
The process involves circling zero entries to match rows and columns effectively.
Multiple solutions can exist for a given cost matrix.

Example of Minimum Cost Assignment

Initial cost matrix:
- [8, 8, 6]
- [2, 3, 7]
- [4, 9, 3]
Through several steps, the algorithm transforms the matrix to find the optimal zero entries leading to minimum costs.

Maximum Cost Assignment

For non-square matrices, the algorithm adjusts entries for calculations.
Example provided:
- Original cost matrix:
  - [3, 5, 2]
  - [6, 4, 1]
- A maximum assignment is derived through similar steps as in the minimum cost assignment.

Flow Network Concepts

Source: Starting point for flow.
Sink: Endpoint where flow concludes.
Flow: Balance of flow into and out of a node; respects edge weights.
Maximum Flow: Greatest feasible flow under given constraints.
Cut: Represents a division in the network that halts flow.
Minimum Cut: The smallest cut quantifying the minimum sum of edge weights.

Maximum Flow Steps

Identify accessible paths from source to sink.
Calculate the maximum flow along chosen paths and adjust edge values accordingly.
Repeat until no further flow can be achieved.

Drawing a Cut

Edges that direct flow towards the source are included in the cut.
Cut capacity is determined by summing the weights of connecting edges.

Max Flow-Min Cut Theorem

The capacity of the maximum flow equals the capacity of the minimum cut, confirming flow limits in a network.

Walks, Paths, and Trails

Walk: Sequence of vertices where repeats are allowed.
Path: Sequence without repeated vertices; open and closed types exist based on endpoints.
Trail: Similar to paths but allows edge repeats; can also be open or closed.

Eulerian Graphs

Defined by having a closed trail that visits every edge once.
All vertices must have even degrees for a graph to be Eulerian.
Semi-Eulerian graphs allow one pair of vertices with an odd degree.

Hamiltonian Graphs

Feature closed paths visiting all vertices once, excluding start and end revisits.
Semi-Hamiltonian graphs contain open paths that cover all vertices exactly once.

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Description

This quiz focuses on the steps involved in solving the Assignment Problem using the Hungarian method. It details the process of finding covering lines, uncovering entries, and selecting matched entries in a cost matrix. Test your understanding of these essential operations research techniques.