CSE860
Final
Due: Saturday 12 noon, May 1.
PART I. Solve problem 1 and 2.
1. For each of the following assertions, state whether they are True, False, or Open according to our current state of knowledge of computability and complexity theory, as described in class.
You do not have to justify your answer choice.
(a) ATM
 m
TQBF
(b) EXPSPACE contains all decidable languages.
(c) NL

IP
(d) P = NP

coNP
(e) coNP = NEXPTIME
(f) COMPOSITE is in P
(g) HAMPATH

coNP
(h) 2SAT
 p
CLIQUE
(i) CLIQUE
 p
2SAT
(j) P contains all context-free languages.
(k) ISO is polynomial time Turing reducible to SAT
(l) MAX-CLIQUE is Turing reducible to CLIQUE (page 274 for MAX-CLIQUE)
2. Prove that the following problem is NP-complete.
Subgraph Isomorphism Problem
Instance : Graph G and H
Question : Is there a subgraph G’ of G such that G’ is isomorphic to H?

PART II. Select any 5 problems and solve them.
1.
Develop an IP protocol to decide if a well formed formula is satisfiable or not.
Your protocol should be a concrete one.
Demonstrate your protocol with the following example:
(x1+x2)*(~x1+x3+~x4)*(x1+x2+~x4) )*(x2+~x2)
How many messages should be sent to have the error less than 40%.
2.
Consider the following problem.
Unique Satisfiability Problem (USat)
Instance : A wff W.
Question : Is W unique satisfiable? That is, there is exactly one assignment that makes W true.
For example W=(x1+x2)(x1+~x2)x2 is uniquely satisfiable (x1=t, x2=t), but
W= (x1+x2)(x1+~x2) is not uniquely satisfiable since (x1=t, x2 = t) (x1=t, x2 = f) are satisfying assignments for W.
What is the complexity of USat problem in the polynomial time hierarchy?
Prove your claim.
3.
Solve 9.13
4.
Solve 5.12
5.
Solve 6.20
6.
Let DIST = {<G, s, t, d> | d is the length of the shortest path from s to t in directed graph G}. In other words, <G, s, t, d>

DIST if and only if there is no path from s to t in G of length d-1, but there is such a path of length d. Show that DIST is NLcomplete.
7.
Solve 4.17