程序代写代做代考 algorithm CS 341: Foundations of Computer Science II

CS 341: Foundations of Computer Science II

Prof. Marvin Nakayama

Homework 13 Solutions

1. The Set Partition Problem takes as input a set S of numbers. The question is whether
the numbers can be partitioned into two sets A and A = S −A such that∑

x∈A

x =

x∈A

x.

Show that SET-PARTITION is NP-Complete. (Hint: Reduce SUBSET-SUM .)

Answer: To show that any problem A is NP-Complete, we need to show four things:
(1) there is a non-deterministic polynomial-time algorithm that solves A, i.e., A ∈ NP,
(2) any NP-Complete problem B can be reduced to A,
(3) the reduction of B to A works in polynomial time,
(4) the original problem A has a solution if and only if B has a solution.

We now show that SET-PARTITION is NP-Complete.

(1) SET-PARTITION ∈ NP: Guess the two partitions and verify that the two have
equal sums.

(2) Reduction of SUBSET-SUM to SET-PARTITION : Recall SUBSET-SUM is de-
fined as follows: Given a set X of integers and a target number t, find a subset
Y ⊆ X such that the members of Y add up to exactly t. Let s be the sum of mem-
bers of X. Feed X ′ = X ∪ {s − 2t} into SET-PARTITION . Accept if and only if
SET-PARTITION accepts.

(3) This reduction clearly works in polynomial time.

(4) We will prove that 〈X, t〉 ∈ SUBSET-SUM iff 〈X ′〉 ∈ SET-PARTITION . Note
that the sum of members of X ′ is 2s− 2t.
⇒: If there exists a set of numbers in X that sum to t, then the remaining numbers
in X sum to s − t. Therefore, there exists a partition of X ′ into two such that each
partition sums to s− t.
⇐: Let’s say that there exists a partition of X ′ into two sets such that the sum over
each set is s− t. One of these sets contains the number s−2t. Removing this number,
we get a set of numbers whose sum is t, and all of these numbers are in X.

2. Let

DOUBLE-SAT = { 〈φ〉 | φ is a Boolean formula with two satisfying assignments }.

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Show that DOUBLE-SAT is NP-Complete. (Hint: Reduce 3SAT .)

Answer:
(1) DOUBLE-SAT ∈ NP: Simply guess two different assignments to all variables and
verify that each clause is satisfied in both cases.

(2) Reduction of 3SAT to DOUBLE-SAT : Given a 3cnf-function ψ, create a new
Boolean function ψ′ by adding a new clause (x ∪ x) to ψ, where x is a new variable
not in ψ. Then check if 〈ψ′〉 ∈ DOUBLE-SAT .
(3) This reduction clearly works in polynomial time.

(4) We now prove that the original 3cnf-function 〈ψ〉 ∈ 3SAT iff the new Boolean func-
tion 〈ψ′〉 ∈ DOUBLE-SAT . If the original 3cnf-function ψ is unsatisfiable, then the
new function ψ′ is also unsatisfiable; i.e., 〈ψ〉 6∈ 3SAT implies 〈ψ′〉 6∈ DOUBLE-SAT .
If 〈ψ〉 ∈ 3SAT , then use the same assignment of variables that are in ψ, and we also
have both x = 0 and x = 1 are valid assignments. Thus, there are at least two
satisfying assignments of the augmented 3cnf-formula ψ′, so 〈ψ′〉 ∈ DOUBLE-SAT .

3. Let G represent an undirected graph. Also let

SPATH = { 〈G, a, b, k〉 | G contains a simple path of length at most k from a to b }

and

LPATH = { 〈G, a, b, k〉 | G contains a simple path of length at least k from a to b }.

(a) Show that SPATH ∈ P.

Answer:
The marking algorithm for recognizing PATH can be modified to keep track of
the length of the shortest paths discovered. Here is a detailed description of the
algorithm.

“On input 〈G, a, b, k〉 where m-node graph G has nodes a and b:
1. Place a mark “0” on node a.

2. For each i from 0 to m:

3. If an edge (s, t) is found connecting s marked “i” to an

unmarked node t, mark node t with “i+1”.

4. If b is marked with a value of at most k, accept. Otherwise, reject.

(b) Show that LPATH is NP-Complete. You may assume the NP-completeness of
UHAMPATH , the Hamiltonian path problem for undirected graphs.

Answer:
First, LPATH ∈ NP because we can guess a simple path of length at least k

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from a to b and verify it in polynomial time. Next UHAMPATH ≤P LPATH,
because the following TM F computes the reduction f .

F = “On input 〈G, a, b〉 where graph G has nodes a and b:
1. Let k be the number of nodes of G.

2. Output 〈G, a, b, k〉.

If 〈G, a, b〉 ∈ UHAMPATH , then G contains a Hamiltonian path of length k
from a to b, so 〈G, a, b, k〉 ∈ LPATH . If 〈G, a, b, k〉 ∈ LPATH , then G contains
a simple path of length k from a to b. But G has only k nodes, so the path is
Hamiltonian. Thus, 〈G, a, b〉 ∈ UHAMPATH .

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