Design Ch.13

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Questions and Answers

What happens in the algorithm when it goes from point c to point e?

It encounters an obstacle that requires rerouting.

It creates a shortcut back to point b.

It proves to be a useless move, necessitating backtracking. (correct)

It successfully reaches its destination without issues.

During the backtracking process in the algorithm, which points are revisited?

From e to c and then to b. (correct)

From a to e and back to d.

From b to a and then forward to c.

From c to e and then to d.

What key concept is illustrated by the movement from point e back to point c?

The effectiveness of forward progress.

The ability to optimize routes dynamically.

The necessity of backtracking in certain situations. (correct)

Efficiency in algorithm design.

What can be inferred about point e in relation to point c?

Point e is a dead-end requiring backtracking. Signup and view all the answers

What does the backtracking from e to c indicate about the algorithm's decision-making?

It reveals the algorithm's need to reevaluate its previous choices. Signup and view all the answers

What is an example of a Hamiltonian circuit starting from vertex a?

a, b, f, e, c, d, a Signup and view all the answers

Which vertices are included in the Hamiltonian circuit mentioned?

a, b, f, e, c, d Signup and view all the answers

Which vertex is the starting point of the Hamiltonian circuit?

a Signup and view all the answers

What is the last vertex visited before returning to the starting point in the circuit described?

c Signup and view all the answers

How many vertices does the Hamiltonian circuit in the question visit?

6 Signup and view all the answers

Which of the following vertices is NOT part of the Hamiltonian circuit?

g Signup and view all the answers

The vertices that the circuit visits in order are:

a, b, f, e, c, d, a Signup and view all the answers

What is the main purpose of the Branch-and-Bound technique?

To optimize solutions by ruling out nonpromising nodes. Signup and view all the answers

In the context of the Branch-and-Bound method, what does a 'bound' refer to?

An estimate of the best possible objective function value for descendants. Signup and view all the answers

Which of the following statements correctly describes the use of a cost matrix in optimization problems?

No two selected elements can be in the same column. Signup and view all the answers

What is an enhancement of backtracking in optimization problems?

Branch-and-Bound. Signup and view all the answers

Which of the following represents a correct lower bound for the given assignment problem?

2 + 3 + 1 + 4. Signup and view all the answers

What characterizes the state-space tree in the Branch-and-Bound method?

Nodes are pruned based on the calculated bounds. Signup and view all the answers

In a Branch-and-Bound approach, what happens when a node's bound is not better than the best solution found so far?

The node is discarded from the search. Signup and view all the answers

What is a key characteristic of the best-first Branch-and-Bound algorithm?

It prioritizes exploring nodes with the most promising bounds. Signup and view all the answers

Which of the following is NOT an objective of Branch-and-Bound?

To evaluate every possible node. Signup and view all the answers

What is the purpose of using the Branch-and-Bound method in the context of the Travel Salesman Problem (TSP)?

To obtain a solution that is close to optimal. Signup and view all the answers

In the context of approximation algorithms for minimization problems, how is the accuracy ratio defined?

r(sa) = f(s*) / f(sa) Signup and view all the answers

What happens in the Nearest-Neighbor algorithm after visiting all cities?

It returns to the starting city. Signup and view all the answers

What does an unbounded accuracy ratio indicate in the context of approximation algorithms?

The solution quality cannot be guaranteed. Signup and view all the answers

In the Multifragment-Heuristic algorithm, what is the first step to be performed?

Sorting the edges by weight. Signup and view all the answers

Which statement is true regarding the Nearest-Neighbor algorithm?

The tour length varies based on the starting city. Signup and view all the answers

What is the performance ratio of an algorithm A?

The worst-case upper bound of the accuracy ratio. Signup and view all the answers

Why might someone choose an approximation algorithm for solving the TSP?

To obtain a satisfactory solution more quickly. Signup and view all the answers

In terms of performance, what does the notation RA indicate in relation to the Nearest-Neighbor algorithm?

RA signifies that the accuracy can be made unbounded. Signup and view all the answers

Study Notes

Course Information

Course: Design and Analysis of Algorithms
Module: 13 Coping with the Limitations of Algorithm Power
University: Saudi Electronic University

Learning Outcomes

Recognize Backtracking algorithms
Express Branch-and-Bound concept
Demonstrate Approximation Algorithms for NP-Hard Problems

Topic Outline

Backtracking
Branch-and-Bound
Approximation Algorithms for NP-Hard Problems

Tackling Difficult Combinatorial Problems

Two main approaches to tackle NP-hard problems:
- Use a strategy to solve the problem exactly but not necessarily in polynomial time.
- Use an approximation algorithm to find a sub-optimal solution in polynomial time.

