# CMPSCI 601: Theory of Computation

### Due Wednesday 28 April 2004 in class or beforehand to CMPSCI office drop box

#### (fax by Fri 7 May 2:00 p.m. EST for off-campus students)

Question 4 changed on 21 April 2004!

• Question 1 (20): The problem BETWEEN is the set {(G,a,b,c): G is a directed graph, a, b, and c are vertices, and G has a path from a to c that passes through b}. Prove that BETWEEN is complete for NL under log-space reductions.

• Question 2 (20): The problem MEMBER-NP is the set {(M,w,1t): M is a nondetermistic multitape TM, w is a string, 1t is a string of 1's, and it is possible for M to accept w in time at most t}. Prove that MEMBER-NP is complete for NP under log-space reductions. Do not use the Fagin or Cook-Levin theorems -- prove directly that MEMBER-NP is in NP and that any NP language reduces to it. (Typo corrected 21 April 2004.)

• Question 3 (15): The decision problem HAMCIRCUIT is the set of directed graphs that have a Hamilton circuit. Prove that if HAMCIRCUIT is in the class P, then there is a poly-time algorithm that inputs a graph G and outputs a Hamilton circuit in G if one exists.

• Question 4 (15): Consider the restriction of the optimization problem TSP where the distance between every pair of distinct nodes is between n and 2n, where n is the number of nodes. Prove that TSP has a (1/2)-approximation algorithm, using the definition in Lecture 21, but is still NP-complete as a decision problem. (The decision problem is the set of inputs (G,k) such that G is a weighted graph meeting the given condition and G has a TSP of total weight at most k.)

(This problem was changed on 21 April, it used to ask for a 1-approximation algorithm which is always the case according to the given definition. Sorry, and thanks to the student who caught this.)

• Question 5 (10): Prove that if the decision problem version of TSP is in P, then there exists a poly-time algorithm that inputs a weighted directed graph G and outputs the length of the optimal TSP tour. (Assume that the weights are integers.)

• Question 6 (20): Under the same assumption as for Question 5, prove that there exists a poly-time algorithm that inputs G and outputs the optimal tour itself.