CIS 262 Fall 2014: Problem Set 6, Due December 4

CIS 262 Fall 2014: Problem Set 6, Due December 4
Please write your answers succinctly and rigorously. Remember that typeset PDF
submissions earn 5% extra credit. If you choose to do this, you may hand-draw complex structures such as DFAs. If you use LATEX, DO NOT SUBMIT YOUR SOURCE
CODE!
Canvas Submission: Create separate PDFs for each of the problems 1, 2, 3, 4 and
5, each with your name and problem number and upload them on Canvas.
Hand Written Submission: Write each of the problems 1, 2, 3, 4 and 5 on separate
sheets and submit 5 separate sheets each with your name and problem number clearly
written.
1. (a) A subset U of nodes in a graph G = (V, E) is called a dominating set if every node 5pts
not in U is adjacent to some node in U (that is, for every node u ∈ V − U , there exists
a node w ∈ U such that (u, w) ∈ E). Prove that the following problem is in NP: {hG, ki |
there exists a dominating set U in the undirected graph G such that U contains k nodes}.
(b) Consider a set A = {a1 , . . . an } and a collection B1 , B2 , . . . , Bm of subsets of A (Bi ⊆ A 5pts
for each i). A set H ⊆ A is called hitting set for the collection B1 , B2 , . . . , Bm , if H
contains atleast one element from each Bi i.e. H ∩ Bi is non empty. Show that the
problem of finding a hitting set H ⊆ A of size at most k for B1 , B2 , . . . , Bm is in NP.
2. Show that the problem of finding vertex cover of size k in a tree (undirected graph with no 10pts
cycles) is in P.
3. Given positive integers x1 , x2 , . . . , xn , show that the problem to decide whether numbers can 10pts
be partitioned into two sets S1 and S2 with the same sum, i.e.
X
xi ∈S1
xi =
X
xj
xj ∈S2
is NP-Complete. (Hint: Reduce from SUBSET-SUM.)
4. Given an undirected graph G = (V, E), a feedback set is a set X ⊆ V with the property that
G − X has no cycles. Show that the problem of finding a feedback set of size at most k in an 10pts
undirected graph G is NP-Complete. (Hint: Reduce from VERTEX-COVER.)
5. (Sipser 7.32)
10pts
6. (Sipser 7.54) (Hint: Reduce from 3SAT.)
10pts