Reductions from 3D Matching: Detailed Proofs
Introduction
3D Matching is a fundamental matching problem proven NP-complete by reduction from 3-SAT. This post provides detailed proofs following the standard template for reducing from 3D Matching to prove other problems are NP-complete.
Problem Definition: 3D Matching
3D Matching Problem:
- Input: Sets X, Y, Z and set T ⊆ X × Y × Z of triples
- Output: YES if there exists matching M ⊆ T covering all elements, NO otherwise
Perfect 3D Matching: A set of triples covering each element exactly once.
Q1: How do you reduce 3D Matching to Set Cover?
Answer: Encode triples as sets covering elements.
Key Insight:
- Each triple covers three elements (one from each set X, Y, Z)
- Perfect matching covers all elements exactly once
- Set cover allows covering elements multiple times, but perfect matching ensures exactly once
Hint: Each triple becomes a set containing its three elements. A perfect 3D matching corresponds to a set cover of size |X| = |Y| = |Z|, where each element is covered exactly once.
1. NP-Completeness Proof of Set Cover: Solution Validation
Set Cover Problem:
- Input: Universe U, collection C of subsets of U, integer k
- Output: YES if there exists subcollection C’ ⊆ C with
|C'|≤ k covering U, NO otherwise
Set Cover ∈ NP:
Verification Algorithm: Given a candidate solution (subcollection C’):
- Check that C’ ⊆ C: O(
|C'|) time - Check that
|C'|≤ k: O(1) time - Check that ∪_{S ∈ C’} S = U: O(
|U|·|C'|) time
Total Time: O(|U| · |C'|), which is polynomial.
Conclusion: Set Cover ∈ NP.
2. Reduce 3D Matching to Set Cover
2.1 Input Conversion
Given a 3D Matching instance: sets X, Y, Z, triples T ⊆ X × Y × Z.
Construction:
- Universe U = X ∪ Y ∪ Z (all elements)
- For each triple t = (x, y, z) ∈ T, create set Sₜ = {x, y, z}
- Collection C = {Sₜ : t ∈ T}
- Target k =
|X|=|Y|=|Z|(assuming equal sizes) - Return Set Cover instance: universe U, collection C, integer k
Key Property: Perfect 3D matching ↔ Set cover of size k
2.2 Output Conversion
Given: Set Cover solution C’ ⊆ C with |C'| ≤ k covering U
Extract Matching:
- M = {t ∈ T : Sₜ ∈ C’}
|M|=|C'|≤ k- Since
|X|=|Y|=|Z|= k and each set covers 3 elements, need exactly k sets - M is perfect 3D matching
3. Correctness Justification
3.1 If 3D Matching has a solution, then Set Cover has a solution
Given: 3D Matching instance has solution M ⊆ T (perfect matching).
Construct Set Cover:
- C’ = {Sₜ : t ∈ M}
|C'|=|M|= k- Since M covers all elements:
- Each x ∈ X appears in exactly one triple t ∈ M
- Each y ∈ Y appears in exactly one triple t ∈ M
- Each z ∈ Z appears in exactly one triple t ∈ M
- Therefore, C’ covers all elements in U
Conclusion: Set Cover has a solution.
3.2a If 3D Matching does not have a solution, then Set Cover has no solution
Given: 3D Matching instance has no perfect matching.
Proof:
- If Set Cover has solution C’ of size k:
- Then M = {t : Sₜ ∈ C’} is matching of size k
- Since
|X|=|Y|=|Z|= k, M covers all elements - M is perfect matching
- Contradiction
Conclusion: Set Cover has no solution.
3.2b If Set Cover has a solution, then 3D Matching has a solution
Given: Set Cover instance has solution C’ ⊆ C with |C'| ≤ k covering U.
Extract Matching:
- M = {t ∈ T : Sₜ ∈ C’}
|M|=|C'|≤ k- Since C’ covers U and
|U|= 3k:- Need at least k sets (each covers 3 elements)
- Have at most k sets
- Therefore, exactly k sets, each covering 3 distinct elements
- M is perfect 3D matching
Conclusion: 3D Matching has a solution.
Polynomial Time: O(|T|) to create sets.
Therefore, Set Cover is NP-complete.
Q2: How do you reduce 3D Matching to Exact Cover?
1. NP-Completeness Proof of Exact Cover: Solution Validation
Exact Cover Problem:
- Input: Universe U and collection C of subsets of U
- Output: YES if there exists subcollection C’ ⊆ C covering each element exactly once, NO otherwise
Exact Cover ∈ NP:
Verification Algorithm: Given a candidate solution (subcollection C’):
- Check that C’ ⊆ C: O(
|C'|) time - Check that each element appears exactly once: O(
|U|·|C'|) time
Total Time: O(|U| · |C'|), which is polynomial.
Conclusion: Exact Cover ∈ NP.
2. Reduce 3D Matching to Exact Cover
2.1 Input Conversion
Given a 3D Matching instance: sets X, Y, Z, triples T ⊆ X × Y × Z.
Construction:
- Universe U = X ∪ Y ∪ Z
- For each triple t = (x, y, z) ∈ T, create set Sₜ = {x, y, z}
- Collection C = {Sₜ : t ∈ T}
- Return Exact Cover instance: universe U, collection C
Key Property: Perfect 3D matching ↔ Exact cover
2.2 Output Conversion
Given: Exact Cover solution C’ ⊆ C
Extract Matching:
- M = {t ∈ T : Sₜ ∈ C’}
- M is perfect 3D matching (each element covered exactly once)
3. Correctness Justification
3.1 If 3D Matching has a solution, then Exact Cover has a solution
Given: 3D Matching instance has solution M ⊆ T (perfect matching).
Construct Exact Cover:
- C’ = {Sₜ : t ∈ M}
- Since M is perfect matching:
- Each element in X appears in exactly one triple t ∈ M
- Each element in Y appears in exactly one triple t ∈ M
- Each element in Z appears in exactly one triple t ∈ M
- Therefore, C’ covers each element exactly once
Conclusion: Exact Cover has a solution.
3.2a If 3D Matching does not have a solution, then Exact Cover has no solution
Given: 3D Matching instance has no perfect matching.
Proof by Contradiction:
- Assume Exact Cover has solution C’
- Then M = {t : Sₜ ∈ C’} is perfect matching
- Contradiction
Conclusion: Exact Cover has no solution.
3.2b If Exact Cover has a solution, then 3D Matching has a solution
Given: Exact Cover instance has solution C’ ⊆ C.
Extract Matching:
- M = {t ∈ T : Sₜ ∈ C’}
- Since C’ covers each element exactly once:
- M covers all elements
- Each element appears in exactly one triple
- M is perfect 3D matching
Conclusion: 3D Matching has a solution.
Polynomial Time: O(1) (trivial reduction).
Therefore, Exact Cover is NP-complete.
Key Takeaways
- Set Encoding: 3D Matching → Set Cover encodes triples as sets
- Exact Coverage: Perfect matching ensures exact coverage
- Triple Structure: Three sets make matching harder than 2D
- Template Structure: All reductions follow rigorous format
3D Matching reductions demonstrate covering structures and exact coverage patterns.