NP-Completeness — P, NP, NP-Hard, Reductions
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# NP-Completeness — P, NP, NP-Hard, Reductions ## 🎯 Learning Objectives - Define complexity classes P, NP, NP-hard, NP-complete - Prove a problem is NP-complete via reduction - Perform reductions between SAT, 3-SAT, Clique, Vertex Cover, Subset Sum - Explain the Cook-Levin theorem * * * ## 1. Complexity Classes ###...
NP-Completeness — P, NP, NP-Hard, Reductions
🎯 Learning Objectives
- Define complexity classes P, NP, NP-hard, NP-complete
- Prove a problem is NP-complete via reduction
- Perform reductions between SAT, 3-SAT, Clique, Vertex Cover, Subset Sum
- Explain the Cook-Levin theorem
1. Complexity Classes
1.1 Intuition
Some problems are easy (can be solved quickly), some are hard (we don't know fast algorithms). P is the set of "tractable" problems. NP is the set of problems with "easily verifiable" solutions. The P vs NP question asks: if we can quickly verify a solution, can we also quickly find one?
1.2 Formal Definitions
(Diagram)
| Class | Definition | Example |
|---|---|---|
| P | Solvable in polynomial time | Sorting, shortest path |
| NP | Verifiable in polynomial time | Sudoku (verify a solution is easy) |
| NP-hard | Every problem in NP reduces to it | Halting problem (can be undecidable!) |
| NP-complete | In NP and NP-hard | SAT, 3-SAT, Clique, Vertex Cover |
1.3 Key Questions
- P = NP? Unknown ($1M Clay Millennium Prize)
- NP = co-NP? Unknown
- Is every NP-hard problem in NP? No (e.g., optimization versions)
2. Polynomial-Time Reductions
2.1 Intuition
A reduction transforms instances of problem A into instances of problem B, such that solving B solves A. If we can solve B efficiently, we can solve A efficiently.
2.2 Formal Definition
(A reduces to B in polynomial time) if there exists a polynomial-time computable function such that:
(Diagram)
2.3 Reduction Examples
3-SAT → Independent Set
Given a 3-CNF formula, construct a graph where:
- Each clause has 3 nodes (one per literal)
- Connect literals in the same clause (triangle)
- Connect each literal to its negation
- Formula satisfiable ↔ graph has independent set of size = number of clauses Independent Set → Clique The complement graph flips edges and non-edges. An independent set in G is a clique in the complement of G.
3. Classic NP-Complete Problems
3.1 SAT (Satisfiability)
Problem: Given a Boolean formula, is there a satisfying assignment?
Cook-Levin Theorem: SAT is NP-complete. Every NP problem can be reduced to SAT.
3.2 3-SAT
Problem: Given a 3-CNF formula (AND of ORs, each OR has 3 literals), is it satisfiable?
Reduction from SAT to 3-SAT: Each clause is converted to CNF with ≤3 literals per clause using auxiliary variables.
3.3 Clique
Problem: Does a graph contain a complete subgraph of size k?
Reduction from 3-SAT: Create a graph where each clause has a triangle of its literals; connect literals that are not contradictory. An independent set of size m (clauses) gives a satisfying assignment.
3.4 Vertex Cover
Problem: Is there a set of k vertices that touches every edge?
Relation to Independent Set: A set C is a vertex cover iff V\C is an independent set.
Reduction from Independent Set:
3.5 Subset Sum
Problem: Given numbers a₁, ..., aₙ and target T, is there a subset summing to T?
Reduction from Vertex Cover: Create numbers encoding which vertices cover which edges.
4. NP-Completeness Proof Template
textTo prove problem X is NP-complete: 1. Show X ∈ NP: - Give a polynomial-time verifier for solutions 2. Show X is NP-hard: - Choose a known NP-complete problem Y - Construct reduction f: Y → X - Prove: x ∈ Y ⇔ f(x) ∈ X - Show f runs in polynomial time
4.1 Worked Proof: Vertex Cover is NP-complete
Step 1: VC ∈ NP Given a graph G=(V,E), integer k, and a candidate set C:
- Check |C| = k: O(n)
- Verify every edge has at least one endpoint in C: O(m)
- Total: O(n + m) — polynomial ✓ Step 2: VC is NP-hard (reduce from Independent Set)
- Given instance of IS: (G, k), ask does G have independent set ≥ k?
- Let n = |V|. Create instance of VC: (G, n - k)
- Claim: G has IS of size k ⇔ G has VC of size n-k
- Proof: If I is an IS of size k, then V\I is a VC (every edge has at most one endpoint in I, so at least one in V\I). Conversely, if C is a VC of size n-k, then V\C is an IS of size k.
5. 📝 Practice Questions
Q1: What is the difference between NP-hard and NP-complete?Answer: NP-complete problems are in NP AND NP-hard. NP-hard problems are at least as hard as any NP problem but may not be in NP themselves. For example, the optimization version "find the minimum vertex cover" is NP-hard but not NP-complete (it's not a decision problem, so not in NP). Q2: Show that if P = NP, then every NP-complete problem can be solved in polynomial time.Answer: If P = NP, then every problem in NP is in P. Since NP-complete problems are in NP, they are also in P. Thus there exists a polynomial-time algorithm for SAT, 3-SAT, Clique, etc. Q3: Reduce 3-SAT to Clique. Given formula (x₁∨x₂∨¬x₃)∧(¬x₁∨x₃∨x₄)∧(x₂∨¬x₃∨¬x₄).Answer: Create graph with 9 vertices (3 per clause). Connect vertices if they're in different clauses and not contradictory. The formula is satisfiable iff there's a 3-clique.
- Clause 1: v₁₁(x₁), v₁₂(x₂), v₁₃(¬x₃)
- Clause 2: v₂₁(¬x₁), v₂₂(x₃), v₂₃(x₄)
- Clause 3: v₃₁(x₂), v₃₂(¬x₃), v₃₃(¬x₄)
Edges connect compatible literals across clauses. A 3-clique picks one literal per clause that can all be true simultaneously. Q4: Why is the Cook-Levin theorem important?Answer: Cook-Levin was the first proof that SAT is NP-complete. It established the technique of reduction for proving NP-hardness and showed that the question "does SAT have a polynomial-time algorithm?" is equivalent to P vs NP. Before Cook-Levin, the concept of NP-completeness didn't exist. Q5: Prove that Dominating Set is NP-complete (hint: reduce from Vertex Cover).Answer: Given graph G=(V,E) and integer k for Vertex Cover, construct G' = (V∪E, E') where each edge e={u,v} in E becomes a vertex in G', and E' connects edge vertices to their endpoint vertices. A vertex cover of size k in G corresponds to a dominating set of size k in G'. The proof follows from the observation that to dominate edge-vertices, at least one endpoint of each edge must be selected.
6. 🔗 Cross-References
- Week 7 - Approximation Algorithms: Coping with NP-hardness
- Week 10 - Parameterized Algorithms: Fixed-parameter tractability
- BSCS3021 (ToC): Undecidability, Turing machines, time complexity Join Discord PreviousString AlgorithmsNextApproximation Algorithms