Lecture 03-31-17

Independent set (Stable set)

• Graph G,K - does G have an independent set of size K?
• Independent set: set of vertices which have no edges between them
• Prove that this problem is NP-complete

Independent set reduction

1. Show that in NP
   • If given the set of size K, then can easily verify
2. CLIQUE is transformable into INDEPENDENT-SET
   • GK -> G’K’
   • G’ = G complement where there is an edge between x and y iff no edge between x and y in G
   • K’ = K
   • Suppose that G has a clique C of size K, then C is an independent set in G’
   • Suppose that G’ has an independent set I of size K, then I must form a clique in G of size K - Easy reduction - Recall: we turned CNF Satisifiability into CLIQUE
   • We didn’t show CLIQUE -> Satisifiability
     • Why not?
     • Since Satisifiability is already in NP-Complete, we don’t have to prove again
       • 3-Satisifiability is CNF-Satisifiability where there are most 3 literals in each clause
   • Also NP-complete - Special case will be hard as well

Cryptography

• If can solve knapsack (which is NP-complete), then can break some common cryptography algorithms

Vertex Cover

• Graph G, number K
• Pick vertices to cover all edges
• NP-complete
• Claim: vertices that are not part of vertex cover must form an independent set
• Reduction of IS -> VC
  • Show that in NP
    • Given a set of vertices, can verify in polynomial time that the set is a vertex cover
  • IS -> VC
    • GK -> G’K’
    • Let G = G’
    • Let K’ = N - K
    • Suppose G has an IS, I of size K
      • V \I forms a VC of G’ since vertices not in IS must have edges
    • Suppose G’ has a VC C of size N - K
      • Therefore, G’ has a IS of V\C = (N - (N - K)) = size K
      • Because any edges between them would not be covered by the vertex cover, which covers all the edges, so IS

Edge Cover

• Pick edges to cover all vertices
• Can be solved in polynomial time

Traveling salesperson problem

• Hard to find order
• Turn covering problem into ordering problem is challenging

Hamilton Cycle problem

• Euler cycle?
• Cycle that visits every vertex exactly once
• NP-complete to determine whether a graph has a hamilton cycle
• Next time: notion of what we do with NP-complete problems once we know they’re NP-complete