Complexity classes XP and FPT

Question 1

What is the general approach to solving computational problems discussed in this chapter?

Answer 1

Try to develop an efficient algorithm or prove the problem is NP-hard to understand its difficulty.

Question 2

Why is proving a problem NP-hard not always satisfactory?

Answer 2

People still want solutions that are as good as possible, even if the exact answer is not feasible.

Question 3

What are some strategies for addressing NP-complete problems?

Answer 3

Approximation algorithms, local search heuristics, and identifying special cases where problems are tractable.

Question 4

How can problem instances sometimes be easier than the worst case?

Answer 4

Instances may have special structures or parameters that allow more efficient algorithms.

Question 5

What is the Vertex Cover Problem?

Answer 5

Given a graph G=(V,E) and an integer k, find a subset S ⊆ V such that every edge has at least one endpoint in S, and |S| ≤ k.

Question 6

What parameters affect the complexity of the Vertex Cover Problem?

Answer 6

The number of vertices n and the allowable size of the vertex cover k.

Question 7

How does the brute-force approach solve the Vertex Cover Problem?

Answer 7

It checks all subsets of size k, taking time O(knk+1), which becomes impractical for even small k.

Question 8

What is the time complexity of the improved Vertex Cover algorithm?

Answer 8

O(2^k * kn), which is practical for small k.

Question 9

What insight helps simplify the Vertex Cover Problem?

Answer 9

If a graph has a small vertex cover, it cannot have many edges, limiting its complexity.

Question 10

What is the significance of edge count in the Vertex Cover Problem?

Answer 10

If G has more than k(n-1) edges, it cannot have a vertex cover of size k.

Question 11

How does the recursive algorithm solve the Vertex Cover Problem?

Answer 11

By removing one endpoint of an edge and checking recursively if a vertex cover exists in the smaller graph.

Question 12

What is the base case for the recursive Vertex Cover algorithm?

Answer 12

If the graph has no edges, the vertex cover is the empty set.

Question 13

What is the branching factor of the recursive Vertex Cover algorithm?

Answer 13

Each recursive call branches into two smaller problems, one for each endpoint of an edge.

Question 14

What is the recurrence relation for the Vertex Cover algorithm?

Answer 14

T(n, k) = 2T(n, k-1) + O(kn), where T(n, 1) = O(n).

Question 15

How does the recursive tree structure affect the algorithm’s runtime?

Answer 15

The tree has at most 2^(k+1) nodes, and each node takes O(kn) time.

Question 16

What happens to the runtime as k increases in the Vertex Cover algorithm?

Answer 16

The runtime grows exponentially with k, making the algorithm impractical for large k.

Question 17

How can the improved Vertex Cover algorithm be used in practice?

Answer 17

It is effective for small k, where exponential scaling is manageable.

Question 18

What is the primary limitation of exponential algorithms like Vertex Cover?

Answer 18

They scale poorly for larger parameter values, becoming infeasible quickly.

Question 19

Why is it important to identify special structures in graphs?

Answer 19

Special structures, like trees or small tree-width graphs, allow NP-complete problems to become tractable.

Question 20

How does the Vertex Cover algorithm improve on brute force?

Answer 20

It uses recursive branching to limit the search space, avoiding examination of all subsets.

Question 21

What is the practical significance of the Vertex Cover algorithm?

Answer 21

It enables solving small instances efficiently, providing insight into NP-complete problems.

Question 22

Can the Vertex Cover algorithm solve all instances efficiently?

Answer 22

No, it becomes infeasible for large k or n due to exponential scaling.

Question 23

Why do we check the number of edges before starting the Vertex Cover algorithm?

Answer 23

To eliminate cases where a vertex cover of size k is impossible due to too many edges.

Question 24

What is the role of maximum degree in analyzing the Vertex Cover Problem?

Answer 24

The maximum degree limits the number of edges a small vertex cover can handle.

Question 25

What does the algorithm do if neither G-{u} nor G-{v} has a vertex cover of size k-1?

Answer 25

It concludes that G does not have a vertex cover of size k.

Question 26

How does the Vertex Cover algorithm use recursive calls?

Answer 26

It explores both possibilities for including one endpoint of an edge in the vertex cover.

Question 27

What is the practical application of the Vertex Cover algorithm?

Answer 27

It can be used in network security, resource allocation, and other optimization problems where coverage is critical.

Question 28

How does moving exponential dependence to k improve practicality?

Answer 28

It separates exponential growth from n, making the algorithm feasible for small k and large n.

Question 29

Why does exponential growth still pose a challenge for the Vertex Cover algorithm?

Answer 29

Even small increases in k result in rapid increases in computation time, limiting scalability.