Aaex7 group2(中英夾雜)
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1) The document discusses problems related to complexity classes P and NP. It shows that several problems are NP-complete, including the Hamiltonian cycle problem, subgraph isomorphism problem, 0-1 integer programming problem, and Hamiltonian path problem. 2) It provides algorithms and reductions to prove several problems are NP-complete, such as reducing Hamiltonian cycle to the subgraph isomorphism problem and reducing 3-SAT to the 0-1 integer programming problem. 3) It also discusses properties of complexity classes P and NP, such as showing P is closed under certain operations and contained within NP intersect co-NP.
![Q1
• Kleene star L* ∈ P
If we can use dynamic programming,
state is that DP[i][j] whether a
substring[i][j] deciding L ∈ P.
• Space O(n2)
• Time O(n3)
2014/1/10
Q1 Q8
7](https://image.slidesharecdn.com/aaex7group2-140110202415-phpapp02/85/Aaex7-group2-7-320.jpg)
Aaex7 group2(中英夾雜)
- 1. Advanced Algorithms Exercise 6 Group 2 陳右緯、楊翔雲、蔡宗衛、BINAYA KAR、林淑慧 2014/1/10 Q1 Q8 1
- 2. Q1 • Show that the class P, viewed as a set of languages is closed under union, intersection, concatenation, complement and Kleene star. That is, if L1, L2 ∈ P, then, L1 ∪ L2 ∈ P, etc. 2014/1/10 Q1 Q8 2
- 3. Q1 • Union L1 ∪ L2 ∈ P f(L) { return f1(L) or f2(L); } • Intersection L1 ∩ L2 ∈ P f(L) { return f1(L) and f2(L); } 2014/1/10 Q1 Q8 3
- 4. Q1 • Concatenation L1 L2 ∈ P f(L) { for i = 0 to L.length L1 = L.substr(0, i) L2 = L.substr(i+1) if(f1(L1) and f2(L2)) return true; return false; } 2014/1/10 Q1 Q8 4
- 5. Q1 • Complement ¬ L1 ∈ P f(L) { return ¬ f1(L); } 2014/1/10 Q1 Q8 5
- 6. Q1 • Kleene star L* ∈ P f(L) { for i = 0 to L.length L1 = L.substr(0, i) L’ = L.substr(i+1) if(f1(L1) and f(L’)) return true; return false; } 2014/1/10 Q1 Q8 6
- 7. Q1 • Kleene star L* ∈ P If we can use dynamic programming, state is that DP[i][j] whether a substring[i][j] deciding L ∈ P. • Space O(n2) • Time O(n3) 2014/1/10 Q1 Q8 7
- 8. Q2 • Show that if the decision version of the Hamiltonian cycle is polynomial-time solvable then so is the problem of finding a Hamiltonian cycle. • 如果有一個演算法可以在多項式時間 內驗證漢米頓環道，就可以在多項式 時間內找到漢米頓環道。 2014/1/10 Q1 Q8 8
- 9. Q2 • 假設 HAM-CYCLE ∈ P • 挑出一個 v ∈ V，Ev 是 v 的所有邊， e1, e2 ∈ Ev，挑 e1, e2 在多項式時間內。 得到 G’= (V, E – Ev ∪ {e1, e2}) 使用驗證的演算法決定是否 G’ 存有 HAMCYCLE，是的話則遞歸下去。 2014/1/10 Q1 Q8 9
- 10. Q2 • HAM-CYCLE(V, E) { v = choose one from V and |Ev| > 2 if(v == null) return DECIDE-HAM-CYCLE(V, E) for choose pair(e1, e2) from Ev G’= (V, E – Ev ∪ {e1, e2}) if(DECIDE-HAM-CYCLE(G’.V, G’.E) return HAM-CYCLE(G’.V, G’.E) return false } 2014/1/10 Q1 Q8 10
- 11. Q3 • Show that the class NP, viewed as a set of languages is closed under union, intersection, concatenation and Kleene star. 2014/1/10 Q1 Q8 11
- 12. Q3 • Same as Q1 ? 2014/1/10 Q1 Q8 12
