# Алгоритмы для NP-трудных задач, осень 2009: NP-полные задачи

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• NP- 3: NP-

.

Computer Science http://logic.pdmi.ras.ru/infclub/

. (Computer Science ) 3. NP- 1 / 42

http://logic.pdmi.ras.ru/~infclub/

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2 NP-

3

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• 1

2 NP-

3

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• 1

2 NP-

3

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• 1

2 NP-

3

. (Computer Science ) 3. NP- 3 / 42

• , , , , ( n, n2,n3) . (,, ) : n! n n , n nn2 , , , s t.

. (Computer Science ) 3. NP- 4 / 42

• , , , , ( n, n2,n3) .

(,, ) : n! n n , n nn2 , , , s t.

. (Computer Science ) 3. NP- 4 / 42

• , , , , ( n, n2,n3) . (,, ) : n! n n , n nn2 , , , s t.

. (Computer Science ) 3. NP- 4 / 42

• , , . , 2n , . , , .

. (Computer Science ) 3. NP- 5 / 42

• , , . , 2n , .

, , .

. (Computer Science ) 3. NP- 5 / 42

• , , . , 2n , . , , .

. (Computer Science ) 3. NP- 5 / 42

• (satisfiabilityproblem, SAT) :

(x y z)(x y)(y z)(z x)(x y z) .

.

. (Computer Science ) 3. NP- 6 / 42

• (satisfiabilityproblem, SAT) :

(x y z)(x y)(y z)(z x)(x y z) .

.

. (Computer Science ) 3. NP- 6 / 42

• (satisfiabilityproblem, SAT) :

(x y z)(x y)(y z)(z x)(x y z) .

.

. (Computer Science ) 3. NP- 6 / 42

• (searchproblem). (instance) I ( , ; ) (solution) S(, ; ). , . S I .

. (Computer Science ) 3. NP- 7 / 42

• (searchproblem).

(instance) I ( , ; ) (solution) S(, ; ). , . S I .

. (Computer Science ) 3. NP- 7 / 42

• (searchproblem). (instance) I ( , ; ) (solution) S(, ; ). , .

S I .

. (Computer Science ) 3. NP- 7 / 42

• (searchproblem). (instance) I ( , ; ) (solution) S(, ; ). , . S I .

. (Computer Science ) 3. NP- 7 / 42

• , , I S |I | . , S I , (I , S) = true.

. (Computer Science ) 3. NP- 8 / 42

• (traveling salesman problem, TSP) n 1, . . . , n n(n 1)/2 , b. , ( ) b, , . , (1), . . . , (n),

d(1),(2) + d(2),(3) + + d(n),(1) b .

. (Computer Science ) 3. NP- 9 / 42

• (traveling salesman problem, TSP) n 1, . . . , n n(n 1)/2 , b.

, ( ) b, , . , (1), . . . , (n),

d(1),(2) + d(2),(3) + + d(n),(1) b .

. (Computer Science ) 3. NP- 9 / 42

• (traveling salesman problem, TSP) n 1, . . . , n n(n 1)/2 , b. , ( ) b, , .

, (1), . . . , (n),

d(1),(2) + d(2),(3) + + d(n),(1) b .

. (Computer Science ) 3. NP- 9 / 42

• (traveling salesman problem, TSP) n 1, . . . , n n(n 1)/2 , b. , ( ) b, , . , (1), . . . , (n),

d(1),(2) + d(2),(3) + + d(n),(1) b .

. (Computer Science ) 3. NP- 9 / 42

• , ? , . TSP, b, ?

. (Computer Science ) 3. NP- 10 / 42

• , ?

, . TSP, b, ?

. (Computer Science ) 3. NP- 10 / 42

• , ? , .

TSP, b, ?

. (Computer Science ) 3. NP- 10 / 42

• , ? , . TSP, b, ?

. (Computer Science ) 3. NP- 10 / 42

• : b T ,

(i ,j)T

dij b .

, , , . .

. (Computer Science ) 3. NP- 11 / 42

• : b T ,

(i ,j)T

dij b .

, , , . .

. (Computer Science ) 3. NP- 11 / 42

• : b T ,

(i ,j)T

dij b .

, , , .

.

. (Computer Science ) 3. NP- 11 / 42

• : b T ,

(i ,j)T

dij b .

, , , . .

