1. CS151 Complexity Theory Lecture 14 May 13, 2015

2. 2 Arthur-Merlin Games IP permits verifier to keep coin-flips private –necessary feature? –GNI protocol breaks without it Arthur-Merlin game: interactive protocol in which coin-flips are public –Arthur (verifier) may as well just send results of coin-flips and ask Merlin (prover) to perform any computation Arthur would have done

3. May 13, 20153 Arthur-Merlin Games Clearly Arthur-Merlin  IP –“private coins are at least as powerful as public coins” Proof that IP = PSPACE actually shows PSPACE  Arthur-Merlin  IP  PSPACE –“public coins are at least as powerful as private coins” !

4. May 13, 20154 Arthur-Merlin Games Delimiting # of rounds: –AM[k] = Arthur-Merlin game with k rounds, Arthur (verifier) goes first –MA[k] = Arthur-Merlin game with k rounds, Merlin (prover) goes first Theorem: AM[k] (MA[k]) equals AM[k] (MA[k]) with perfect completeness. – i.e., x  L implies accept with probability 1 –proof on problem set

5. May 13, 20155 Arthur-Merlin Games Theorem: for all constant k  2 AM[k] = AM[2]. Proof: –we show MA[2]  AM[2] –implies can move all of Arthur’s messages to beginning of interaction: AMAMAM…AM = AAMMAM…AM … = AAA…AMMM…M

6. May 13, 20156 Arthur-Merlin Games Proof (continued): –given L  MA[2] x  L  9 m Pr r [(x, m, r)  R] = 1  Pr r [ 9 m (x, m, r)  R] = 1 x  L  8 m Pr r [(x, m, r)  R]  ε  Pr r [ 9 m (x, m, r)  R]  2 |m| ε –by repeating t times with independent random strings r, can make error ε < 2 -t –set t = m+1 to get 2 |m| ε < ½. order reversed

7. May 13, 20157 MA and AM Two important classes: –MA = MA[2] –AM = AM[2] definitions without reference to interaction: –L  MA iff 9 poly-time language R x  L  9 m Pr r [(x, m, r)  R] = 1 x  L  8 m Pr r [(x, m, r)  R]  ½ –L  AM iff 9 poly-time language R x  L  Pr r [ 9 m (x, m, r)  R] = 1 x  L  Pr r [ 9 m (x, m, r)  R]  ½

8. May 13, 20158 MA and AM L  AM iff 9 poly-time language R x  L  Pr r [ 9 m (x, m, r)  R] = 1 x  L  Pr r [ 9 m (x, m, r)  R]  ½ Relation to other complexity classes: –both contain NP (can elect to not use randomness) –both contained in ∏ 2. L  ∏ 2 iff 9 R  P for which: x  L  Pr r [ 9 m (x, m, r)  R] = 1 x  L  Pr r [ 9 m (x, m, r)  R] < 1 –so clear that AM  ∏ 2 –know that MA  AM

9. May 13, 20159 MA and AM P NP coNP Σ2Σ2 Π2Π2 AMcoAM MAcoMA

10. May 13, 201510 MA and AM Theorem: coNP  AM  PH = AM. Proof: –suffices to show Σ 2  AM (and use AM  Π 2 ) –L  Σ 2 iff 9 poly-time language R x  L  9 y 8 z (x, y, z)  R x  L  8 y 9 z (x, y, z)  R –Merlin sends y –1 AM exchange decides coNP query: 8 z (x, y, z)  R ? –3 rounds; in AM

11. May 13, 201511 MA and AM We know Arthur-Merlin = IP. –“public coins = private coins” Theorem (GS): IP[k]  AM[O(k)] –stronger result –implies for all constant k  2, IP[k] = AM[O(k)] = AM[2] So, GNI  IP[2] = AM

12. May 13, 201512 Back to Graph Isomorphism The payoff: –not known if GI is NP-complete. –previous Theorems: if GI is NP-complete then PH = AM –unlikely! –Proof: GI NP-complete  GNI coNP- complete  coNP  AM  PH = AM

