MAX-3SAT

The decision version of MAX-3SAT is NP-complete. Therefore, a polynomial-time solution can only be achieved if P = NP. An approximation within a factor of 2 can be achieved with this simple algorithm, however:

• Output the solution in which most clauses are satisfied, when either all variables = TRUE or all variables = FALSE.
• Every clause is satisfied by one of the two solutions, therefore one solution satisfies at least half of the clauses.

The Karloff-Zwick algorithm runs in polynomial-time and satisfies ≥ 7/8 of the clauses. While this algorithm is randomized, it can be derandomized using, e.g., the techniques from [1] to yield a deterministic (polynomial-time) algorithm with the same approximation guarantees.

The PCP theorem implies that there exists an ε > 0 such that (1-ε)-approximation of MAX-3SAT is NP-hard.

Proof:

Any NP-complete problem ${\displaystyle L\in {\mathsf {PCP}}(O(\log(n)),O(1))}$ by the PCP theorem. For x ∈ L, a 3-CNF formula Ψ_x is constructed so that

• x ∈ L ⇒ Ψ_x is satisfiable
• x ∉ L ⇒ no more than (1-ε)m clauses of Ψ_x are satisfiable.

The Verifier V reads all required bits at once i.e. makes non-adaptive queries. This is valid because the number of queries remains constant.

• Let q be the number of queries.
• Enumerating all random strings R_i ∈ V, we obtain poly(x) strings since the length of each string ${\displaystyle r(x)=O(\log |x|)}$.
• For each R_i
  • V chooses q positions i₁,...,i_q and a Boolean function f_R: {0,1}^q->{0,1} and accepts if and only if f_R(π(i₁,...,i_q)). Here π refers to the proof obtained from the Oracle.

Next we try to find a Boolean formula to simulate this. We introduce Boolean variables x₁,...,x_l, where l is the length of the proof. To demonstrate that the Verifier runs in Probabilistic polynomial-time, we need a correspondence between the number of satisfiable clauses and the probability the Verifier accepts.

• For every R, add clauses representing f_R(x_i1,...,x_iq) using 2^q SAT clauses. Clauses of length q are converted to length 3 by adding new (auxiliary) variables e.g. x₂ ∨ x₁₀ ∨ x₁₁ ∨ x₁₂ = ( x₂ ∨ x₁₀ ∨ y_R) ∧ ( y_R ∨ x₁₁ ∨ x₁₂). This requires a maximum of q2^q 3-SAT clauses.
• If z ∈ L then
  • there is a proof π such that V^π (z) accepts for every R_i.
  • All clauses are satisfied if x_i = π(i) and the auxiliary variables are added correctly.
• If input z ∉ L then
  • For every assignment to x₁,...,x_l and y_R's, the corresponding proof π(i) = x_i causes the Verifier to reject for half of all R ∈ {0,1}^r(|z|).
    • For each R, one clause representing f_R fails.
    • Therefore, a fraction ${\displaystyle {\frac {1}{2}}{\frac {1}{q2^{q}}}}$ of clauses fails.

It can be concluded that if this holds for every NP-complete problem then the PCP theorem must be true.

Håstad [2] demonstrates a tighter result than Theorem 1 i.e. the best known value for ε.

He constructs a PCP Verifier for 3-SAT that reads only 3 bits from the Proof.

For every ε > 0, there is a PCP-verifier M for 3-SAT that reads a random string r of length ${\displaystyle O(\log(n))}$ and computes query positions i_r, j_r, k_r in the proof π and a bit b_r. It accepts if and only if 'π(i_r) ⊕ π(j_r) ⊕ π(k_r) = b_r.

The Verifier has completeness (1−ε) and soundness 1/2 + ε (refer to PCP (complexity)). The Verifier satisfies

${\displaystyle z\in L\implies \exists \pi Pr[V^{\pi }(x)=1]\geq 1-\epsilon }$

${\displaystyle z\not \in L\implies \forall \pi Pr[V^{\pi }(x)=1]\leq {\frac {1}{2}}+\epsilon }$

If the first of these two equations were equated to "=1" as usual, one could find a proof π by solving a system of linear equations (see MAX-3LIN-EQN) implying P = NP.

• If z ∈ L, a fraction ≥ (1 − ε) of clauses are satisfied.
• If z ∉ L, then for a (1/2 − ε) fraction of R, 1/4 clauses are contradicted.

This is enough to prove the hardness of approximation ratio

${\displaystyle {\frac {1-{\frac {1}{4}}({\frac {1}{2}}-\epsilon )}{1-\epsilon }}={\frac {7}{8}}+\epsilon '}$

MAX-3SAT(B) is the restricted special case of MAX-3SAT where every variable occurs in at most B clauses. Before the PCP theorem was proven, Papadimitriou and Yannakakis[3] showed that for some fixed constant B, this problem is MAX SNP-hard. Consequently, with the PCP theorem, it is also APX-hard. This is useful because MAX-3SAT(B) can often be used to obtain a PTAS-preserving reduction in a way that MAX-3SAT cannot. Proofs for explicit values of B include: all B ≥ 13,[4][5] and all B ≥ 3[6] (which is best possible).

Moreover, although the decision problem 2SAT is solvable in polynomial time, MAX-2SAT(3) is also APX-hard.[6]

The best possible approximation ratio for MAX-3SAT(B), as a function of B, is at least ${\displaystyle 7/8+\Omega (1/B)}$ and at most ${\displaystyle 7/8+O(1/{\sqrt {B}})}$,[7] unless NP=RP. Some explicit bounds on the approximability constants for certain values of B are known.[8] [9] [10] Berman, Karpinski and Scott proved that for the "critical" instances of MAX-3SAT in which each literal occurs exactly twice, and each clause is exactly of size 3, the problem is approximation hard for some constant factor.[11]

MAX-EkSAT is a parameterized version of MAX-3SAT where every clause has exactly k literals, for k ≥ 3. It can be efficiently approximated with approximation ratio ${\displaystyle 1-(1/2)^{k}}$ using ideas from coding theory.

It has been proved that random instances of MAX-3SAT can be approximated to within factor 8/9.[12]

Lecture Notes from University of California, Berkeley Coding theory notes at University at Buffalo