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MAX 3SAT

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MAX-3SAT is a problem in the computational complexity subfield of computer science. It generalises the Boolean satisfiability problem (SAT) which is a decision problem considered in complexity theory. It is defined as:

Contents

Given a 3-CNF formula Φ (i.e. with at most 3 variables per clause), find an assignment that satisfies the largest number of clauses.

MAX-3SAT is a canonical complete problem for the complexity class MAXSNP (shown complete in Papadimitriou pg. 314).

Approximability

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.

    Theorem 1 (inapproximability)

    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 L ∈ P C P ( O ( log ⁡ ( n ) ) , O ( 1 ) ) by the PCP theorem. For x ∈ L, a 3-CNF formula Ψx is constructed so that

  • x ∈ L ⇒ Ψx is satisfiable
  • x ∉ L ⇒ no more than (1-ε)m clauses of Ψx 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 Ri ∈ V, we obtain poly(x) strings since the length of each string r ( x ) = O ( log ⁡ | x | ) .
  • For each Ri
  • V chooses q positions i1,...,iq and a Boolean function fR: {0,1}q->{0,1} and accepts if and only if fR(π(i1,...,iq)). 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,...,xl, 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 fR(xi1,...,xiq) using 2q SAT clauses. Clauses of length q are converted to length 3 by adding new (auxiliary) variables e.g. x2 ∨ x10 ∨ x11 ∨ x12 = ( x2 ∨ x10 ∨ yR) ∧ ( yR ∨ x11 ∨ x12). This requires a maximum of q2q 3-SAT clauses.
  • If z ∈ L then
  • there is a proof π such that Vπ (z) accepts for every Ri.
  • All clauses are satisfied if xi = π(i) and the auxiliary variables are added correctly.
  • If input z ∉ L then
  • For every assignment to x1,...,xl and yR's, the corresponding proof π(i) = xi causes the Verifier to reject for half of all R ∈ {0,1}r(|z|).
  • For each R, one clause representing fR fails.
  • Therefore a fraction 1 2 1 q 2 q of clauses fails.

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

    Theorem 2

    Håstad 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 O ( log ⁡ ( n ) ) and computes query positions ir, jr, kr in the proof π and a bit br. It accepts if and only if

    π(ir) ⊕ π(jr) ⊕ π(kr) = br.

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

    z ∈ L ⟹ ∃ π P r [ V π ( x ) = 1 ] ≥ 1 − ϵ z ∉ L ⟹ ∀ π P r [ V π ( x ) = 1 ] ≤ 1 2 + ϵ

    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

    1 − 1 4 ( 1 2 − ϵ ) 1 − ϵ = 7 8 + ϵ ′

    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 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, and all B ≥ 3 (which is best possible).

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

    The best possible approximation ratio for MAX-3SAT(B), as a function of B, is at least 7 / 8 + Ω ( 1 / B ) and at most 7 / 8 + O ( 1 / B ) , unless NP=RP. Some explicit bounds on the approximability constants for certain values of B are known. 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.

    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 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 .

    References

    MAX-3SAT Wikipedia
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