Universität Koblenz-Landau FB 4 Informatik
Prof. Dr. Viorica Sofronie-Stokkermans∗1 12.07.2021 M.Ed. Dennis Peuter∗2
Exercises for Advances in Theoretical Computer Science Exercise Sheet 13
Due at 20.07.2021, 12:00 s.t.
Exercise 13.1
Consider the following alphabet and languages:
Σ ={a, b,0,1};
L1 ={anbn|n≥0};
L2 ={1}.
Give a functionf : Σ∗ →Σ∗ which polynomially reducesL1 to L2. We require that for all w∈Σ∗\L1,f(w) = 0.
Exercise 13.2
a) We know that SAT is NP-complete. In the previous exercise we saw that satis- fiability of formulae in DNF can be checked in polynomial time, so DNF-SAT = {F | F is a satisfiable formula of propositional logic in disjunctive normal form} is in complexity class P.
If we could construct a polynomial reduction of SAT to DNF-SAT (i.e. if we could prove that SAT≺pol DNF-SAT) then we could show that P = NP.
Formulae in propositional logic can be transformed to DNF using distributivity:
A∧(B1∨ · · · ∨Bk)≡(A∧B1)∨ · · · ∨(A∧Bk).
Why does this not lead to a polynomial reduction?
b) Consider the following propositional logic formula:
F = (P∨ ¬Q∨ ¬(R∨ ¬S))∧(Q∨ ¬R∨S)
Apply Steps 1-4 on pages 32-36 of the slides from 13.07.2021 (complexity2.pdf) to this formula for computing the formula in 3-CNF associated to F (formula which is satisfiable iff F is satisfiable).
Exercise 13.3
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?
c) Consider the following formula in 3-CNF:
F = (¬P1∨P2∨P3)∧(P1∨ ¬P2∨P4)∧(P2∨ ¬P3∨ ¬P4) 1) Is the formula satisfiable? If yes then give a satisfying assignment.
2) Starting fromF construct the pair(GF, kF)as explained on pages 42-43 of the slides from 13.07.2021 (complexity2.pdf).
3) Does the graphGF have a clique of sizekF? If so indicate such a clique and recon- struct from it an assignment which makesF true.
Exercise 13.4
Consider the following problem:
SET PACKING = {(C, l)|C ={S1, . . . , Sn}, everySi is a finite 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).
c) Prove that Ghas a clique of sizekif and only if there exists a subset DofC withk elements such that the elements of Dare pairwise disjoint.
d) Infer thatClique (the problem whether a graph has a clique of size k) can be poly- nomially reduced to SET PACKING.
e) Is SET PACKING NP-complete? Justify your answer.
∗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 20.07.2021, 12:00 s.t. via e-mail (with “Homework ACTCS” in the subject) todpeuter@uni-koblenz.de.