13. Derandomization revisited L  MA iff 9 poly-time language R x  L  9 m Pr r [(x, m, r)  R] = 1 x  L  8 m Pr r [(x, m, r)  R]  ½ Recall PRGs: –for all circuits C of size at most s : |Pr y [C(y) = 1] – Pr z [C(G(z)) = 1]| ≤ ε May 13, 201513 NP AM MA seedoutput string G t bitsu bits

14. Using PRGs for MA L  MA iff 9 poly-time language R x  L  9 m Pr r [(x, m, r)  R] = 1 x  L  8 m Pr r [(x, m, r)  R]  ½ produce poly-size circuit C such that C(x, m, r) = 1, (x,m,r) 2 R for each x, m can hardwire to get C x,m 9 m Pr y [C x,m (y) = 1] = 1 (“yes”) 8 m Pr y [C x,m (y) = 1] ≤ 1/2 (“no”) May 13, 201514

15. Using PRGs for MA can compute Pr z [C x,m (G(z)) = 1] exactly –evaluate C x (G(z)) on every seed z  {0,1} t –running time (O(|C x,m |)+(time for G))2 t x  L  9 m [ Pr z [C x,m (G(z)) = 1] = 1 ] x  L  8 m [ Pr z [C x,m (G(z)) = 1]  ½ + ² ] – L 2 NP if PRG with t = O(log n), ² < 1/4 Theorem: E requires exponential size circuits  MA = NP. May 13, 201515 poly(n) poly-time

16. May 13, 201516 MA and AM P NP coNP Σ2Σ2 Π2Π2 AMcoAM MAcoMA (under a hardness assumption)

17. May 13, 201517 MA and AM P NP coNP Σ2Σ2 Π2Π2 AMcoAM MA = = coMA (under a hardness assumption) What about AM, coAM?

18. May 13, 201518 Derandomization revisited Theorem (IW, STV): If E contains functions that require size 2 Ω(n) circuits, then E contains functions that are 2 Ω(n) - unapproximable by circuits. Theorem (NW): if E contains 2 Ω(n) -unapp- roximable functions there are poly-time PRGs fooling poly(n)-size circuits, with seed length t = O(log n), and error ² < 1/4.

19. May 13, 201519 Oracle circuits A-oracle circuit C –also allow “A-oracle gates” circuit C –directed acyclic graph –nodes: AND (  ); OR (  ); NOT (  ); variables x i   x1x1 x2x2   x3x3 …xnxn  A 1 iff x 2 A x

20. May 13, 201520 Relativized versions Theorem: If E contains functions that require size 2 Ω(n) A-oracle circuits, then E contains functions that are 2 Ω(n) - unapproximable by A-oracle circuits. Recall proof: –encode truth table to get hard function –if approximable by s(n)-size circuits, then use those circuits to compute original function by size s(n)  -size circuits. Contradiction.

21. May 13, 201521 Relativized versions 01001010 m:m: 01001010 Enc(m): 00010 01100010 R: 01000 D C f:{0,1} log k  {0,1} f ’:{0,1} log n  {0,1} small A-oracle circuit C approximating f’ decoding procedure i 2 {0,1} log k small A-oracle circuit computing f exactly f(i)

22. May 13, 201522 Relativized versions Theorem: if E contains 2 Ω(n) -unapp- roximable fns., there are poly-time PRGs fooling poly(n)-size A-oracle circuits, with seed length t = O(log n), and error ² < 1/4. Recall proof: –PRG from hard function on O(log n) bits –if doesn’t fool s-size circuits, then use those circuits to compute hard function by size s ¢ n ± - size circuits. Contradiction.

23. Relativized versions May 13, 201523 G n (y)=f log n (y |S 1 )◦f log n (y |S 2 )◦…◦f log n (y |S m ) 010100101111101010111001010 f log n : –doesn’t fool A-oracle circuit of size s: |Pr x [C(x) = 1] – Pr y [C( G n (y) ) = 1]| > ε –implies A-oracle circuit P of size s’ = s + O(m): Pr y [P(G n (y) 1 … i-1 ) = G n (y) i ] > ½ + ε/m

