CS 470/570 Exam 6 Spring 2017 Solution

Transcription

1 CS 470/570 Exam 6 Spring 2017 Solution CS 470 Score is based on your best 4 out of 6 problems. CS 570 Score is based on your best 5 out of 6 problems. Extra credit will be awarded if you can solve additional problems correctly within the allotted time. 1. The Venn diagram below shows that X is a subset of Y, X is disjoint from Z, and Y intersects Z. Draw a Venn diagram that shows the relationships between these seven sets: P, NP, NP-hard, NP-complete, co-np, co-np-hard, and co-np-complete. To make the diagram as general as possible, do not assume that any two sets are equal unless it is known that they are definitely equal. X Y Z undecidable NP-hard NP-complete co-nphard co-npcomplete NP P co-np

2 2. This graph G represents the friendship relations of some individuals on a particular social network. Every vertex starts with a distinct letter, so you can abbreviate each vertex by just its first letter. Katniss Morpheus Peeta Trinity Rue Neo Albus Hermione Luna Dobby Squirtle Buzz Eevee Woody Charizard Jessie a. Write the vertices of a maximum clique: {Albus, Hermione, Luna, Dobby} b. Write the vertices of a maximum independent set: {Katniss, Trinity, Albus, Eevee, Jessie} c. Write the vertices of a minimum vertex cover: Vertices not in max independent set = {Rue, Peeta, Morpheus, Neo, Hermione, Luna, Dobby, Squirtle, Charizard, Woody, Buzz} d. Write the vertices of a Hamiltonian cycle in order: Katniss Rue Peeta Albus Hermione Trinity Neo Morpheus Jessie Buzz Woody Dobby Luna Squirtle Eevee Charizard Katniss

3 3. Prove that the Set Cover problem defined below is NP-complete. Set Cover Input: Collection C of subsets of a finite universal set U, and integer k. Question: Is there a subset S C such that S k and every element in U belongs to at least one member of S? If so, such a set S is called a set cover for U. Example: Let U = {1,2,3,4,5}, C = {{1,2,3},{1,4},{2,4},{2,5},{3,4,5}}, and k = 2. The answer is yes with solution S = {{1,2,3},{3,4,5}}. Set Cover NP: Guess a subset S of C. Verify that S k. Take the union of all the sets in S, and verify that this union equals U. Set Cover NP-hard: Reduce Vertex Cover p Set Cover. Vertex Cover input is graph G = (V,E) and integer k. Let U = E and C = {Xv v V} where Xv = {e E e is incident to v}. So G has a vertex cover S of size k iff C contains a set cover {Xv v S} for U of size k. Example: G a b d e k=2 2 4 c Min vertex cover of G = {b,d}. U = {1,2,3,4,5}. C = {Xa,Xb,Xc,Xd,Xe} = {{1},{1,2,3},{2,4},{3,4,5},{5}}. k=2. Min set cover of U = {Xb,Xd} = {{1,2,3},{3,4,5}}.

4 4. Prove that the Dominating Set problem defined below is NP-complete. Dominating Set Input: Graph G = (V,E) and integer k. Question: Is there a subset S V such that S k and for every x V S there exists some y S for which (x,y) E? If so, such a set S is called a dominating set of G. Example: Let G be the graph from problem 2, and let k = 4. The answer is yes, and one possible dominating set is S = {Rue, Neo, Squirtle, Buzz}. Dominating Set NP: Guess a subset S of V. Verify that S k. For every vertex x, verify that either x S or x has some neighbor y S. Dominating Set NP-hard: Reduce Vertex Cover p Dominating Set. Vertex Cover input is graph G = (V,E) and integer k. Let G = (V,E ) where V = V {xy (x,y) E} and E = E {(x,xy), (y,xy) (x,y) E}. In any dominating set of G, we can replace any vertex xy by either x or y. So G has a vertex cover S of size k iff G has a dominating set S of size k. Example: G a b d e k=2 Min vertex cover of G = {b,d}. c ab bd de G a b d e k=2 bc Min dominating set of G = {b,d}. c cd

5 5. Prove that the Degree-Bounded Spanning Tree (DBST) problem defined below is NP-complete. Degree-Bounded Spanning Tree Input: Graph G = (V, E) and integer k. Question: Is there a subset S E such that subgraph (V,S) forms a spanning tree of G and for all x V, the number of edges incident to x in S is at most k? Example: Let G be the graph from problem 2, and let k = 3. The answer is yes, and one possible solution is when S consists of the red edges in G. DBST NP: Guess a subset S of E. Use DFS or BFS to verify that subgraph (V,S) is connected and has no cycles. For every vertex x V, verify that x has at most k incident edges in subgraph (V,S). DBST NP-hard: Reduce Hamiltonian Path p DBST. Hamiltonian Path input is graph G = (V,E). Let k = 2. G has a Hamiltonian path iff G has a spanning tree (V,S) in which each vertex has degree 2.

6 6. Define a pattern as a string over alphabet {0,1,x}. Say pattern S covers binary string B if S can be transformed into B by replacing each x with either 0 or 1. Example: pattern 0xx1 covers binary strings 0001, 0011, 0101, and Prove that the Pattern Cover problem defined below is NP-complete. Input: List of m patterns [S1,, Sm] each having length n. Question: Does there exist any binary string of length n that is not covered by at least one of the given m patterns? Example: The patterns [00x, x11, 1x0] do not cover these two binary strings: 010, 101. Pattern Cover NP: Guess a binary string B of length n. For 1 j m, verify that Sj[k] is the negation of B[k] for at least one position k where 1 k n. Pattern Cover NP-hard: Reduce CNF Satisfiability p Pattern Cover. CNF Satisfiability input is a CNF formula with clauses C1,, Cm and variables a1,, an. Transform each clause Cj into pattern Sj as follows: If Cj contains term ak then let Sj [k]=0, else if Cj contains term aa kk then let Sj [k]=1, else let Sj [k]=x. The CNF formula with clauses C1,, Cm is satisfiable iff some binary string B is not covered by S1,, Sm. To obtain a satisfying truth assignment, let each variable ak = B[k]. Example: CNF formula = (a b ) (a c ) (b c) (b d) (c d ) with variables {a,b,c,d}. Solution has a = b = false, c = d = true. List of patterns = [01xx, 1x1x, x10x, x0x0, xx01]. Does not cover the pattern B = Extra credit (mainly for fun): List the franchises (book, film, gaming, and/or tv) whose characters appear in the graph G of problem 2. Hunger Games The Matrix Harry Potter Pokemon Toy Story