- 13. Q4 • Prove that P ⊆ NP ∩ co-NP and if NP ≠ co-NP, then P ≠ NP. • Problem NP ≠ co-NP → P ≠ NP ⇒ P = NP → NP = co-NP • Because class P is closed under complement(NOT), P = NP ⇒ ¬ P = ¬ NP ⇒ P = co-NP ⇒ NP = co-NP 2014/1/10 Q1 Q8 13
- 14. Q4 • Prove that P ⊆ NP ∩ co-NP and if NP ≠ co-NP, then P ≠ NP. • Problem NP ≠ co-NP → P ≠ NP ⇒ P = NP → NP = co-NP • Because class P is closed under complement(NOT), P = NP ⇒ ¬ P = ¬ NP ⇒ P = co-NP ⇒ NP = co-NP 2014/1/10 Q1 Q8 14
- 15. Q5 • Show that 2-CNF-SAT is polynomial time solvable. 2014/1/10 Q1 Q8 15
- 16. Q5 • Clause (x ∨ y) if x = false, y = true. if y = false, x = true. • Build DAG. Each variable v has v and ¬ v // true and false. • (x ∨ y) have two direct edge (¬ x, y) and (¬ y, x) // if x is false, y must true. // if y is false, x must true. • (¬ x ∨ y) have two direct edge (¬ (¬ x), y) and (¬ y, ¬ x) 2014/1/10 Q1 Q8 16
- 17. Q5 • Each variable v has v and ¬ v // true and false. • 建造 SCC 強連通元件，同一個 SCC 表示可以互 相推論得到，如果 v 和 v’ 同時在同一個 SCC 中， 則表示怎麼推論都會矛盾，意即不滿足。 2014/1/10 Q1 Q8 17
- 18. Q6 • The subgraph isomorphism problem takes two graphs G1 and G2 and asks whether G1 is isomorphic to a subgraph of G2. Show that this problem is NPcomplete. • HAM-CYCLE ≤p Subgraph-Isomorphism Problem 2014/1/10 Q1 Q8 18
- 19. Q6 • HAM-CYCLE G(V, E) Let G2 =G, G1 = cycle with v nodes. 藉由轉換 HAM-CYCLE 相當於子圖同構問題， 由於 HAM-CYCLE 是 NP-Complete， 因此子圖同構問題也是 NP-Complete。 • HAM-CYCLE ≤p Subgraph-Isomorphism Problem 2014/1/10 Q1 Q8 19
- 20. Q7 • Given an integer m-by-n matrix A, and an integer mvector b, the integer programming problem asks whether there is an integer n-vector x such that Ax ≤ b. • Prove that this problem is NP-complete. • 3-CNF-SAT ≤p 0-1 integer-programming problem 2014/1/10 Q1 Q8 20
- 21. Q7 • Given an integer m-by-n matrix A, and an integer mvector b, the integer programming problem asks whether there is an integer n-vector x such that Ax ≤ b. • Prove that this problem is NP-complete. • 3-CNF-SAT ≤p 0-1 integer-programming problem • // 總覺得題目好像少給了什麼 2014/1/10 Q1 Q8 21
- 22. Q7 • 3-CNF-SAT Ex. (x1 ∨ x2 ∨ x3) ∧ (x1 ∨ ¬ x3 ∨ ¬ x4) • (x1 ∨ x2 ∨ x3) : x1 + x2 + x3 ≥ 1 - x1 - x2 - x3 ≤ -1 • (x1 ∨ ¬ x3 ∨ ¬ x4) : x1 + (1-x3) + (1-x4) ≥ 1 - x1 + x 3 + x 4 ≤ 1 1 • Ax ≤ b - 1 - 1 - 1 0 x -1 0 1 1 1 • 以此類推，可以在多項式時間內轉換，得證。 0-1 integer-programming problem is NPC. 2014/1/10 Q1 Q8 22
- 23. Q8 • Given a set S of integers, determine whether there exists a subset A ⊆ S, such that • sigma(x)|x ∈ A = sigma(y)| y ∈ S - A Show this problem is NP-complete. 2014/1/10 Q1 Q8 23
- 24. Q8 • SUBSUM-SET(SOS) ≤p SET-PARTITION • 令 K = 一半即可 ? 未解決 2014/1/10 Q1 Q8 24
- 25. Q10 • Show that the Hamiltonian path problem is NP-complete. • HAM-CYCLE ≤p HAM-PATH 2014/1/10 Q1 Q8 25
- 26. Q10 • HAM-CYCLE ≤p HAM-PATH 2014/1/10 Q1 Q8 26
- 27. Q10 • HAM-CYCLE ≤p HAM-PATH • 將其中一個點拆成兩個(入點、出點)， 並且多增加兩個點連到入點和出點。 • 如果 HAM-PATH 存在，只要將這兩個 點合併再一起即可(合併成一條路) 2014/1/10 Q1 Q8 27