Complexity Theory: Overview of definitions (D), theorems (T) and languages (L)

Type Definitions Book Slides Tutorials D Asymptotic Upper Bound big-O Def. 7.2 p. 253 L 9 sl. 3 D Strict Asymptotic Upper Bound small-o Def. 7.5 p. 254 L 9 sl. 4 Running time or (worst-case) time com- D Def. 7.1 p. 252 L 9 sl. 6 plexity of TM D TIME(t(n)) Def. 7.7 p. 255 L 9 sl. 8 D P Def. 7.12 p. 262 L 9 sl.12 Running time or (worst-case) time com- D Def. 7.9 p. 259 L 10 sl. 8 plexity of nondeterministic TM D NTIME(t(n)) Def. 7.21 p. 271 L 10 sl. 9 Thm. 7.20 p. 270 D NP (see also Def. 7.19 L 10 sl. 11 p. 270) D Polynomial Time Verifiable Languages Def. 7.18 p. 269 L 11 sl. 3 D co-NP p. 273 L 11 sl. 5 D EXPTIME p. 312 L 11 sl. 6 D Polynomial Time Computable Function Def. 7.28 p. 276 L 11 sl. 8 D Polynomial Time Mapping Reduction Def. 7.29 p. 276 L 11 sl. 8 D NP-completeness Def. 7.34 p. 280 L 12 sl. 3 D Space complexity of TM Def. 8.1 p. 307 L 14 sl. 3 D Space complexity of Nondeterministic TM Def. 8.1 p. 307 L 14 sl. 3 D SPACE(t(n)) Def. 8.2 p. 308 L 14 sl. 4 D NSPACE(t(n)) Def. 8.2 p. 308 L 14 sl. 4 D PSPACE Def. 8.6 p. 312 L 14 sl. 11 D PSPACE-completeness Def. 8.8 p. 313 L 15 sl. 2 Type Theorems Book Slides Tutorials T Time complexity of simulating k-tape TM Thm. 7.8 p. 258 L 9 sl. 10 Polynomial Time Equivalence of Deter- T p. 261 L 9 sl. 11 ministic Models T PATH ∈ P p. 264 L 9 sl. 13 T ALLDFA ∈ P T 9 ex. 5 L 10 sl. 2– T RELPRIME ∈ P Thm. 7.15 p. 265 3 Thm. 7.16 p. 266– L 10 sl. 4– T Every context-free language is in P T 10 ex. 1 267 5 L 10 sl. 6– T Closure properties of P T 10 ex. 2 7 T HAMPATH ∈ NTIME(n2) p. 268–269 L 10 sl. 10 Poly. Time Verifiers vs. Nondeterministic T Thm. 7.20 p 270 L 11 sl. 4 Poly. Time TMs T CLIQUE ∈ NP Thm. 7.24 p. 272 L 11 sl. 5 T SUBSET-SUM ∈ NP Thm. 7.25 p. 273 L 11 sl. 5 T Solving problems in P by reductions Thm. 7.31 p. 277 L 11 sl. 9 Thm. 7.32 p. 278– T 3SAT ≤P CLIQUE L 11 sl. 12 279 T BINPACK ∈ NP T 11 ex. 2 T co-NP ⊆ EXPTIME T 11 ex. 4 T Poly. Time Reductions are Transitive T 11 ex. 5 T If B ∈ P is NP-complete then P = NP Thm. 7.35 p. 280 L 12 sl. 3 If B is NP-complete and B poly. time T Thm. 7.36 p. 280 L 12 sl. 3 reduces to C ∈ NP then C is NP-complete Thm. 7.37 p. 280– L 12 sl. 4– T SAT is NP-complete (Cook-Levin) T 12 ex. 3 286 11 Proof of Cor. 7.42 p. L 13 sl. 3– T CNF-SAT ≤ 3SAT T 13 ex. 4 P 286 4 T 3SAT is NP-complete Cor. 7.42 p. 286 L 13 sl. 3 T CLIQUE is NP-complete Cor. 7.43 p. 287 L 13 sl. 5 Proof of Thm. 7.44 L 13 sl. 6– T 3SAT ≤ VERTEX-COVER T 13 ex. 5 P p. 288–289 7 T VERTEX-COVER is NP-complete Thm. 7.44 p. 288 L 13 sl. 6 Proof of Thm. 7.46 L 13 sl. 8– T 3SAT ≤ HAMPATH P p. 290–294 9 T HAMPATH is NP-complete Thm. 7.46 p. 290 L 13 sl. 8 T UHAMPATH is NP-complete Thm. 7.55 p. 295 L 13 sl. 10 Test 4 T SUBSET-SUM is NP-complete Thm. 7.56 p. 296 L 13 sl. 11 Thm. 7.56 p. 296– L 13 sl. 11– T 3SAT ≤ SUBSET-SUM Test 4 P 298 12 T Closure properties of NP T 13 ex. 1 T NOTA is NP-complete T 13 ex. 3 T SAT ∈ SPACE(n) Exa. 8.3 p. 308 L 14 sl. 5 L 14 sl. 6– T ALL ∈ NSPACE(n) read slides NFA 7 NSPACE(t(n)) ⊆ SPACE(t2(n)) (Sav- L 14 sl. 9– T Thm. 8.5 p. 310–311 itch) 10 T NP ⊆ PSPACE p. 312 L 14 sl. 12 T co-NP ⊆ PSPACE L 14 sl. 12 T 14 ex. 2 T PSPACE ⊆ EXPTIME p. 312 L 14 sl. 13 T 14 ex. 4 Overview of Time and Space Complexity T p. 312–313 L 14 sl. 14 Classes T Closure properties of PSPACE T 14 ex. 1 A is PSPACE-complete (padding T LBA L 15 sl. 3 technique) TQBF is PSPACE-complete (without T p. 315 L 15 sl. 4 proof) Type Languages Book Slides Tutorials L PATH p. 263 L 9 sl. 13 L RELPRIME p. 265 L 10 sl. 2 L HAMPATH p. 268 L 10 sl. 9 L CLIQUE p. 272 L 11 sl. 5 L SUBSET-SUM p. 273 L 11 sl. 5 L SAT p. 275 L 11 sl. 10 L 3SAT p. 278 L 11 sl. 11 L CNF-SAT and 3SAT p. 278 L 13 sl. 3 L VERTEX-COVER p. 288 L 13 sl. 6 L UHAMPATH p. 295 L 13 sl. 10 L SUBSET-SUM p. 295 L 11 sl. 11 L 14 sl. 6 L ALLNFA p. 309 and L 15 sl. 5 L ALBA L 15 sl. 3 L TQBF p. 315 L 15 sl. 4