Backtracking and Optimization Problems MCQs

Correct Answer: B. NP-Complete

Explanation:
The Vertex Cover problem is a classic NP-Complete problem, which means: 1) It is in NP (solutions can be verified in polynomial time) 2) Every problem in NP can be reduced to it in polynomial time This makes it one of the fundamental problems in computational complexity theory.

Correct Answer: A. Cover all edges with minimum number of vertices

Explanation:
In the Vertex Cover problem, we need to find the smallest set of vertices such that every edge in the graph has at least one endpoint in this set. The goal is to "cover" all edges while using the minimum number of vertices. This is distinct from edge cover, which covers all vertices with edges.

Correct Answer: B. Greedy Matching

Explanation:
The Greedy Matching approach for Vertex Cover works by repeatedly selecting an edge and including both its endpoints in the solution, then removing all edges incident to these vertices. This simple algorithm guarantees a 2-approximation, meaning the solution is at most twice the size of the optimal vertex cover.

Correct Answer: B. NP-Hard

Explanation:
The Set Covering Problem is NP-Hard, which means it's at least as hard as the hardest problems in NP. In fact, it's one of the original 21 NP-complete problems identified by Richard Karp in 1972. This classification explains why we typically use approximation algorithms for solving it.

Correct Answer: A. Covering all elements with the minimum number of sets

Explanation:
In the Set Covering Problem, we have a universe of elements U and a collection of subsets S of U. The goal is to find the smallest sub-collection of sets from S whose union equals the universe U. In other words, we want to select the minimum number of sets that cover all elements.

Correct Answer: B. Set-Cover

Explanation:
For the Set Cover problem, the standard greedy algorithm (which repeatedly selects the set that covers the most uncovered elements) achieves a logarithmic approximation ratio of ln(n) where n is the number of elements. This is notably better than the linear approximation of many other NP-hard problems, but it has been proven that this logarithmic approximation is essentially the best possible unless P=NP.

Correct Answer: B. 2

Explanation:
The First-Fit algorithm for Bin Packing has an approximation factor of 2, meaning it will never use more than twice the optimal number of bins. This makes it a decent practical algorithm despite its simplicity. The First-Fit Decreasing variant (which sorts items by decreasing size first) performs even better with an approximation factor of 11/9 ≈ 1.22.

Correct Answer: B. NP-Hard

Explanation:
The Bin Packing Problem is NP-Hard, which means finding an exact solution is computationally intractable for large inputs. The decision version ("Can all items be packed in k bins?") is NP-Complete, but the optimization version ("What is the minimum number of bins needed?") is NP-Hard. This explains why approximation algorithms and heuristics are commonly used for practical applications.

Correct Answer: B. First-Fit

Explanation:
The First-Fit algorithm scans through bins in order and places each item in the first bin that can accommodate it. If no such bin exists, it opens a new bin. This differs from Best-Fit (which chooses the bin with minimal remaining capacity) and Next-Fit (which only checks the most recently used bin).

Correct Answer: C. Exact solution requires exponential time

Explanation:
Finding the exact solution to the Bin Packing Problem requires exponential time (unless P=NP), as it's an NP-Hard problem. This is why approximation algorithms are so important in practice. Many polynomial-time approximation algorithms exist with different approximation ratios, and the problem is known to be APX-hard, meaning it cannot be approximated arbitrarily well in polynomial time unless P=NP.