P, NP, and Beyond. Hengfeng Wei. May 01 May 04, Hengfeng Wei P, NP, and Beyond May 01 May 04, / 29
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1 P, NP, and Beyond Hengfeng Wei May 01 May 04, 2017 Hengfeng Wei P, NP, and Beyond May 01 May 04, / 29
2 P, NP, and Beyond 1 2 Reductions: Tetris is NP-complete
3 P P = c>0 DTIME(n c ) TC f(n) = O(n c ) t(n) = O(n d ) T (n) = kf(n) + t(n) k T k (n) = f (i) (n) + t(n) i=1 k = O(1) vs k = Θ(n O(1) ) O vs Θ, Ω Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
4 NP Definition (NP) L NP if polynomial-time verifier V (x, c) such that x {0, 1}, x L c {0, 1}, V (x, c) = 1 NP-problems has short certificates that are easy to verify TC HAM-PATH NP Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
5 NP TC NP is closed under,,, L 1 NP, L 2 NP = L = L 1 L 2 NP Question: Is NP-complete closed under,,,? Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
6 NP Theorem NP is closed under A (x, y) : 1 k x c = c 1 #c 2 # #c k #m 1 &m 2 & &m k 1 c =c 1 #c 2 # #c k #m 1 &m 2 & &m k 1 i=k i=1a(x i, c i ) x L c, A(x, c) = 1 Reference Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
7 conp L NP? = L NP SAT = {ϕ : ϕ is not satisfiable} TAUT = {ϕ : ϕ is a tautology} conp = {L : L NP} Definition (conp) L conp if polynomial-time verifier V (x, c) such that x {0, 1}, x L c {0, 1}, V (x, c) = 1 Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
8 NP vs conp conp {0, 1} \ NP P NP conp P = NP = NP = conp NP conp = P NP Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
9 NP-hard and NP-complete L NP, L p L = L is NP-hard NP-complete = NP NP-hard Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
10 NP-hard and NP-complete TC HAM-PATH is NP-complete HAM-CYCLE p HAM-PATH p : split v into v 1, v 2 ; add s, t, (s, v 1 ), (v 2, t) Question: HAM-PATH p HAM-CYCLE p : add v ; (v, v), v V Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
11 P vs NP solve vs verify exhaustive search avoidable? P NP = P NP-complete Theorem (NP-intermediate: Ladner s theorem, 1975) P NP = L NP \ P L / NP-complete Factoring, Graph (group) isomorphism (vs Subgraph isomorphism) Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
12 EXP EXP = c>0 DTIME(2 nc ) P NP EXP Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
13 Time Hierarchy Theorem P EXP Theorem (Time Hierarchy Theorem, 1965) f(n) log f(n) = o(g(n)) = DTIME(f(n)) DTIME(g(n)) Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
14 R R = DTIME(< ) #undecidable #decidable #algs = N #problems = 2 N = R Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
15 PSPACE PSPACE = c>0 SPACE(n c ) P PSPACE NP PSPACE EXP Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
16 PSPACE-complete Definition (QBF: Quantified Boolean Formula) Q 1 x 1 Q 2 x 2 Q n x n φ(x 1, x 2,, x n ) Q i :, TQBF = {True QBF} PSPACE-complete SAT : ϕ = x 1,, x n φ(x 1, x 2,, x n ) NP-complete TAUT : ϕ = x 1,, x n φ(x 1, x 2,, x n ) conp-complete Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
17 PSPACE-complete The QBF game φ(x 1, x 2,, x 2n ) Player 1 wins φ(x 1, x 2,, x 2n ) is true Does player 1 has a winning strategy? x 1 x 2 x 3 x 4 x 2n φ(x 1, x 2,, x 2n ) Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
18 NP vs PSPACE NP? = PSPACE Short certificate for winning strategy? Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
19 PH Definition (Polynomial Hierarchy) L p i if polynomial-time decidable relation R(x, u 1, u 2,, u i ) such that x {0, 1}, x L u 1 {0, 1} u 2 {0, 1} Q i u i {0, 1} R(x, u 1, u 2,, u i ) = 1 p 1 = co p 1 p 1 = NP p 1 = conp PH = i p i Unique-SAT p 2 Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
20 Summary P NP PH PSPACE EXP P EXP References Computational Complexity A Modern Approach by Arora and Barak (the first 5 chapters) Computer and Intractability A Guide to the Theory of NP-Completeness by Garey and Johnson Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
21 If HAM-CYCLE P TC HAM-CYCLE P = HAM-CYCLE-LIST P 1 starting from v 2 removing each edge e on v 3 checking G \ e 4 restoring and marking the critical edge e = (v, u) 5 v = u Reference Question remove e E in arbitrary order if (G \ e) HAM-CYCLE? Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
22 G 3 HAM-CYCLE TC (Karaganis, 1968) Theorem G 3 HAM-CYCLE Let T = (V, E) be a tree For any edge e E, there is a Hamilton cycle on T 3 that contains e References On the Cube of a Graph by Jerome J Karaganis, 1968 The Cube of Every Connected Graph is 1-Hamiltonian by Gary Chartrand and S F Kapoor, documents/comp-fa14solpdf Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
23 P, NP, and Beyond 1 2 Reductions: Tetris is NP-complete
24 Reductions: Tetris is NP-complete Tetris is NP-complete References 6890 Algorithmic Lower Bounds: Fun with Hardness Proofs, by Prof Erik Demaine, Fall 2014 (Lecture 03, from 00:51:00) Tetris is Hard, Made Easy by Ron Breukelaar, Hendrik Jan Hoogeboom, and Walter A Kosters, 2003 Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
25 Reductions: Tetris is NP-complete Tetris Hengfeng Wei P, NP, and Beyond May 01 May 04, / 29
26 Reductions: Tetris is NP-complete TETRIS Definition (TETRIS: The Tetris Problem) TETRIS NP Hengfeng Wei P, NP, and Beyond May 01 May 04, / 29
27 Reductions: Tetris is NP-complete 3-PARTITION Definition (3-PARTITION) Hengfeng Wei P, NP, and Beyond May 01 May 04, / 29
28 Reductions: Tetris is NP-complete 3-PARTITION p TETRIS: the initial board Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
29 Reductions: Tetris is NP-complete 3-PARTITION p TETRIS: the piece sequence Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
30 Reductions: Tetris is NP-complete 3-PARTITION p TETRIS: = Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
31 Reductions: Tetris is NP-complete 3-PARTITION p TETRIS: = Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, / 29
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