1. Prabhas Chongstitvatana
NP-complete proofs
The circuit satisfiability proof of NP-completeness relies on a direct proof that L p CIRCUIT-SAT for every L  NP.
Lemma 36.8
If L is a language such that L’ p L for some L’  NPC, then L is NP-hard. Moreover, if L  NP, then L  NPC.
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2. Prabhas Chongstitvatana
Prove L  NP
Select a known NP-complete language L’
Descrive an algorithm that compute f mapping every instance of L’ to an instance of L.
Prove f satisfies x  L’ iff f(x)  L x  {0,1}*
Prove that the algorithm computing f runs in polynomial time.
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3. Prabhas Chongstitvatana
NP-complete problems
SAT = { <Phi> | Phi is a satisfiable boolean formula }
Theorem 36.9
SAT  NP-complete
Proof : SAT  NP (show a certificate can be verified in polynomial time),
CIRCUIT-SAT p SAT
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4. Prabhas Chongstitvatana
3-CNF-SAT
CNF conjunctive normal form (AND of Ored-clauses) to simplify reduction algorithm
3-CNF-SAT  NP, SAT  p 3-CNF-SAT
CLIQUE = { <G,K> | G is a graph with a clique of size k }
A clique in an undirected graph G = (V,E) is a subset V’  V of vertices each pair of which is connected by an edge in E (a clique is a complete subgraph of G).
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5. Prabhas Chongstitvatanau w x v y z
CLIQUE  NP; 3-CNF-SAT p CLIQUE
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6. Prabhas Chongstitvatana
VERTEX-COVER = { <G,k> | graph G has vertex cover of size k }
A vertex cover of an undirected graph G = (V,E) is a subset V’ V such that if (u,v) E, then u  V’ or v  V’ (or both).
VERTEX-COVER  NP;
CLIQUE p VERTEX-COVER
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7. Prabhas Chongstitvatana
SUBSET-SUM = { <S, t> | there exists a subset S’  S such that t =
Example : if S = {1, 4, 16, 64, 256, 1040, 1041, 1093, 1284, 1344} and t = The subset S’ = {1, 6, 64, 256, 1040, 1093, 1284 } is a solution.
SUBSET-SUM  NP;
VERTEX-COVER p SUBSET-SUM
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8. Prabhas Chongstitvatana
HAM-CYCLE = {<G> | G is a hamitonian graph }
HAM-CYLE  NP;
3-CNF-SAT p HAM-CYCLE
TSP = {<G,c,k> | G = (V,E) is a complete graph, c is a function from V  V  Z, k  Z, and G has a traveling-saleman tour with cost at most k }
An integer cost c(i,j) to travel from city i to city j. A tour is a hamiltonian cycle.
TSP  NP; HAM-CYCLE p TSP
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9. Prabhas Chongstitvatana
CIRCUIT-SAT
3-CNF-SAT
SAT
TSP
HAM-CYCLE
CLIQUE
VERTEX-COVER
SUBSET-SUM
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10. Prabhas Chongstitvatana
Unsolvable
A set with as many elements as the integers is called countably infinite.
Not every infinite set is countable.
Let’s show that there are computationally unsolvable problems. The subset of problems can be described by boolean functions on the integers; f such that f(n) is 0/1.
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11. Prabhas Chongstitvatana
Suppose the set of all such functions is countably infinite. There is a correspondence between each function and each integer.
. . .
f i n
f i ( n )
By diagonalization, we can show that at least one boolean function cannot be in this list.
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12. Prabhas Chongstitvatana
If every program is a finite string of symbols, each chosen from a single finite alphabet, then it is possible to show that the set of all programs is countably infinite.
So there is more problems than there are programs to solve them. Thus, at least one function is not describable as the output of any program, so it is computationally unsolvable.
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13. Prabhas Chongstitvatana
Example : A simple system such as Presburger arithmetic, is a formal system of the positive integers together with addition and equality alone. Deciding whether a statement in this system is true is in the length of the statement.
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14. Prabhas Chongstitvatana
Unsolvable
relativized unsolvable (oracle machines)
Solvable
provably infeasible (Presburger arithmetic)
probably infeasible (satisfiability)
feasible
hard (normal optimization problems)
easy (normal computer science problems)
randomized easy (primality testing)
really easy (sorting)
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15. Prabhas Chongstitvatana
Solving hard problems
Relax the problem -- use approx. algo.
Relax the method -- use probabilistic algo and give up total correctness
Relax the architecture -- use parallel
Relax the machine -- use analog computer ?
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