1 Slides by Asaf Shapira & Michael Lewin & Boaz Klartag & Oded Schwartz. Adapted from things beyond us.

2 Introduction In this lecture we’ll cover: Definition of PCP Prove some classical inapproximabillity results. Give a review on some other recent ones.

3 Review: Decision, Optimization Problems A 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) } An 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) ] A 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)

4 Review: 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

5 Review: NP-Hardness A 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 That is referred to as the more general, Cook reduction. An efficient algorithm, translating any NP problem to a single instance of L thereby showing that L NP-hard is referred to as Karp reduction.

6 Review: 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  and 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 Approximation - Some definitions Definition:  -approximation An  -approximation of a maximization (similar for minimization) function f, is a function, g, such that on input X, outputs g(X) such that: g(X)   f(X). Definition: PTAS (polynomial time approximation scheme) We say that a maximization function f, has a PTAS, if for every 1    0, there is a polynomial  -approximation for f, where the algorithm is polynomial in |X| and .

8 Approximation - NP-hard? We know that by using Cook/Karp reductions, we can show many decision problems to be NP-hard. Can an approximation problem be NP-Hard? One can easily show, that if there is ,for which there is an  -approximating for TSP, P=NP.

9 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

10 Probabilistically-Checkable-Proofs Hence, Cook-Levin theorem states that a verifier can efficiently verify membership-proofs for any NP language PCP characterization of NP, in contrast, states that a membership-proof can be verified probabilistically –by choosing randomly one local-test, –accessing the small set of variables it depends on, –accept or reject accordingly erroneously accepting a non-member only with small probability

11 Gap Problems A gap-problem is a maximization (or minimization) problem ƒ(X, Y), and two thresholds t 1 > t 2 X must be accepted if max Y [ ƒ(X, Y) ]  t 1 X must be rejected if max Y [ ƒ(X, Y) ]  t 2 other X’s may be accepted or rejected (don’t care) (almost a decision problem, relates to approximation)

12 Reducing gap-Problems to Approximation Problems Using 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 Simply 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 )

13 gap-SAT Def: 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 D, v and  may be a function of l

14 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

15 Optimal Characterization One cannot expect the error-probability to be less than exponentially small in the number of bits each local-test looks at –since a random assignment would make such a fraction of the local-tests satisfied One cannot hope for smaller than polynomially small error-probability –since 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]

16 Approximating MAX-CLIQUE is NP-hard We will reduce gap-SAT to gap -CLIQUE. Given an expression  = {    l } of Boolean- functions over variables y 1,...,y m of range 2 V, Each of  1  l depends on at most D variables, We must determine whether all the functions can be satisfied or only a fraction less than . We will construct a graph, G , such that it has a clique of size r  there exists an assignment, satisfying r of the functions y 1,...,y m.

17 Definition of G  For each  i  , G  has a vertex for every satisfying assignment of  i  1.  i..  l       All assignments to ’s variables All assignments to  i ’s variables Not satisfying Not satisfying  i Satisfying Satisfying  i

18 Definition of G  Two vertices are connected if the assignments are consistent  1.  i..  l       Consistent values NOT Consistent Different values of same variable

19 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 Properties of G 

20 Each of the following theorems gives a hardness of approximation result of Max-Clique: –Thm [AS,ALMSS] NP  PCP[ O(1), 1, ½] –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 Hardness of approximation of Max-Clique

21 We will show that if Life Is Meaningful (P  NP) Max-3Sat does not have a PTAS. Given an instance of gap-SAT,  = {    l }, we will transform each of the  i ‘s into a 3-SAT expression  i. As each of the  i ‘s depends on up to D variables. The equivalent  i expressions require exp(D) clauses. Since D = O(1) we still remain with a blow up of O(1) We define the equivalent 3-SAT expresion to be:  = The number of clauses in   exp(D)  l Hardness of approximation of Max-3SAT

22 If X  L then there is an assignment satisfying all l boolean functions of . Such an assignment satisfies all clauses of . If X  L then no assignment satisfies more then  l boolean functions of . Therefore no assignment satisfies more than |  | -  l. Therefore solving Gap-3SAT with thresholds t 1 = 1 and t 2 = 1 -  l/|  |  1 -  /exp(D) is NP-Hard. We conclude that there can be no PTAS for Max- 3SAT. Gap-3SAT is NP-Hard with thresholds 1 and 7/8+ . Can be solved with thresholds 1 and 7/8. Hardness of approximation of Max-3SAT

23 The PCP theorem has ushered in a new era of hardness of approximation results. Here we list a few: We showed that Max-Clique ( and equivalently Max- Independet-Set ) do not has a PTAS. It is known in addition, that to approximate it with a factor of n 1-  is hard unless co-RP = NP. Chromatic Number - It is NP-Hard to approximate it within a factor of n 1-  unless co-RP = NP. There is a simple reduction from Max-Clique which shows that it is NP-Hard to approximate with factor n . Chromatic Number for 3-colorable graph - NP-Hard to approximate with factor 5/3-  (i.e. to differentiate between 4 and 3). Can be approximated within O(n   log O(1) n). More Results Related to PCP

24 Set Cover - NP-Hard to approximate it within a factor of c  logn for some constant c. Can not be approximated within factor (c-  )  logn unless NP  Dtime(n loglogn ). More Results Related to PCP

25 Maximum Satisfying Linear Sub-System - The problem: Given a linear system Ax=b (A is n x m matrix ) in field F, find the largest number of equations that can be satisfied by some x. –If all equations can be satisfied the problem is in P. –If F=Q NP-Hard to approximate by factor m . Can be approximated in O(m/logm). –If F=GF(q) can be approximated by factor q (even a random assignment gives such a factor). NP-Hard to approximate within q- . Also NP-Hard for equations with only 3 variables. –For equations with only 2 variables. NP-Hard to approximated within but can be approximated within More Results Related to PCP