solutions2

# solutions2 - Solutions 2 Some solutions are only sketches...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solutions 2 Some solutions are only sketches: you should be able to fill in the details. Note again that if you are asked to prove NP-completeness the first thing you must do is prove membership of NP . As this is normally very easy, it is omitted here. Question 1 Show that if a language L is coNP-complete, then L is NP-complete. We need to show that there is a reduction from every language in NP to L . Choose any language L ∈ NP . Then L is in coNP . So there is a polynomial-time reduction R from L to L . Then R is also a reduction from L to L : x ∈ L ⇔ x 6∈ L ⇔ R ( x ) 6∈ L ⇔ R ( x ) ∈ L . Question 2 Prove that Graph Expanders is in coNP . Recall that a problem is in coNP if, when the answer is “No”, we can find a certificate that can be verified in polynomial time. If a graph is not an expander, then there is a set of vertices S , | S | ≤ | V | / 2, such that there are fewer than | S | vertices with neighbours in S . The set S is the certificate and it is easy to verify that it has this property. Question 3 Prove that Minimum Leaf Spanning Tree is NP-complete. Let K = 2. A spanning tree with exactly two leaves (obviously you cannot have less than two leaves) is a Hamiltonian path. So Hamiltonian Path is a special case of Minimum Leaf Spanning Tree . As we know Hamiltonian Path is NP-complete it immediately follows that Minimum Leaf Spanning Tree is NP-complete (this is the restriction method of proof)....
View Full Document

## This note was uploaded on 01/29/2012 for the course ECE 345 taught by Professor Veneris during the Fall '10 term at University of Toronto.

### Page1 / 3

solutions2 - Solutions 2 Some solutions are only sketches...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online