Complexity Theory: Overview of definitions (D), theorems (T) and languages (L) Chronological order Type Lecture 9 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 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 D P Def. 7.12 p. 262 L 9 sl.12 L PATH p. 263 L 9 sl. 13 T PATH 2 P p. 264 L 9 sl. 13 T ALLDFA 2 P T 9 ex. 5 Type Lecture 10 Book Slides Tutorials L RELPRIME p. 265 L 10 sl. 2 Thm. 7.15 p. L 10 sl. 2{ T RELPRIME 2 P 265 3 Thm. 7.16 p. L 10 sl. 4{ T Every context-free language is in P T 10 ex. 1 266{267 5 L 10 sl. 6{ T Closure properties of P T 10 ex. 2 7 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 L HAMPATH p. 268 L 10 sl. 9 T HAMPATH 2 NTIME(n2) p. 268{269 L 10 sl. 10 Thm. 7.20 p. 270 (see also D NP Def. 7.19 p. L 10 sl. 11 270) Type Lecture 11 Book Slides Tutorials D Polynomial Time Verifiable Languages Def. 7.18 p. 269 L 11 sl. 3 Poly. Time Verifiers vs. Nondeterministic T Thm. 7.20 p 270 L 11 sl. 4 Poly. Time TMs L CLIQUE p. 272 L 11 sl. 5 Thm. 7.24 p. T CLIQUE 2 NP L 11 sl. 5 272 L SUBSET-SUM p. 273 L 11 sl. 5 Thm. 7.25 p. T SUBSET-SUM 2 NP L 11 sl. 5 273 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 Thm. 7.31 p. T Solving problems in P by reductions L 11 sl. 9 277 L SAT p. 275 L 11 sl. 10 L 3SAT p. 278 L 11 sl. 11 Thm. 7.32 p. T 3SAT ≤P CLIQUE L 11 sl. 12 278{279 T BINPACK 2 NP T 11 ex. 2 T co-NP ⊆ EXPTIME T 11 ex. 4 T Poly. Time Reductions are Transitive T 11 ex. 5 Type Lecture 12 Book Slides Tutorials D NP-completeness Def. 7.34 p. 280 L 12 sl. 3 Thm. 7.35 p. T If B 2 P is NP-complete then P = NP L 12 sl. 3 280 If B is NP-complete and B poly. time Thm. 7.36 p. T L 12 sl. 3 reduces to C 2 NP then C is NP-complete 280 Thm. 7.37 p. L 12 sl. 4{ T SAT is NP-complete (Cook-Levin) T 12 ex. 3 280{286 11 Type Lecture 13 Book Slides Tutorials L CNF-SAT and 3SAT p. 278 L 13 sl. 3 Proof of Cor. L 13 sl. 3{ T CNF-SAT ≤ 3SAT T 13 ex. 4 P 7.42 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 L VERTEX-COVER p. 288 L 13 sl. 6 Proof of Thm. L 13 sl. 6{ T 3SAT ≤ VERTEX-COVER T 13 ex. 5 P 7.44 p. 288{289 7 Thm. 7.44 p. T VERTEX-COVER is NP-complete L 13 sl. 6 288 Proof of Thm. L 13 sl. 8{ T 3SAT ≤ HAMPATH P 7.46 p. 290{294 9 Thm. 7.46 p. T HAMPATH is NP-complete L 13 sl. 8 290 L UHAMPATH p. 295 L 13 sl. 10 Thm. 7.55 p. T UHAMPATH is NP-complete L 13 sl. 10 Test 4 295 L SUBSET-SUM p. 295 L 11 sl. 11 Thm. 7.56 p. T SUBSET-SUM is NP-complete L 13 sl. 11 296 Thm. 7.56 p. L 13 sl. 11{ T 3SAT ≤ SUBSET-SUM Test 4 P 296{298 12 T Closure properties of NP T 13 ex. 1 T NOTA is NP-complete T 13 ex. 3 Type Lecture 14 Book Slides Tutorials 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 T SAT 2 SPACE(n) Exa. 8.3 p. 308 L 14 sl. 5 L 14 sl. 6 L ALLNFA p. 309 and L 15 sl. 5 L 14 sl. 6{ T ALL 2 NSPACE(n) read slides NFA 7 NSPACE(t(n)) ⊆ SPACE(t2(n)) (Sav- Thm. 8.5 p. L 14 sl. 9{ T itch) 310{311 10 D PSPACE Def. 8.6 p. 312 L 14 sl. 11 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 Type Lecture 15 Book Slides Tutorials D PSPACE-completeness Def. 8.8 p. 313 L 15 sl. 2 L ALBA L 15 sl. 3 A is PSPACE-complete (padding T LBA L 15 sl. 3 technique) L TQBF p. 315 L 15 sl. 4 TQBF is PSPACE-complete (without T p. 315 L 15 sl. 4 proof).