24. Relativized versions May 13, 2015 G n (y)=f log n (y |S 1 )◦f log n (y |S 2 )◦…◦f log n (y |S m ) 010100101111101010111001010 f log n : P output f log n (y ’) hardwired tables y’ A-oracle circuit approximating f 24

25. Using PRGs for AM L  AM iff 9 poly-time language R x  L  Pr r [ 9 m (x, m, r)  R] = 1 x  L  Pr r [ 9 m (x, m, r)  R]  ½ produce poly-size SAT-oracle circuit C such that C(x, r) = 1, 9 m (x,m,r) 2 R for each x, can hardwire to get C x Pr y [C x (y) = 1] = 1(“yes”) Pr y [C x (y) = 1] ≤ ½(“no”) May 13, 201525 1 SAT query, accepts iff answer is “yes”

26. Using PRGs for AM x  L  [ Pr z [C x (G(z)) = 1] = 1 ] x  L  [ Pr z [C x (G(z)) = 1]  ½ + ² ] C x makes a single SAT query, accepts iff answer is “yes” if G is a PRG with t = O(log n), ² < ¼, can check in NP: –does C x (G(z)) = 1 for all z? May 13, 201526

27. May 13, 201527 Relativized versions Theorem: If E contains functions that require size 2 Ω(n) A-oracle circuits, then E contains functions that are 2 Ω(n) -unapproximable by A- oracle circuits. Theorem: if E contains 2 Ω(n) -unapproximable functions there are PRGs fooling poly(n)-size A- oracle circuits, with seed length t = O(log n), and error ² < ½. Theorem: E requires exponential size SAT- oracle circuits  AM = NP.

28. May 13, 201528 MA and AM P NP coNP Σ2Σ2 Π2Π2 MA = = coMA (under a hardness assumption) coAMAM

29. May 13, 201529 MA and AM P NP coNP Σ2Σ2 Π2Π2 MA = = coMA (under a hardness assumption) = coAMAM =

30. May 13, 201530 New topic(s) Optimization problems, Approximation Algorithms, and Probabilistically Checkable Proofs

31. May 13, 201531 Optimization Problems many hard problems (especially NP-hard) are optimization problems –e.g. find shortest TSP tour –e.g. find smallest vertex cover –e.g. find largest clique –may be minimization or maximization problem –“opt” = value of optimal solution

32. May 13, 201532 Approximation Algorithms often happy with approximately optimal solution –warning: lots of heuristics –we want approximation algorithm with guaranteed approximation ratio of r –meaning: on every input x, output is guaranteed to have value at most r*opt for minimization at least opt/r for maximization

33. May 13, 201533 Approximation Algorithms Example approximation algorithm: –Recall: Vertex Cover (VC): given a graph G, what is the smallest subset of vertices that touch every edge? –NP-complete

34. May 13, 201534 Approximation Algorithms Approximation algorithm for VC: –pick an edge (x, y), add vertices x and y to VC –discard edges incident to x or y; repeat. Claim: approximation ratio is 2. Proof: –an optimal VC must include at least one endpoint of each edge considered –therefore 2*opt  actual

35. May 13, 201535 Approximation Algorithms diverse array of ratios achievable some examples: –(min) Vertex Cover: 2 –MAX-3-SAT (find assignment satisfying largest # clauses): 8/7 –(min) Set Cover: ln n –(max) Clique: n/log 2 n –(max) Knapsack: (1 + ε) for any ε > 0

36. May 13, 201536 Approximation Algorithms (max) Knapsack: (1 + ε) for any ε > 0 called Polynomial Time Approximation Scheme (PTAS) –algorithm runs in poly time for every fixed ε>0 –poor dependence on ε allowed If all NP optimization problems had a PTAS, almost like P = NP (!)

