Introduction to Automata Theory, Languages, and Computation

Solutions for Chapter 10

Revised 6/30/01.

Solutions for Section 10.1

Solutions for Section 10.2

Solutions for Section 10.3

Solutions for Section 10.4

Solutions for Section 10.1

Exercise 10.1.1(a)

The MWST would then be the line from 1 to 2 to 3 to 4.

Exercise 10.1.3

For every problem P in NP there would be some polynomial p that bounded the running time (and therefore the output length) of the reduction from P to the NP-complete problem in question. That would imply an algorithm for P that ran in time O(p(n) + [p(n)]log2p(n)). The first term, p(n), can be neglected. The exponent is k log n for some constant k. Moreover, p(n)k is at most nk' for some other constant k'. Thus, we can say that there would be some constant c such that problem P could be solved in time O(nc log2n).

Exercise 10.1.5(a)

Given (G,A,B), construct G1 and G2 to be G, with start symbols A and B, respectively. Since this transformation is essentially copying, it can be performed in linear time, and is therefore surely a polynomial-time reduction.

Exercise 10.1.5(c)

Absolutely nothing! Part of any NP-completeness proof is a part that shows the problem to be in NP. These problems are, in fact, undecidable, and therefore surely not in NP.

Exercise 10.1.6(b)

Test for membership in one language and then, if the input is not in the first, test for membership in the second. The time taken is no more than the sum of the times taken to recognize each language. Since both ar in P, then can each be recognized in polynomial time, and the sum of polynomials is a polynomial. Thus, their union is in P.

Exercise 10.1.6(c)

Let L1 and L2 be languages in P, and suppose we want to recognize their concatenation. Suppose we are given an input of length n. For each i between 1 and n-1, test whether positions 1 through i holds a string in L1 and positions i+1 through n hold a string in L2. If so, accept; the input is in L1L2. If the test fails for all i, reject the input.

The running time of this test is at most n times the sum of the running times of the recognizers for L1 and L2. Since the latter are both polynomials, so is the running time for the TM just described.

Exercise 10.1.6(f)

Given a polynomial-time TM M for L, we can modify M to accept the complement of L as follows:

  1. Make each accepting state of M a nonaccepting state from which there are no moves. Thus, if M accepts, the new TM will halt without accepting.

  2. Create a new state q, which is the only accepting state in the new TM. For each state-symbol combination that has no move, hte new TM enters state q, whereupon it accepts and halts.

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Solutions for Section 10.2

Exercise 10.2.1(a)

Choosing x = y = z = 1 makes the expression satisfiable. Thus, the expression is in SAT.

Exercise 10.2.2(a)

There are actually only three distinct Hamilton circuits, once we account for the differences in direction and differences in starting point. These three circuits are (1,2,3,4), (1,3,2,4), and (1,3,4,2). We can express the existence of one of these three circuits (using the simplified notation of Section 10.3) by: x12x23x34x14 + x13x23x24x14 + x13x34x24x12.

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Solutions for Section 10.3

Exercise 10.3.1(a)

In what follows, [-x] stands for x-bar, the complement of x. We'll begin by using the construction to put it into CNF. xy is already the product of clauses (x)(y), and [-x]z is the product of clauses ([-x])(z). When we use the OR construction to combine these, we get (x+u)(y+u)([-x]+[-u])(z+[-u]).

Now, to put this expression into 3-CNF, we have only to expand the four clauses, each of which has only two literals, by introducing four new variables and doubling the number of clauses. The result: (x+u+v1)(x+u+[-v1])(y+u+v2)(y+u+[-v2])([-x]+[-u]+v3)([-x]+[-u]+[-v3])(z+[-u]+v4)(z+[-u]+[-v4]).

Exercise 10.3.3(a)

It is satisfiable. Let any two variables be assigned TRUE, say x1 and x2, and let the other two variables be assigned FALSE. Then in any set of three variables, there must be at least one true and at least one false. Thus, none of the clauses can be false.

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Solutions for Section 10.4

Exercise 10.4.1

For part (a): There are triangles (3-cliques), such as {1,2,3}. However, there is no 4-clique, since there are only 4 nodes, and one edge is missing. Thus k = 3 is the answer.

For part (b): All pairs of nodes must have an edge between them, and the number of pairs of k nodes is k choose 2, or k(k-1)/2.

