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Definition 0.1. The p-Set-Packing problem is :
Instance : a family F and k ∈ N
Parameter : k
Problem : Decide whether F contains at least k pairwise disjoint sets.
Proposition 0.2. Show that p-Set-Packing is A[1]-complete.
Definition 0.3. An homomorphism from a structure A to B is a mapping that preserve the
membership in all relations. A strong homomorphism also preserve non-membership. An embedding
is an isomorphism of substructures. A strong embedding is an embedding wich is also a strong
homomorphism.
The p-X problem is :
Instance : Structures A and B
Parameter : ||A||
Problem : Decide whether there exist a X from A to B.
Proposition 0.4. All those p-X problems are A[1]-complete.
Definition 0.5. The p-short-PCP is :
Instance : Pairs of strings over Σ (a1 , b1 ) . . . (an , bn ), k ∈ N
Parameter : k
Problem : Decide if there are i1 , . . . , ik ∈ [1, n] such that ai1 .ai2 . . . ain = bi1 .bi2 . . . bin
Proposition 0.6. All p-short-PCP is A[1]-complete.
Definition 0.7. The p-WD[φ] is :
Instance : A structure A and k ∈ N
Parameter : k
Problem : Decide whether A |= ϕ(S) for a relation S of size k.
Define Clique, vertex-cover, dominating-set and hitting-set that way.
Definition 0.8. a circuit C is k-satisfiable if it is satisfied by a tuple of weight k, ie with k true
variables.
The p-WSAT is :
Instance : A circuit C and k ∈ N
Parameter : k
Problem : Decide whether C is k satisfiable.
Definition 0.9. The p-Bounded-NTM-Halt problem is :
Instance : A non-deterministic Turing machine M , n ∈ N in unary, and k ∈ N
Parameter : k
Problem : Decide whether M accepts the empty string in at most n steps and using at most k
non-deterministic steps.
show that p − W SAT ≡FPT p − Bounded − N T M − Halt.