1. 1 INTRODUCTION NP, NP-hardness Approximation PCP

2. 2 Decision, Optimization Problems zA decision problem is a Boolean function ƒ(X), or alternatively a language L  {0, 1} * comprising all strings for which ƒ is TRUE :L = { X  {0, 1} * | ƒ(X) } zAn optimization problem is a function ƒ(X, Y) which, given X, is to be maximized (or minimized) over all possible Y’s: max y [ ƒ(X, Y) ] zA threshold version of max-ƒ(X, Y) is the language L t of all strings X for which there exists Y such that ƒ(X, Y)  t transforming an optimization problem into decision (transforming an optimization problem into decision)

3. 3 The Class NP The classical definition of the class NP is as follows We say that a language L  {0, 1} * belongs to the class NP, if there exists a Turing machine V L [referred to as a verifier] such that X  L  there exists a witness Y such that V L (X, Y) accepts, in time |X| O(1) That is, V L can verify a membership-proof of X in L in time polynomial in the length of X

4. 4 NP-Hardness zA language L is said to be NP-hard if an efficient (polynomial-time) procedure for L can be utilized to obtain an efficient procedure for any NP-language zThat is referred to as the more general, Cook reduction. An efficient algorithm, translating any NP problem to a single instance of L -- thereby showing L NP-hard -- is referred to as Karp reduction.

5. 5 Motivation Proving a set L to be NP-hard implies that L, for all practical purposes, is unsolvable

6. 6 Characterizing NP Thm [Cook, Levin]: For any L  NP there is an algorithm that, on input X, constructs in time |X| O(1), a set of Boolean functions, local-tests  L,X = {  1  l } over variables y 1,...,y m s.t.:  each of  1  l depends on o(1) variables  X  L  there exists an assignment A: { y 1,..., y m }  { 0, 1 } satisfying all    l [ note that m and l must be at most polynomial in |X| ]

7. 7 NP-hard Problems Def: MAX-CLIQUE of a graph G - denoted  (G) - is the size of the largest complete-subgraph of G Thm[Cook, Karp]: MAX-CLIQUE is NP-hard Proof: Given  above we construct a graph G  whose MAX-CLIQUE determines whether  is satisfiable or not

8. 8 GGGG has one vertex for each  i   and an assignment satisfying  i  1.  i..  l         All assignments to ’s variables All assignments to  i ’s variables Not satisfying Not satisfying  i Satisfying Satisfying  i

9. 9 GGGG two vertexes connected if assignments consistent  1...  i..  l         Consistent values Different value same variable

10. 10 Lemma:  (G  ) = l  X  L  Consider an assignment A satisfying For each i consider A's restriction to  i ‘s variables The corresponding l vertexes form a clique in G   Any clique of size m in G  implies an assignment satisfying m of  1  l

11. 11 Approximation We’ve just seen that computing MAX-CLIQUE is NP-hard How about approximating it ?? [ I.e., coming up with a number that deviate from it by at most some specified (small as possible) factor ]

12. 12 Strong, PCP Characterizations of NP Thm[AS,ALMSS]: For any L  NP there is a polynomial-time algorithm that, on input X, outputs  L,X = {    l } over y 1,...,y m s.t.  each of    l depends on O(1) variables  X  L  assignment A: { y 1,..., y m }  { 0, 1 } satisfying all  L,X  X  L  assignment A: { y 1,..., y m }  { 0, 1 } satisfies < ½ fraction of  L,X

13. 13 Probabilistically-Checkable- Proofs zHence, Cook-Levin theorem states that a verifier can efficiently verify membership- proofs for any NP language zPCP characterization of NP, in contrast, states that a membership-proof can be verified probabilistically yby choosing randomly one local-test, yaccessing the small set of variables it depends on, yaccept or reject accordingly zerroneously accepting a non-member only with small probability

14. 14 Gap Problems zA gap-problem is [can be defined similarly for minimization] a maximization [can be defined similarly for minimization] problem ƒ(X, Y), and two thresholds t 1 > t 2 X must be accepted ifmax Y [ ƒ(X, Y) ]  t 1 X must be rejected ifmax Y [ ƒ(X, Y) ]  t 2 [don’t care] other X’s may be accepted or rejected [don’t care] [almost a decision problem, relates to approximation]

15. 15 Reducing gap-Problems to Approximation Problems zUsing an efficient approximation algorithm for ƒ(X, Y) to within a factor g, one can efficiently solve the corresponding gap problem gap-ƒ(X, Y), as long as t 1 / t 2 > g 2 zSimply run the approximation algorithm. The outcome clearly determines which side of the gap the given input falls in. [ Hence, proving a gap problem NP-hard translates to its approximation version, for appropriate factors ]

16. 16 gap-SAT zDef: gap-SAT[D, v,  ] is as follows:  instance: a set  = {    l } of Boolean- functions (local-tests) over variables y 1,...,y m of range 2 V  locality: each of  1  l depends on at most D variables  Maximum-Satisfied-Fraction is the fraction of  satisfied by an assignment A: { y 1,..., y m }  2 v if this fraction 4 = 1  accept 8 <   reject zD, v and  may be a function of l

17. 17 The PCP Hierarchy Def: L  PCP[ D, V,  ] if L is efficiently reducible to gap-SAT[ D, V,  ]  Thm [AS,ALMSS] NP  PCP[ O(1), 1, ½] [ The PCP characterization theorem above ]  Thm [ RaSa ] NP  PCP[ O(1), m, 2 -m ] for m  log c n for some c > 0  Thm [ DFKRS ]NP  PCP[ O(1), m, 2 -m ] for m  log c n for any c > 0  Conjecture [BGLR] NP  PCP[ O(1), m, 2 -m ] for m  log n Focus of the rest of this presentation

18. 18 Optimal Characterization zOne cannot expect the error-probability to be less than exponentially small in the number of bits each local-test looks at ysince a random assignment would make such a fraction of the local-tests satisfied zOne cannot hope for smaller than polynomially small error-probability ysince it would imply less than one local-test satisfied, hence each local-test, being rather easy to compute, determines completely the outcome [ the BGLR conjecture is hence optimal in that respect ]

19. 19 Approximating MAX-CLIQUE is NP-hard Apply the same reduction above from SAT to MAX-CLIQUE (yielding G  ), to the gap-SAT of any of the above theorems The outcome is a gap-CLIQUE problem, which is therefore NP-hard for the corresponding factor We may conclude that  approximating MAX-CLIQUE is NP-hard