. (Computer Science ) 3. NP- 11 / 42

• ,

(independent set problem) g g , , . (vertex cover problem) b, b , ( b , ). (clique problem) g g , .

. (Computer Science ) 3. NP- 12 / 42

• ,

(independent set problem) g g , , .

(vertex cover problem) b, b , ( b , ). (clique problem) g g , .

. (Computer Science ) 3. NP- 12 / 42

• ,

(independent set problem) g g , , . (vertex cover problem) b, b , ( b , ).

(clique problem) g g , .

. (Computer Science ) 3. NP- 12 / 42

• ,

(independent set problem) g g , , . (vertex cover problem) b, b , ( b , ). (clique problem) g g , .

. (Computer Science ) 3. NP- 12 / 42

• , (longest path problem)? , s t, g . s t g .

. (Computer Science ) 3. NP- 13 / 42

• , (longest path problem)?

, s t, g . s t g .

. (Computer Science ) 3. NP- 13 / 42

• , (longest path problem)? , s t, g .

s t g .

. (Computer Science ) 3. NP- 13 / 42

• , (longest path problem)? , s t, g . s t g .

. (Computer Science ) 3. NP- 13 / 42

• (knapsack problem): w1, . . . ,wn N v1, . . . , vn N n . W g . , W , g . , O(nW ). , W , logW .

. (Computer Science ) 3. NP- 14 / 42

• (knapsack problem): w1, . . . ,wn N v1, . . . , vn N n . W g .

, W , g . , O(nW ). , W , logW .

. (Computer Science ) 3. NP- 14 / 42

• (knapsack problem): w1, . . . ,wn N v1, . . . , vn N n . W g . , W , g .

, O(nW ). , W , logW .

. (Computer Science ) 3. NP- 14 / 42

• (knapsack problem): w1, . . . ,wn N v1, . . . , vn N n . W g . , W , g . , O(nW ). , W , logW .

. (Computer Science ) 3. NP- 14 / 42

• , , , ., 12 IIIIIIIIIIII . , , , (unaryknapsack). , .

. (Computer Science ) 3. NP- 15 / 42

• , , , .

, 12 IIIIIIIIIIII . , , , (unaryknapsack). , .

. (Computer Science ) 3. NP- 15 / 42

• , , , ., 12 IIIIIIIIIIII .

, , , (unaryknapsack). , .

. (Computer Science ) 3. NP- 15 / 42

• , , , ., 12 IIIIIIIIIIII . , , , (unaryknapsack).

, .

. (Computer Science ) 3. NP- 15 / 42

• , , , ., 12 IIIIIIIIIIII . , , , (unaryknapsack). , .

. (Computer Science ) 3. NP- 15 / 42

• . , W g . , (subset sum problem), W . , . ?

. (Computer Science ) 3. NP- 16 / 42

• . , W g .

, (subset sum problem), W . , . ?

. (Computer Science ) 3. NP- 16 / 42

• . , W g . , (subset sum problem), W . , . ?

. (Computer Science ) 3. NP- 16 / 42

• NP-

1

2 NP-

3

. (Computer Science ) 3. NP- 17 / 42

• NP-

(NP-) ( P)

3-SAT 2-SAT, Horn SAT

. (Computer Science ) 3. NP- 18 / 42

• NP-

, , , , . . , . : , .

. (Computer Science ) 3. NP- 19 / 42

• NP-

, , , , . .

, . : , .

. (Computer Science ) 3. NP- 19 / 42

• NP-

, , , , . . , .

: , .

. (Computer Science ) 3. NP- 19 / 42

• NP-

, , , , . . , . : , .

. (Computer Science ) 3. NP- 19 / 42

• NP-

P NP

NP .P , .

. (Computer Science ) 3. NP- 20 / 42

• NP-

P NP

NP .

P , .

. (Computer Science ) 3. NP- 20 / 42

• NP-

P NP

NP .P , .

. (Computer Science ) 3. NP- 20 / 42

• NP-

, P =NP, ? , , ( , )? (reductions), . , , , . , , , NP: , NP. , , P =NP, .

. (Computer Science ) 3. NP- 21 / 42

• NP-

, P =NP, ? , , ( , )?

(reductions), . , , , . , , , NP: , NP. , , P =NP, .

. (Computer Science ) 3. NP- 21 / 42

• NP-

, P =NP,

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