Cs6515 Exam 3 Newest Version Complete 46+ Questions And Correct Detailed Answers (Verified Answers) Already Graded A+

CS6515 EXAM 3 NEWEST VERSION COMPLETE 46+ QUESTIONS AND CORRECT DETAILED ANSWERS (VERIFIED ANSWERS) ALREADY GRADED A+ CS 6515 EXAM 3 Enhanced Graduate Algorithms Study Guide — 2026 Focus: NP-completeness, polynomial reductions, Independent Set, 3SAT, approximation algorithms, and exam-style reasoning. Original study resource: This guide does not reproduce the paid Stuvia document, its claimed 46+ questions, answer key, or any protected course assessment. 1. Exam 3 Map: Complexity & Intractability Georgia Tech's CS 6515 is a graduate algorithms course covering algorithm design and analysis; the official course description includes NP complexity theory. The public preview associated with the linked Stuvia item emphasizes P/NP, NP-hardness, reductions, and graph/3SAT relationships. • P: decision problems solvable in polynomial time. • NP: decision problems whose YES certificates can be verified in polynomial time. • NP-hard: at least as hard as every problem in NP under polynomial-time reductions; it need not itself be in NP. • NP-complete: both in NP and NP-hard. • Containment: P ⊆ NP; whether P = NP remains unresolved. Q. Does NP mean 'not polynomial'? Answer / rationale: No. NP means polynomial-time verification of a certificate; a problem can be in both P and NP. Q. What proves NP-completeness? Answer / rationale: Show the problem is in NP and show it is NP-hard, usually by reducing a known NP-complete problem to it. 2. Polynomial-Time Reductions Reductions are a core exam skill: the arrow direction determines which problem inherits hardness. • A ≤p B: transform any instance of A into an instance of B in polynomial time while preserving YES/NO. • Direction matters: A → B means B is at least as hard as A. • Transitivity: A ≤p B and B ≤p C implies A ≤p C. • Hardness transfer: if A is NP-hard and A ≤p B, then B is NP-hard. • NP-complete target: after proving hardness, also prove B ∈ NP. Q. If A is NP-complete and A ≤p B, what follows? Answer / rationale: B is NP-hard; B is NP-complete only if B is also shown to be in NP. Q. Why is reversing the arrow dangerous? Answer / rationale: A reduction from B to a known hard A does not show B is hard. Hardness transfers when the known hard problem reduces to B. 3. Reduction Proof Template • 1. Start with arbitrary instance I of source A. • 2. Construct f(I), an instance of target B. • 3. Show f(I) is computable in polynomial time. • 4. Prove I is YES ⇒ f(I) is YES. • 5. Prove f(I) is YES ⇒ I is YES. • 6. Conclude A ≤p B and transfer hardness. Q. Why prove both directions? Answer / rationale: A valid reduction must preserve the decision answer; one direction alone can allow false positives or false negatives. 4. 3SAT Fundamentals • Literal: x or ¬x. • Clause: OR of literals. • 3CNF: AND of clauses, with three literals per clause in standard 3SAT. • Satisfying assignment: every clause contains at least one true literal. • 3SAT: a canonical NP-complete source problem used in many reductions. Q. When is a clause satisfied? Answer / rationale: When at least one literal in that clause is true. 5. 3SAT → Independent Set • Create one vertex for each literal occurrence in each clause. • Clause edges: connect literal vertices from the same clause. • Conflict edges: connect complementary literals such as x and ¬x. • If there are m clauses, target an Independent Set of size m. • Clause edges enforce at most one chosen literal per clause. • Conflict edges prevent contradictory truth assignments. • A size-m independent set therefore chooses one mutually compatible literal from each clause. Q. Why does size m matter? Answer / rationale: There are m clauses, and clause edges prevent multiple selections within one clause. An independent set of size m must select one from every clause. Q. What do conflict edges accomplish? Answer / rationale: They prevent the construction from selecting both x and ¬x. 6. Independent Set, Clique & Vertex Cover • Independent Set: no two selected vertices are adjacent. • Clique: every two selected vertices are adjacent. • Complement graph: adjacency is reversed. • IS ↔ Clique: an independent set of size k in G is a clique of size k in the complement. • Vertex Cover: a set touching every edge. • Identity: α(G)+τ(G)=|V|. Q. How does complementing a graph relate IS and Clique? Answer / rationale: Nonadjacent pairs in G become adjacent in the complement, turning an independent set into a clique of equal size. Q. What is the relationship between maximum Independent Set and minimum Vertex Cover? Answer / rationale: Their sizes sum to the total number of vertices. 7. NP-Completeness Decision Checklist • Is the problem a decision problem? • Can a proposed YES certificate be verified in polynomial time? • Which known NP-complete problem resembles the target? • Can every source instance be transformed into the target in polynomial time? • Does the transformation preserve YES iff YES? • Is the construction polynomial in input size? Q. Does a polynomial verifier prove NP-completeness? Answer / rationale: No. It proves only membership in NP; hardness still requires a reduction. 8. Approximation Algorithms • Optimization: seek the best feasible solution rather than only YES/NO. • Approximation: efficiently finds a feasible solution with a provable quality guarantee. • For minimization, a c-approximation commonly satisfies ALG ≤ c·OPT. • For maximization, a common form is ALG ≥ OPT/c. • Always state the convention: ratio terminology varies. • Greedy ≠ automatically approximate: the guarantee must be demonstrated. Q. Why use approximation algorithms? Answer / rationale: They provide efficient solutions with mathematical guarantees for problems where exact optimization may be computationally difficult. 9. Bounds Used in Approximation Proofs • For minimization, a lower bound L ≤ OPT can help show ALG/L ≤ c and hence ALG ≤ c·OPT. • For maximization, an upper bound U ≥ OPT can help establish a guaranteed fraction of OPT. • A proof identifies a benchmark, bounds the algorithm, then relates the benchmark to OPT. • Always distinguish feasibility from optimality. 10. Classic Approximation Patterns • Vertex Cover 2-approximation: repeatedly take an uncovered edge and select both endpoints. • The selected edges form a matching; its size is a lower bound on OPT. • Metric TSP: MST-based approaches exploit triangle inequality so shortcutting does not increase tour length. • Set Cover: greedy coverage arguments lead to logarithmic-style guarantees. • Knapsack: common approximation strategies compare greedy choices with a strong single-item candidate. Q. Why does the matching argument prove a factor-2 bound for Vertex Cover? Answer / rationale: Each matching edge needs a distinct covered endpoint, so matching size is a lower bound on OPT; selecting both endpoints costs at most twice that bound. Q. What assumption is crucial for metric TSP shortcutting? Answer / rationale: The triangle inequality. 11. High-Yield Exam Traps Trap Correction Reduction direction Known hard A must reduce to target