COP-4534 Algorithm Techniques: Combinatorial Optimization Quiz

11 Questions

What does the term 'Combinatorial Optimization' refer to?

Finding an object of the solution space maximizing a given objective function

What is the main challenge associated with Combinatorial Optimization problems?

They often involve maximizing or minimizing objective functions

How is the Traveling Salesman Problem (TSP) defined?

Finding the shortest route that visits each city exactly once and returns to the starting city

In the context of TSP, what does 'n cities and pairwise distances' refer to?

The cities involved and the distances between each pair of cities

How is the Traveling Salesman Problem (TSP) represented in terms of a graph?

As a complete weighted graph with n vertices

What is the purpose of the 'printPermutations' method in the given code snippet?

To generate all permutations of an array a

What is the primary purpose of the 'constructCandidateSet' method in the code snippet?

To determine the possible candidates for the next position in a permutation

In the context of combinatorial optimization, what role does an exhaustive search play?

It finds an optimal solution by considering all possible combinations.

What is the key characteristic of a Hamiltonian cycle in a graph?

It visits each vertex exactly once and returns to the starting vertex.

What distinguishes an approximation algorithm from an exhaustive search in combinatorial optimization?

It provides a suboptimal solution with less computational effort.

How does the 'printPermutations' method contribute to solving combinatorial optimization problems like the TSP?

By systematically exploring all possible permutations for finding an optimal solution.

Study Notes

Combinatorial Optimization

Combinatorial Optimization is the process of finding an object in the solution space that maximizes or minimizes a given objective function.
CO problems are often hard problems.

Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is a classic example of a Combinatorial Optimization problem.
Given n cities and their pairwise distances, the goal is to find a shortest route that visits each city exactly once and returns to the starting city.
TSP can be represented as a graph problem, where we find a shortest route that visits every vertex exactly once and returns to the starting vertex, equivalent to a Hamiltonian cycle with minimum cost.

Approaching Combinatorial Optimization

There are three common approaches to solving Combinatorial Optimization problems:
- Exhaustive search (brute-force)
- Heuristic method
- Approximation Algorithms

Exhaustive Search Solution to TSP

One way to solve TSP is through exhaustive search, where we:
- Generate all permutations of the vertices of the graph
- Compute the cost of each permutation (cycle)
- Determine the cycle with the smallest cost

Generating All Permutations

To generate all permutations, we can use a recursive function:
- printPermutations(int n): initializes the recursion
- printPermutations(int[] a, int k): recursive function to generate all permutations
- constructCandidateSet(int[] a, int k): constructs the candidate set for the next permutation