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Question: Does there exist a variable assignment that satisfies at leastkclauses ? In this exercise we will construct a kernelization forMAXSAT and show that the kernel has size bounded by O(k2)

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WS 2015/2016 03.02.2016 Exercises to the lecture

Complexity Theory Sheet 13 Prof. Dr. Roland Meyer

M.Sc. Peter Chini Delivery until 10.02.2016 at 12h

Exercise 13.1 (Maximal satisfiability)

Consider the following problem, calledMAXSAT: Input:A formula ϕ=Vm

i=1Ci, where the Ci are the clauses, andk∈N. Parameter:k.

Question: Does there exist a variable assignment that satisfies at leastkclauses ? In this exercise we will construct a kernelization forMAXSAT and show that the kernel has size bounded by O(k2). To this end, recall the following:

• A literalis either a variable or a negated variable.

• A clauseis a disjunction of literals.

• The size of a CNF-formula ϕ is the sum of the numbers of literals used in the clauses ofϕ.

Let (ϕ, k) be a problem instance. Our first reduction step is to delete alltrivial clauses.

A clause inϕ is calledtrivial if it contains a variable and its negation.

a) Show that, by removing all trivial clauses, we can reduce (ϕ, k) to a problem instance (ϕn, k0) so that: k0 ≤kand (ϕn, k0)∈MAXSAT if and only if (ϕ, k)∈MAXSAT. For a further reduction, we look at long clauses: these are clauses that contain more thank0 literals.

b) Prove the following: If ϕn contains more than k0 long clauses, then (ϕn, k0) ∈ MAXSAT.

Hint: We only need one true literal for a clause to be satisfied.

c) Denote by (ϕs,k), where ˆˆ k =k0−t, the instance that we get if we remove all t long clauses from (ϕn, k0). Show that we have: (ϕn, k0)∈MAXSAT if and only if (ϕs,ˆk)∈MAXSAT.

Hence, we obtain an instance that only consists of clauses of size at most k0. Now we show that the size of such an instance is also bounded by the parameter:

d) If (ϕs,k) has more than 2ˆˆ kclauses, then show that (ϕs,ˆk) is inMAXSAT. Hint: Look at any variable assignment and its complement.

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Finally, we can construct a kernelization forMAXSAT:

e) Summarize the reduction steps in an algorithm and show that the size of the kernel (the size of the formulas, obtained from the reduction steps) is bounded byO(k2).

Exercise 13.2 (Karp’s reduction) Consider the problem 3SAT:

Given:A formula ψ in CNF so that the clauses ofψ contain at most 3 literals.

Question: Isψ satisfiable ?

The reduction of Karp maps SAT-instancesϕto 3SAT-instances by the following rule:

ifC=`1∨ · · · ∨`t is a clause ofϕ, it is mapped to the formula:

(`1∨`2∨z1)∧(¬z1∨`3∨z2)∧(¬z2∨`4∨z3)∧ · · · ∧(¬zt−3∨`t−1∨`t) This creates a 3SAT-instance.

a) Ifϕdenotes a SAT-instance andf denotes Karp’s reduction, show that we have:

ϕ∈SAT if and only if f(ϕ)∈3SAT. Hint: You may use the concept of resolution to show this.

Now consider the parametrized problemWEIGHTEDSAT: Given:A formula ϕin CNF and a natural number k.

Parameter:k∈N.

Question: Is there a satisfying assignment forϕthat evaluatesexactly kvariables to 1.

The problem WEIGHTED3SAT is defined similarly.

b) Argue why Karp’s reduction is not a parametrized many-one reduction from WEIGHTEDSAT toWEIGHTED3SAT.

Exercise 13.3 (Reduction toFPT)

Let L, L0 ⊆Σ∗×Nbe parametrized languages so thatLreduces toL0 by a parametrized many-one reduction andL0 isFPT. Show that also Lis FPT.

Delivery until 10.02.2016 at 12h into the box next to 34-401.4

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