template6-steps

a) if 3-SAT ∈ P, then NP = co-NP true b) EXPTIME = PSPACE unknown c) ∀ A,B,C, if A is NP-hard and C ∈ NP and A <p B and B <p C, then B is NP-complete true d) EXPTIME = P false e) P ≠ NP unknown f) 2-SAT ∈ NP-P false
(step 2,4) Input: (G,k)
Define Reduction f: 1. f: graphs x integers → graphs x integers 2. f is poly-time computable
Output: (H,n):
3. ∀ (G,k),
(G,k) ∈ Vertex_cover ↔ f(G,k) = (H,n) ∈ Dominating_Set
Hint: Use 3-SAT, follow the template/6-steps, and think about: True/False ↔ Blue/Red Φ ∈ 3-SAT ↔ ∃ assignment s.t. each variable is true xor false and each clause has one or more true literals (add something else to the “clause” that will be ...). You must state explicitly what reduction f is.
Proof:
(step 1)
1) Dominating_Set = {(H,n): ∃ T ⊆ VH s.t. every
vertex in VH-T is adjacent to a vertex of T} ∈ Σ1,p =
NP 2) Dominating_Set is NP-hard we will show Vertex_Cover <p Dominating_Set
Hint: Use Vertex Cover, follow the template/6-steps, and think about: “edge covered by one vertex in Dominating_Set You must state explicitly what reduction f is.
(step 2,4) Define Reduction f: 1. f: formulas → sets x sets of sets 2. f is poly-time computable Input: Boolean formula Φ in 3-SAT form e.g. Φ = (a or b or not c) AND ((not a) or b or d) AND … Let k be the number of clauses and N be the number of variables Output: (S,C):
25 points
Proof:
(step 1)
1) SET_SPLITTING = {(S,C): ∃σ: S → {blue, red} ∈ Σ1,p = NP
∀Ci ∈ C, ∃ x,y ∈ Ci s.t. σ(x) ≠ σ(y)}
2) SET_SPLITTING is NP-hard we will show 3-SAT <p SET_SPLITTING
Showing (→)
Claim: the coloring σ is a valid coloring
(step 6)
Given a coloring σ, define satisfying assignment x of φ: Without loss of generality, assume that the color of the “False” element is red. In that case, let each red literal be false and each blue literal be true
Test 3 Solutions
Computer Science 500: Spring 2000 Wednesday, April 26
This test is closed-book, 75 minutes Please make sure that you understand what the questions ask for. Do not hesitate to ask Pete for clarification. Please check your work and be as precise as reasonably possible.
(step 5)
Showing (→)
Given a cover S of G of size k, define T: Let = S
Claim: T is a dominating set for H of size n
(step 6)
Showing (←)
Given a dominating set T of size n define cover S: Without loss of generality, we can assume that T contains none of the vertices {ze}. ( If it does, we can replace with either of the vertices of e.) Given T of the above form, let S = T
Showing (←)
Claim: the assignment x is a consistent and satisfying assignment
Problem 4: (True or False or Unknown/not yet proven or dis-proven)
3 points each
25 points
Problem 2: Define Dominating_Set = {(H,n): graph H = (VH,EH) and there is a subset of vertices V’ of size n such that every vertex in VH-V’ is adjacent to a vertex of V’} Prove that Dominating_Set is NP-complete
3. ∀ Φ,
Φ ∈ 3-SAT ↔ f(Φ) = (S,C) ∈ SET_SPLITTING
∀ φ, f (φ) = (S,C) where S = all 2N possible literals and a special element called “F” C = {Cv1 … CvN, Cc1, … Cck} where: ∀ i: 1 < i < N, Cvi = {xi, xi} ∀ j: 1 < j < k, Ccj = {λi1, λi2, λi3} where clausej = (λi1 ∨ λi2∨ λi3)
(step 3)
(step 5)
Given a satisfying assignment x of φ, define coloring σ: Color the “False” element red. Color each false literal red. Color each true literal blue.
F) Lf = {M: L(M) ∈ co-NP} Yes. RT case 1 and so the language is not recognizable G) Lg = {M: L(M) is recognizable} No. The set is all T.M.’s H) Lh = {M: M accepts every input} Yes. RT case 2 and so the language is not co-recognizable
Claim: S is a vertex cover of G of size k
Problem 3: Define SET-SPLITTING = {(S,C): S is a finite set and C = {C1, … Ck} is a collection of subsets of S such that the elements of S can be colored red or blue so that no Ci has all its elements colored the same color.} Prove that SET-SPLITTING is NP-complete
∀ (G,k), f (G,k) = (H,n) where
(step 3)
H = a graph, s.t. for every edge e = (u,v) of G, H also has that edge as well as a new vertex ze and two new edges (u, ze) and (v, ze) n=k
4pts each
For each language, state: 1) Does Rice’ Theorem apply (directly) to the language?
Problem 1:
2a) If your answer to (1) is “yes”, what fact does Rice’s Theorem imply? 2b) If your answer to (1) is “no”, why does Rice’s Theorem not apply? A) La = {M: L(M) is NP-complete} Yes - φ is not NP-complete and so the language is not co-recognizable (case 2) B) Lb = {M: M halts on all of M’s inputs} No. Membership does not depend only on the language computed (e.g. A rejects all inputs, B DNH on all inputs) C) Lc = VERTEX_COVER No. Not a set of Turing Machines D) Ld = {M: L(M) ⊆ Lu AND L(M) ⊆ Lu and L(M) is decidable} Yes. RT case 1 so the language is not recognizable E) Le = {M: L(M) = ATM} Yes. RT case 2 and so the language is not co-recognizable