CMPSCI 601: Theory of Computation

Offered through the PEEAS distance learning program

Homework Assignment #3

David Mix Barrington

Assignment posted Fri 25 July 2003

Make sure you have read the statement on academic honesty on the homework assignment index page.

• Question 1 (30): For each of the following three problems, either prove that it is complete for NL under L-reductions or prove that it is in L:

  • (a,10) REACH-OUT-2 = {(G,s,t): G is a directed graph with out-degree at most two and there is a path from s to t in G}

  • (b,10) REACH-ACYCLIC = {(G,s,t): G is a directed acyclic graph and there is a path from s to t in G} Here you may assume that G is presented as a levelled graph (as in Spring 2003 HW#6, Question 12), so your algorithm knows that G is actually acyclic.

  • (c,10) REACH-IN-1 = {(G,s,t): G is a directed graph with in-degree at most 1 and there is a path from s to t in G}.

  (Recall that the in-degree of a directed graph is the largest number of edges that enter any one vertex, and that the out-degree is the largest number of edges that leave any single vertex.)

• Question 2 (20): A group is a set with a binary operation that is associative (for any x, y, and z, (x times y) times z = x times (y times z), has an identity element (there exists e such that for any x, e time x = x times e = x) and has inverses (for any x there exists y such that x times y = e). The ITERATED-PRODUCT problem is as follows:

  • Input: a multiplication table for a group (a function from G times G to G obeying the group properties), and a sequence of elements of the group (a string w in G^*). Note that if n is the input size, G has at most n elements and the length of w is at most n.

  • Output: the product of the elements in the sequence.

  Here is your problem:

  • (a,15) Prove that the ITERATED-PRODUCT problem can be solved in log-space, that is, the function is in FL.

  • (b,5) Your argument for (a) should not have required both the associativity condition and the inverses condition. Prove that with the correct one of these conditions omitted, the problem is still in FL.

• Question 3 (15): Two directed graphs G and H are isomorphic if there exists a one-to-one, onto function f from the vertices of G to the vertices of H such that for any two vertices x and y in G, there is an edge from x to y in G iff there is an edge from f(x) to f(y) in H. Prove that the language GRAPH-ISOMORPHISM = {(G,H): G and H are isomorphic} is in NP. (It is unknown whether GRAPH-ISOMORPHISM is NP-complete or whether it is in P, though it is thought at least not to be NP-complete.)

• Question 4 (15): Consider the restriction of the HAMILTON-PATH problem where the given graph is guaranteed to be acyclic. Prove that the set {G: G is a directed acyclic graph with a Hamilton path} is in P. Here, unlike in Question 1 above, assume that the graph is merely given to you as an adjacency list or matrix -- it is known that in P you can check whether it is acyclic. (Hint: Look up "topological sort" on the Web or in an algorithms book.)

• Question 5 (20): It was shown in lecture that the language 3-COLOR = {G: G is an undirected graph with a 3-coloring} is NP-complete. Consider the language 5-COLOR = {G: G is an undirected graph with a 5-coloring}.

  • (a,10) Describe a log-space reduction from 3-COLOR to 5-COLOR.

  • (b,5) Complete the proof that 5-COLOR is NP-complete.

  • (c,5) It follows from (b) that there is a log-space reduction from 5-COLOR to 3-COLOR. Describe how you would construct this reduction.

Last modified 25 July 2003