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CS221: Computational Complexity Prof. Salil Vadhan
Lecture 34: More Hardness of Approximation,
NP ⊆ PCP[polylog, polylog]
12/16 Scribe: Vitaly Feldman
Contents
1 More Tight Hardness of Approximation Results 1
2 Open Problems 2
3 Sketch of the Proof of the PCP Theorem 2
Additional Readings. Papadimitriou, Ch. 13 has a detailed treatment of approximability, with less emphasis on the PCP theorem (which is stated in different language in Thm 13.12).
1 More Tight Hardness of Approximation Results
Last time we saw the tight results for hardness of approximation of Max3SAT and E3LIN2. Today we will survey some other tight hardness of approximation results.
Definition 1 MaxClique — given a graph G = (V, E) find the largest clique.
This problem is completely equivalent to finding the maximum independent set. Best known approximation algorithm gives (^) logn (^2) n -approximation (which is not much better than the trivial n-approximation).
Theorem 2 (H˚astad) It is NP-hard to approximate MaxClique within n^1 −≤^ factor for any positive ≤.
The result is proven by tailoring the PCP Theorem for this specific problem (namely, optimizing the so-called “amortized free bit complexity” of the PCP).
Definition 3 Set Cover problem: given a collection of sets S 1 ,... , Sm which are subsets of the set { 1 ,... , n}, find the smallest subcollection of sets Si 1 ,... , Sit whose union is equal to { 1 ,... , n} (that is, they cover the “universe”).
The greedy algorithm gives H(n) ≈ ln n approximation to this problem (H(n) is the n-th harmonic number).
Theorem 4 (Feige) It is NP-hard to approximate Set Cover to within (1 − ≤) ln n factor for any positive ≤.
These results show that NP-complete problems, while equivalent as languages, exhibit wide variety of properties with respect to approximation of their optimization versions. In particular, the best achievable approximation factors for various problems ranged from constant (Max3SAT, E3LIN2), to logarithmic (Set Cover), to polynomial (MaxClique).
2 Open Problems
Besides the tight results that we saw, there are several problems for which the current results are not tight. We will survey some of the most important ones.
Definition 5 Vertex Cover: given a graph G find the smallest set of vertices that cover all the edges.
There is a long known algorithm for this problem that approximates the solution within a factor of 2. On the other hand, the best known hardness of approximation result for this problem is for ≈ 1 .36 (in a recent paper by Dinur and Safra [DS02]).
Definition 6 Graph Coloring: given a graph G, find a coloring of the vertices of this graph that uses the minimum number of colors.
H˚astad’s clique paper also shows that this problem is hard to approximate to within n^1 −≤. However, a more interesting problem is that of trying to color a graph which is promised to be 3-colorable. There is a very nontrivial n 143 -approximation algorithm for this problem. On the other hand the best known hardness of approximation factor is 43. (This result does not use the PCP theorem.)
Definition 7 Shortest Vector problem: given a lattice in Rn^ (specified by a basis consisting of n linearly independent vectors), find the shortest nonzero vector in the lattice. (Similarly, there is the Closest Vector problem, where you are also given a target vector, and the problem is to find the lattice vector closest to the target.)
These problems are of particular significance in cryptography (partly because of a worst-case/average- case equivalence established for a variant of approximating Shortest Vector). The best known polynomial-time approximation algorithm for these problems achieves a ≈ 2 n/^ log^ n^ approximation factor, but the best known hardness of approximation results are
2 − ≤ and nO(1/loglogn)^ respec- tively.
3 Sketch of the Proof of the PCP Theorem
We are now going to show some of the ideas that are involved in the proof the PCP theorem. We will “almost” show the following result which was proved independently by Babai et al. [BFLS91] and Feige et al. [FGLSS91].
And finally, the verifier chooses a random αm and verifies that pm(αm) = P (α 1 ,... , αm). In order to do this, the verifier needs to evaluate P , which in turn requires 3 evaluations of Aˆ. This can be done by making the complete description of Aˆ part of the PCP.
The parameters. Even though this protocol will not work as is, let us examine the parameters of this protocol. The verifier needs to choose m random αi’s. By our definitions, m log |F| = O(log n log log n). Thus the randomness complexity is polylogarithmic, as desired. For the query complexity, we need to specify the coefficient of the pi’s. The degree of each pi is bounded by a degree of P in variable number i, which, as can be easily seen from the definition of P , is O(log n). Each coefficient is just a field element. Three more field elements are required for the evaluations of Aˆ. The complete computation yields that the total number of bits read is O(log^2 n log log n). Thus the algorithm achieves the parameters of the theorem but does not actually work ...
The problem. The problem with this protocol is that its soundness relies on the fact that the polynomial P for which we do the sum check protocol is a low degree polynomial (in the IP = PSPACE protocol the verifier knew the polynomial itself while here it depends on Aˆ which is given to the verifier as a table in the PCP). Unfortunately, the verifier cannot know whether the table for Aˆ is actually a multilinear (or at least low-degree) polynomial. This allows the dishonest prover to use any function as Aˆ and thus to cheat the verifier in the last step.^1 This gives rise to the following problem which is exactly the missing component.
Definition 9 Low degree testing: given an oracle access to an arbitrary function f : Fm^ → F check whether f is a low-degree polynomial and do so using only a “small” number of queries to the oracle.
We will discuss this problem in the next lecture.
References
[BFLS91] Laszlo Babai, Lance Fortnow, Leonid Levin, and Mario Szegedy. Checking Computations in Polylogarithmic Time. STOC 1991, pp. 21–31.
[DS02] Irit Dinur, Shmuel Safra. The importance of being biased. STOC 2002, pp: 33–42.
[FGLSS91] Uriel Feige, Shafi Goldwasser, Laszlo Lovasz, Shmuel Safra, Mario Szegedy. Approxi- mating Clique is Almost NP-Complete. FOCS 1991, pp. 2–
(^1) Another problem is that the prover may write a low-degree polynomial for Aˆ, but one which does not have { 0 , 1 } values on { 0 , 1 }m, making it possible that the sum-check Equation (2) holds even though the Equations (1) do not. A fix for this is to instead have the prover prove that a “random” linear combination of {P (x) : x ∈ { 0 , 1 }m} is zero. (The coefficients are chosen by the verifier in a way that uses few random bits and guarantees that the sum will be nonzero with high probability if any of the P (x)’s are nonzero.)
