1. Lecture 24 Classical NP-hard Problems

2. Outline Hamiltonian Path to TSP Cycle 3-SAT to Quadratic Program
Tripartite Matching to SUBSET SUM

3. General Recipe for reductions
In order to prove B is NP-hard, given that we know A is NP hard.
Reduce A to B.
What is a complete reduction?
Given an instance X of A, construct an instance Y of B.
Prove that if answer to X is YES, answer to Y is also YES.
Prove that if answer to X is NO, answer to Y is also NO. (Usually: prove the contrapositive: if answer to Y is YES, answer to X is also YES)

4. Hamiltonian Path
Given a graph G, a starting vertex s and an ending vertex t.
A Hamiltonian path from s to t is a path from s to t that visits every vertex in G exactly once.
Hamiltonian Path problem asks if there exists a Hamiltonian path from s to t.

5. Travelling Salesman Problem
Given a weighted graph G, a salesman wants to start at vertex s, visit all vertices in the graph, and then come back to s.
Given a threshold L, TSP cycle problem asks whether there is a way to visit all vertices (and come back to s) with total edge length at most L.
(Starting vertex does not matter, so s is not part of input)

6. Reduction from Hamiltonian Path to TSP cycle
Step 1: Given an instance X of Hamiltonian Path, construct an instance Y of TSP cycle.
Idea: Find similarities/differences of the problems.
Similarities: visit every vertex in graph.
Differences:
No weights in Hamiltonian Path
Path vs. cycle.

7. Outline Hamiltonian Path to TSP Cycle 3-SAT to Quadratic Program
Tripartite Matching to SUBSET SUM

8. 3-SAT
Literals: A variable ( 𝑥 1 ), or the negation of a variable ( 𝑥 1 ).
Clause: Logic OR of literals ( 𝑥 1 ∨ 𝑥 3 ∨ 𝑥 5 )
Conjunctive Normal Form: “and” of “or”s.
𝑥 1 ∨ 𝑥 3 ∨ 𝑥 5 ∧ 𝑥 2 ∨ 𝑥 3 ∨ 𝑥 4 ∧⋯∧ 𝑥 1 ∨ 𝑥 4 ∨ 𝑥 5
3-SAT: Given a conjunctive normal form, where each clause contains at most 3 literals, decide whether there is a value of variables such that the formula is satisfied
(A clause is satisfied if one of its literals is true; The formula is satisfied if ALL clauses are satisfied.)

9. Quadratic Programming
Same as linear programming, except you are allowed to have quadratic constraints (e.g. x12+x2x3 + 3x4≤ 3)
Decision problem: Given a set of quadratic constraints, decide whether there is a feasible solution.

10. Reduction from 3-SAT to Quadratic Programming
Step 1: Given an instance X of 3-SAT, construct an instance Y of Quadratic Programming.
Two problems are very different, needs to use gadgets.

11. Gadget for variables Variable xi: either true or false
Gadget: A variable in quadratic program that can only take two values (say 0 or 1)
yi2 = yi
xi = true  yi = 1; xi = false  yi = 0
Negation of a variable: (1-yi)

12. Gadget for Clauses Clause: ( 𝑥 1 ∨ 𝑥 3 ∨ 𝑥 5 )
Interpretation: x1 = true or x3 = true or x5 = false
 y1 = 1 or y3 = 1 or (1-y5) = 1
Can we write this as a constraint?
y1+y3+(1-y5) ≥ 1

13. Outline Hamiltonian Path to TSP Cycle 3-SAT to Quadratic Program
Tripartite Matching to SUBSET SUM

14. Tripartite Matching
There are three sets of vertices U, V, W, each of size n.
A hyper-edge is a 3-tuple (u,v,w) where u, v, w are vertices in U, V, W respectively.
A tripartite matching is a way of selecting n hyper- edges, so that every vertex is adjacent to a hyper- edge.
The decision version of the problem asks whether a tripartite matching exists.

15. SUBSET SUM/KNAPSACK
In the SUBSET SUM problem, you are given n numbers a1, a2, …, an, and a target m.
The answer is YES if and only if there is a subset of these n numbers that sums up to m.

16. Reduction from Tripartite Matching to SUBSET SUM
Step 1: Given an instance X of Tripartite Matching, construct an instance Y of SUBSET SUM.
Similarity: Tripartite Matching wants to select hyper-edges, SUBSET SUM wants to select numbers.
Question: how to map a hyper-edge to a number?

17. Idea: Use an intermediate problem
SUBSET VECTORS
Given n vectors v1, v2, …, vn, and a target u.
The answer is YES if and only if there is a subset of these n vectors that sums up to u.
Reduce Tripartite Matching to SUBSET VECTORS
Reduce SUBSET VECTORS to SUBSET SUM