Universität Koblenz-Landau FB 4 Informatik
Prof. Dr. Viorica Sofronie-Stokkermans∗1 31.01.2018
M.Ed. Dennis Peuter∗2
Exercises for Advances in Theoretical Computer Science Exercise Sheet 13
Due at 05.02.2018, 10:00 s.t.
Exercise 13.1
Consider the following propositional logic formula:
F = (P∨ ¬Q∨ ¬(R∨ ¬S))∧(Q∨ ¬R∨S)
Apply Steps 1-4 on pages 15-19 of the slides from 31.01.2018 (complexity3.pdf) to this formula for computing the formula in 3-CNF associated to F (formula which is satisable i F is satisable).
Exercise 13.2
a) Draw the complete graphs with 3,4 and 5vertices.
b) Consider the undirected graphG= (V, E), whereV ={a, b, c, d, e, f}and E={(a, b),(a, c),(a, e),(a, f),(b, c),(b, d),(b, e),(c, e),(c, f)}.
(Note that in an undirected graph the edge(x, y)is identical to the edge(y, x), i.e. they are not ordered pairs but sets (or 2-multisets) of vertices.)
1) Draw the graphG.
2) DoesGhave a clique of size 3? DoesGhave a clique of size 4? DoesGhave a clique of size 5?
Exercise 13.3
Consider the following formula in 3-CNF:
F = (¬P1∨P2∨P3)∧(P1∨ ¬P2∨P4)∧(P2∨ ¬P3∨ ¬P4) a) Is the formula satisable? If yes then give a satisfying assignment.
b) Starting from F construct the pair (GF, kF) as explained on pages 25-26 of the slides from 31.01.2018 (complexity3.pdf).
c) Has the graphGF a clique of sizekF? If so indicate such a clique and reconstruct from it an assignment which makesF true.
Exercise 13.4
Consider the following problem:
SET PACKING = {(C, l)|C ={S1, . . . , Sn}, everySi is a nite set and there exists D⊆C withl elements such that the elements ofD are pairwise disjoint}
a) Prove that SET PACKING∈NP.
For every pair (G, k), whereG= (V, E) is an undirected graph with vertices{v1, . . . , vm} and edges in E we associate the pair (C, l), where l = k and C = {S1, . . . , Sm}, with Si ={(vi, vj),(vj, vi)|(vi, vj)6∈E}.
b) Estimate the time needed for constructing(C, l) from (G, k). Prove:
c) Si∩Sj 6=∅if and only if there is no edge between vi andvj inG.
d) IfG0 is a clique ofGwith size k, with vertices {vi1, . . . , vik}then the sets in D={Si1, . . . , Sik}are pairwise disjoint.
e) Ghas a clique of sizeki there exists a subset DofCwithlelements such that the elements ofD are pairwise disjoint.
f) Infer that Clique (the problem whether a graph has a clique of size k) can be poly- nomially reduced to SET PACKING.
g) Is SET PACKING NP-complete? Justify your answer.
Exercise 13.5
A zoo acquires for the rst time n animals x1, . . . , xn. A list L of enemies is provided, containing sets{xi, xj}consisting of two animals which cannot be placed in the same cage.
The zoo has k cages.
We are interested in the problem of deciding whether the animals can be placed in the zoo such that they are all safe. Thus,
ZOO = {(L, n, k)| thenanimals with enemy list Lcan be placed onk cages such that they are all safe. } (1) One of the two triples (Li, ni, ki) is an instance of ZOO. Which is this?
({{1,2},{1,3},{2,3},{3,4}},4,3) ({{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}},4,3)
(2) Let f be the function which associates with every undirected graph G = (V, E) whereV ={v1, . . . , vm} the tuple(LG, nG, kG) where:
LG=E, nG=m, kG = 3
Prove thatf denes a polynomial reduction of 3-colorability toZOO.
(3) Is ZOO in NP? Briey justify your answer (you do not need to construct a Turing machine for this).
(4) In the lecture we will study thek-colorability problem:
k-colorability ={G|Gundirected graph that can be colored with k colors}. We know that the 3-colorability problem is an NP-complete problem.
Prove or refute the following: ZOO is an NP-complete problem.
∗1 B 225 sofronie@uni-koblenz.de https://userpages.uni-koblenz.de/~sofronie/
∗2 B 223 dpeuter@uni-koblenz.de https://userpages.uni-koblenz.de/~dpeuter/
If you want to submit solutions, please do so until 05.02.2018, 10:00 s.t. via the cardboard box in the shelf in room B 222 or via e-mail (with Homework ACTCS in the subject) to dpeuter@uni-koblenz.de.
