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1. A 2-CNF stands for a conjunction normal form boolean formula in which each clause consists 2

literals. For example, the formula (x1∨x2)∧(x3∨¬x2) is a 2-CNF. The 2-SAT problems asks given

a 2-CNF formula ϕ, decide whether it is satisfiable or not. Give a polynomial-time algorithm for

2-SAT:

(a) The boolean formula p ⇒ q is false only when p = True and q = False.

Show that (x1 ∨ x2) iff (¬x1 ⇒ x2) ∨ (¬x2 ⇒ x1) .

(b) Show that if (p ⇒ q) ∧ (q ⇒ r), then if the formula is evaluated for True

when p = True then it must be that r = True.

(c) Think of the implication relation is an edge between two literals and build

an implication directed graph G for the boolean formula ϕ.

(d) Prove that ϕ is satisfied iff for all xi we have that xi and ¬xi are not in the

same strongly connected component of G.

(e) Use an algorithm to detect a strongly connect component in the implication

graph to solve the 2-SAT problem.

(f) If ϕ is satisfiable, find a satisfying assigment.

2. Given a weighted graph G = (V, E) and an integer k. The Longest Path problem asks whether

there exists a simple path of length greater or equal to k. The Shortest Path problem asks whether

there exists a simple path of length smaller or equal to k.

(a) Show a polynomial reduction from the Hamiltonian Path problem to the Longest Path problem.

(b) Show a polynomial reduction from the Hamiltonian Path problem to the Shortest Path problem.

3. Show that the following three problems are polynomial time reducible to each other.

• Set-Cover: Given a collection of sets, and a number k, the Set-Cover problem asks if there

are at most k sets from the collection of sets such that their union contains every element in

the union of all sets.

• Hitting-Set: Given a collection of sets, and a number k, the Hitting-Set problem asks if are

there at most k elements of the sets such that there is at least one element from each set? (PS:

In class Prof. Gafni actually proved the Hitting-Set to be NP-complete, rather than Set-Cover.

In this problem you see that Set-Cover is also therefore NP-complete)

• Dominating-Set: Given an undirected graph G, and a number k, the Dominating-Set problem asks if there is a subset of vertices of size ≤ k such that every vertex in graph is either

in the subset or has a neighbor that is in the subset.

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(a) Show a polynomial reduction from Hitting-Set to Dominating-Set.

(b) Show a polynomial reduction from Dominating-Set to Hitting-Set.

(c) Show a polynomial reduction from Set-Cover and Dominating-Set.

(d) Show a polynomial reduction from Dominating-Set to Set-Cover.

4. The degree bounded spanning tree problem is given by a undirected graph G = (V, E), and a

number k. The problem is to decide whether or not G has a spanning tree T such that each vertex

in T has a degree of at most k. Prove that the degree bounded Spanning Tree is NP-hard. (Hint:

reduce a well known NP-complete problem you have seen to the degree bounded Spanning Tree

problem)

! Homework assignments are STRICTLY due on the exact time indicated. Please submit a hard copy

of your homework solution with your Name, Bruin ID, Discussion Number, clearly indicated on

the first page. If your homework consists of multiple pages, please staple them together. Email

attachments or other electronic delivery methods are not acceptable.

! We recommend to use LATEX, LYX or other word processing software for submitting the homework.

This is not a requirement but it helps us to grade the homework and give feedback. For grading,

we will take into account both the correctness and the clarity. Your answer are supposed to be in

a simple and understandable manner. Sloppy answers are expected to receiver fewer points.

! Unless specified, you should justify your algorithm with proof of correctness and time complexity.

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