Exact Solution Strategies

Exhaustive search (brute force): useful only for small instances
Dynamic programming: applicable to some problems (e.g., knapsack problem)
Backtracking: eliminates some unnecessary cases, yields solutions in reasonable time for many instances but worst case is still exponential
Branch-and-bound: further refines the backtracking idea for optimization problems

Backtracking

Construct the state-space tree:
- Nodes represent partial solutions
- Edges represent choices in extending partial solutions
Explore the state space tree using depth-first search
"Prune" non-promising nodes: Stop exploring subtrees that cannot lead to a solution, backtrack to the parent node.

Example: n-Queens Problem

Place n queens on an n-by-n chessboard such that no two queens share the same row, column, or diagonal.
A visual representation of the four-queens problem is included to display the arrangement.

Example: Hamiltonian Circuit Problem

Starts at a vertex (e.g., a).
Selects next vertex based on alphabetization to break ties.
Traces possible paths until a dead end is reached, which leads to backtracking.
Finds a Hamiltonian circuit (if one exists) by tracing paths and backtracking until a solution is found.
Displays a state-space tree for illustrating the process for finding a Hamiltonian circuit for a given graph.

Example: Hamiltonian Circuit Problem -Useless Case

Demonstrates a scenario where a path proves useless, leading to backtracking.
Explains how the algorithm might backtrack and explore alternative paths to find a Hamiltonian circuit (if one exists) by tracing paths and backtracking until a solution is found.

Branch-and-Bound

An enhancement of backtracking, applicable to optimization problems.
For each node (partial solution) in a state-space tree, computes bounds on the value of the objective function for all descendants of the node (each possible extension of the partial solution).
Uses the bounds to:
- "Prune" nodes that cannot lead to a better solution
- Guide the search by prioritizing more promising extensions.
Illustrates the concept with an assignment problem example.

Example: Assignment Problem

Examples of the level of state space trees for an assignment problem
Illustrates the use of lower bounds to prune the search space

Approximation Approach

Apply a fast (polynomial-time) approximation algorithm to find a near-optimal solution
Accuracy measures:
- Accuracy ratio (for minimization problems): approximate solution value / optimal solution value
- Accuracy ratio (for maximization problems): optimal solution value / approximate solution value
Performance ratio: the lowest upper bound of the accuracy ratio for all instances of the problem

Nearest-Neighbor Algorithm for TSP

Starts at a city, travels to the nearest unvisited city, repeating until all cities are visited and returns to the starting city.
Accuracy is potentially unbounded, as the optimal tour may not be found.

Multifragment Heuristic Algorithm

Sorts edges by weight in non-decreasing order.
Adds edges to the tour, avoiding cycles of length less than the number of vertices.
Repeats this process until a complete tour is formed
This algorithm shows improvements in accuracy compared to some others

Twice-Around-the-Tree Algorithm

Constructs a Minimum Spanning Tree (MST).
Creates a path that traverses the MST twice, returning to the starting vertex.
Makes shortcuts to create a tour visiting each vertex only once, to reduce redundancy.
This algorithm usually generates better tours than nearest neighbor but accuracy is unbounded.

Greedy Algorithm for Knapsack Problem

Sorts items by value-to-weight ratio in descending order.
Selects items in this order, skipping those that exceed the knapsack capacity.
The algorithm provides an exact solution in the continuous version.

Bin Packing Problem: First-Fit Algorithm

Sorts items in decreasing order of their value-to-weight ratio.
Generates all subsets of items that fit the knapsack,
Finds the most valuable subset among these subsets and return it.
Time efficiency is O(kn^k + 1)
Accuracy is f(s*) / f(s₂) ≤ 1 + 1/ k

Activities

Activities encourage group work to apply the introduced concepts to particular problems (like Travel Salesman Problem, Knapsack Problem).

Summary

Backtracking and branch-and-bound are techniques for solving problems with exponentially growing choices.
Approximation algorithms are used for difficult optimization problems, providing near-optimal but not necessarily perfect solutions.
Examples of approximation algorithms for the traveling Salesman Problem and knapsack problem are illustrated.

List of Readings

Required reading: Chapter 12 from Anany Levitin (2012)
Recommended reading: Chapter 18 from Michael T. Goodrich, Roberto Tamassia (2014)

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Test your knowledge on coping with the limitations of algorithm power in this quiz focused on Backtracking algorithms, Branch-and-Bound concepts, and Approximation Algorithms for NP-Hard problems. Understand various strategies to tackle combinatorial problems and their efficiencies. Perfect for students in the Design and Analysis of Algorithms course.