For part (c): We reduce NC to CLIQUE as follows. Suppose we are given an instance (G,k) of NC. Construct the instance (G',n-k) of CLIQUE, where n is the total number of nodes of G, and G' is G with the set of edges complemented; that is, G' has edge (u,v) if and only if G does not have that edge.

We must show that G has a node cover of size k if and only if G' has a clique of size n-k. First, let C be a node cover of G of size k. We claim that C', the complement of the nodes in C, is a clique in G' of size n-k. Surely C' is of size n-k. Suppose it is not a clique. Then there is a pair of nodes (u,v) that do not have an edge in G'. Thus this edge is in G. But neither u nor v is in C, contradicting the assumption that is is a node cover.

Conversely, if C' is a clique of size n-k in G', then we claim that C the complement of C', is a node cover of size k in G. The argument is similar: if (u,v) is an edge of G not covered by C, then both u and v are in C', but the edge (u,v) is not in G', contradicting the assumption that C' is a clique.

Exercise 10.4.2

For each clause, we add one node, and connect it so that it can only be colored in one of the n+1 available colors if the clause is made true. Suppose the clause consists of literals with variables xi, xj, and xk, possibly negated. The node for the clause is connected to:

  1. xm for all m = 0, 1,..., n, except for i, j, and k. Thus, the only possible colors for the nodes are the ones used for its literals.

  2. If the literal with xi is positive (not negated), connect the node for the clause to the node for xi. If the literal is negated, connect the node for the clause to the node for xi-bar.

  3. Connect to nodes for xj and xk, analogously.

Now, if at least one of the literals of the clause is made true by the assignment where the color c0 corresponds to truth, then that literal will not be colored with the color for its variable, and we can use that color for the clause's node. However, if the truth assignment makes all three literals false, then the clause's node is connected to nodes of all n+1 colors, and we cannot complete the coloring. Thus, coloring the complete graph with n+1 colors is possible if and only if there is a satisfying truth assignment for the 3-CNF expression.

Exercise 10.4.3(a)

Yes; a Hamilton circuit can be found by going around the inner circle, say from 11 to 20, clockwise, then to 10, around the outer circle counterclockwise, to 1, and then back to 11.

Exercise 10.4.4(f)

Let (G,k) be an instance of the clique problem, and suppose G has n nodes. We produce an instance of the half-clique problem, as follows:

  1. If k = n/2, just produce G. Note that G has a half-clique if and only if it has a clique of size k. in this case.

  2. If k > n/2, add 2k - n isolated nodes (nodes with no incident edges). The resulting graph has a half-clique (whose size must be (n + (2k-n))/2 = 2k, if and only if G has a clique of size k.

  3. If k < n/2, add n - 2k nodes, and connect them in all possible ways to each other and to the original nodes of G. The new graph thus has 2(n-k) nodes. The new nodes, plus a clique of size k in G form a clique of size (n-2k) + k = n-k, which is half the number of nodes in the new graph. Conversely, if the new graph has a half-clique, then it must include at least (n-k) - (n-2k) = k nodes of the graph G, implying that G has a clique of size k.

These steps complete a reduction of CLIQUE to HALF-CLIQUE. It is evidently performable in polynomial time, since the number of new nodes and edges is at most the square of the original number of nodes, and the rules for adding nodes and edges are simple to carry out.

Exercise 10.4.5(a)

Following the hint, pick any node x in graph G. Add a duplicate node y that is adjacent to exactly those nodes to which x is adjacent. Then, add new nodes u and v that are adjacent to x and y, respectively, and no other nodes. Call the resulting graph G'.

We claim G' has a Hamilton path if and only if G has a Hamilton circuit. If G has a Hamilton circuit, the following is a Hamilton path in G': start at u, go to x, follow the Hamilton circuit, but end at y instead of x, and then go to v.

If G' has a Hamilton path, it must start at u and end at v, or vice-versa (which is really the same path. Moreover, the path must go from xy, visiting all the nodes of G as it does. Thus, if we replace y by x along this path, we get a Hamilton circuit in G.

Exercise 10.4.5(c)

A spanning tree with two leaf nodes is a Hamilton path. Thus, the Hamilton path problem reduces to the question of whether a graph has a spanning tree with only 2 leaf nodes. Surely, then, Hamilton path reduces to the more general problem stated in the question, where the number of leaf nodes is a parameter of the problem.

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