37. May 13, 201537 Approximation Algorithms A job for complexity: How to explain failure to do better than ratios on previous slide? –just like: how to explain failure to find poly- time algorithm for SAT... –first guess: probably NP-hard –what is needed to show this? “gap-producing” reduction from NP- complete problem L 1 to L 2

38. May 13, 201538 Approximation Algorithms “gap-producing” reduction from NP- complete problem L 1 to L 2 no yes L1L1 L 2 (min. problem) f opt k rk

39. May 13, 201539 Gap producing reductions r-gap-producing reduction: –f computable in poly time –x  L 1  opt(f(x))  k –x  L 1  opt(f(x)) > rk –for max. problems use “  k” and “< k/r” Note: target problem is not a language –promise problem (yes  no not all strings) –“promise”: instances always from (yes  no)

40. May 13, 201540 Gap producing reductions Main purpose: –r-approximation algorithm for L 2 distinguishes between f(yes) and f(no); can use to decide L 1 –“NP-hard to approximate to within r” no yes L1L1 f k rk yes no L1L1 f k/r k L 2 (min.)L 2 (max.) yes noyes no

41. May 13, 201541 Gap preserving reductions gap-producing reduction difficult (more later) but gap-preserving reductions easier f k rk k’ r’k’ Warning: many reductions not gap-preserving yes no yes no L 1 (min.) L 2 (min.)

42. May 13, 201542 Gap preserving reductions Example gap-preserving reduction: –reduce MAX-k-SAT with gap ε –to MAX-3-SAT with gap ε’ –“MAX-k-SAT is NP-hard to approx. within ε  MAX-3-SAT is NP-hard to approx. within ε’ ” MAXSNP (PY) – a class of problems reducible to each other in this way –PTAS for MAXSNP-complete problem iff PTAS for all problems in MAXSNP constants

43. May 13, 201543 MAX-k-SAT Missing link: first gap-producing reduction –history’s guide it should have something to do with SAT Definition: MAX-k-SAT with gap ε –instance: k-CNF φ –YES: some assignment satisfies all clauses –NO: no assignment satisfies more than (1 – ε) fraction of clauses

44. May 13, 201544 Proof systems viewpoint k-SAT NP-hard  for any language L  NP proof system of form: –given x, compute reduction to k-SAT:  x –expected proof is satisfying assignment for  x –verifier picks random clause (“local test”) and checks that it is satisfied by the assignment x  L  Pr[verifier accepts] = 1 x  L  Pr[verifier accepts] < 1

45. May 13, 201545 Proof systems viewpoint MAX-k-SAT with gap ε NP-hard  for any language L  NP proof system of form: –given x, compute reduction to MAX-k-SAT:  x –expected proof is satisfying assignment for  x –verifier picks random clause (“local test”) and checks that it is satisfied by the assignment x  L  Pr[verifier accepts] = 1 x  L  Pr[verifier accepts] ≤ (1 – ε) –can repeat O(1/ε) times for error < ½

46. May 13, 201546 Proof systems viewpoint can think of reduction showing k-SAT NP-hard as designing a proof system for NP in which: –verifier only performs local tests can think of reduction showing MAX-k-SAT with gap ε NP-hard as designing a proof system for NP in which: –verifier only performs local tests –invalidity of proof* evident all over: “holographic proof” and an  fraction of tests notice such invalidity

47. May 13, 201547 PCP Probabilistically Checkable Proof (PCP) permits novel way of verifying proof: –pick random local test –query proof in specified k locations –accept iff passes test fancy name for a NP-hardness reduction

48. May 13, 201548 PCP PCP[r(n),q(n)]: set of languages L with p.p.t. verifier V that has (r, q)-restricted access to a string “proof” –V tosses O(r(n)) coins –V accesses proof in O(q(n)) locations –(completeness) x  L   proof such that Pr[V(x, proof) accepts] = 1 –(soundness) x  L   proof* Pr[V(x, proof*) accepts]  ½

49. May 13, 201549 PCP Two observations: –PCP[1, poly n] = NP proof? –PCP[log n, 1]  NP proof? The PCP Theorem (AS, ALMSS): PCP[log n, 1] = NP.

50. May 13, 201550 PCP Corollary: MAX-k-SAT is NP-hard to approximate to within some constant . –using PCP[log n, 1] protocol for, say, VC –enumerate all 2 O(log n) = poly(n) sets of queries –construct a k-CNF φ i for verifier’s test on each note: k-CNF since function on only k bits –“YES” VC instance  all clauses satisfiable –“NO” VC instance  every assignment fails to satisfy at least ½ of the φ i  fails to satisfy an  = (½)2 -k fraction of clauses.