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CHAPTER 11. NP-COMPLETENESS
11-8. [7] Design and implement a backtracking algorithm to test whether a set of
clause sets is satisfiable. What criteria can you use to prune this search?
11-9. [8] Implement the vertex cover to satisfiability reduction, and run the resulting
clauses through a satisfiability solver code. Does this seem like a practical way
to compute things?
Basic Reductions
11-10. [4] An instance of the set cover problem consists of a set X of n elements, a
family F of subsets of X, and an integer k. The question is, does there exist k
subsets from F whose union is X?
For example, if X = {1, 2, 3, 4} and F = {{1, 2}, {2, 3}, {4}, {2, 4}}, there
does not exist a solution for k = 2, but there does for k = 3 (for example,
{1, 2}, {2, 3}, {4}).
Prove that set cover is NP-hard with a reduction from vertex cover.
11-11. [4] The baseball card collector problem is as follows. Given packets P1 , . . . , Pm ,
each of which contains a subset of this year’s baseball cards, is it possible to
collect all the year’s cards by buying ≤ k packets?
For example, if the players are {Aaron, M ays, Ruth, Skiena} and the packets
are
{{Aaron, M ays}, {M ays, Ruth}, {Skiena}, {M ays, Skiena}},
there does not exist a solution for k = 2, but there does for k = 3, such as
{Aaron, M ays}, {M ays, Ruth}, {Skiena}
Prove that the baseball card collector problem is NP-hard using a reduction
from vertex cover.
11-12. [4] The low-degree spanning tree problem is as follows. Given a graph G and an
integer k, does G contain a spanning tree such that all vertices in the tree have
degree at most k (obviously, only tree edges count towards the degree)? For
example, in the following graph, there is no spanning tree such that all vertices
have a degree at most three.
(a) Prove that the low-degree spanning tree problem is NP-hard with a reduction from Hamiltonian path.
(b) Now consider the high-degree spanning tree problem, which is as follows.
Given a graph G and an integer k, does G contain a spanning tree whose
highest degree vertex is at least k? In the previous example, there exists
a spanning tree with a highest degree of 7. Give an efficient algorithm to
solve the high-degree spanning tree problem, and an analysis of its time
complexity.
11.10. EXERCISES
385
11-13. [5] In the minimum element set cover problem, we seek a set cover S ⊆ C of
a universal set U = {1, . . . , n} such that sum of the sizes of the subsets in S is
at most k. (a) Show that C = {{1, 2, 3}, {1, 3, 4}, {2, 3, 4}, {3, 4, 5}} has a cover
of size 6, but none of size 5 because of a repeated element. (b) Prove that this
problem is NP-hard. (Hint: set cover remains hard if all subsets are of the same
size.)
11-14. [3] The half-Hamiltonian cycle problem is, given a graph G with n vertices,
determine whether G has a simple cycle of length exactly #n/2$, where the floor
function rounds its input down to the nearest integer. Prove that this problem
is NP-hard.
11-15. [5] The 3-phase power balance problem asks for a way!
to partition
! a set!of n
positive integers into three sets A, B, or C such that i ai = i bi = i ci .
Prove that this problem is NP-hard using a reduction from integer partition or
subset sum (see Section 10.5 (page 329)).
11-16. [4] Show that the following problem is NP-hard:
Problem: Dense Subgraph
Input: A graph G, and integers k and y.
Output: Does G contain a subgraph of exactly k vertices and at least y edges?
11-17. [4] Show that the following problem is NP-hard:
Problem: Clique, No-clique
Input: An undirected graph G = (V, E) and an integer k.
Output: Does G contain both a clique of size k and an independent set of size
k?
11-18. [5] An Eulerian cycle is a tour that visits every edge in a graph exactly once.
An Eulerian subgraph is a subset of the edges and vertices of a graph that has
an Eulerian cycle. Prove that the problem of finding the number of edges in the
largest Eulerian subgraph of a graph is NP-hard. (Hint: the Hamiltonian circuit
problem is NP-hard even if each vertex in the graph is incident upon exactly
three edges.)
11-19. [5] Show that the following problem is NP-hard:
Problem: Maximum Common Subgraph
Input: Two graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ), and a budget b.
Output: Two sets of nodes S1 ⊆ V1 and S2 ⊆ V2 whose deletion leaves at least
b nodes in each graph, and makes the two graphs identical.
11-20. [5] A strongly independent set is a subset of vertices S in a graph G such that for
any two vertices in S, there is no path of length two in G. Prove that strongly
independent set is NP-hard.
11-21. [5] A kite is a graph on an even number of vertices, say 2n, in which n of the
vertices form a clique and the remaining n vertices are connected in a tail that
consists of a path joined to one of the vertices of the clique. Given a graph and
a goal g, the max kite problem asks for a subgraph that is a kite and contains
2g nodes. Prove that max kite is NP-hard.