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:

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[edit]

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)[edit]

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

  • xL ⇒ Ψx is satisfiable
  • xL ⇒ 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 RiV, 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. x2x10x11x12 = ( x2x10yR) ∧ ( \bar{y_R}x11x12). This requires a maximum of q2q 3-SAT clauses.
  • If zL 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 zL 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 \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.

Theorem 2[edit]

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

z \not \in L \implies \forall \pi Pr[V^{\pi} (x) = 1] \le \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

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

Related problems[edit]

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[1] 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,[2][3] and all B ≥ 3[4] (which is best possible).

Moreover, although the decision problem 2SAT is solvable in polynomial time, MAX-2SAT(3) is also APX-hard.[4]

The best possible approximation ratio for MAX-3SAT(B), as a function of B, is at least 7/8+\Omega(1/B) and at most 7/8+O(1/\sqrt{B}),[5] unless NP=RP. Some explicit bounds on the approximability constants for certain values of B are known.[6] [7] [8] 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.[9]

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 9/8.[10]

References[edit]

  1. ^ Christos Papadimitriou and Mihalis Yannakakis, Optimization, approximation, and complexity classes, Proceedings of the twentieth annual ACM symposium on Theory of computing, p.229-234, May 02–04, 1988.
  2. ^ Rudich et al., "Computational Complexity Theory," IAS/Park City Mathematics Series, 2004 page 108 ISBN 0-8218-2872-X
  3. ^ Sanjeev Arora, "Probabilistic Checking of Proofs and Hardness of Approximation Problems," Revised version of a dissertation submitted at CS Division, U C Berkeley, in August 1994. CS-TR-476-94. Section 7.2.
  4. ^ a b Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti Spaccamela, A., and Protasi, M. (1999), Complexity and Approximation. Combinatorial Optimization Problems and their Approximability Properties, Springer-Verlag, Berlin. Section 8.4.
  5. ^ Luca Trevisan. 2001. Non-approximability results for optimization problems on bounded degree instances. In Proceedings of the thirty-third annual ACM symposium on Theory of computing (STOC '01). ACM, New York, NY, USA, 453-461. DOI=10.1145/380752.380839 http://doi.acm.org/10.1145/380752.380839
  6. ^ On some tighter inapproximability results, Piotr Berman and Marek Karpinski, Proc. ICALP 1999, pages 200--209.
  7. ^ P. Berman and M. Karpinski, Improved Approximation Lower Bounds on Small Occurrence Optimization, ECCC TR 03-008 (2003)
  8. ^ P. Berman, M. Karpinski and A. D. Scott, Approximation Hardness and Satisfiability of Bounded Occurrence Instances of SAT, ECCC TR 03-022 (2003).
  9. ^ P. Berman, M. Karpinski and A. D. Scott, Approximation Hardness of Short Symmetric Instances of MAX-3SAT, ECCC TR 03-049 (2003).
  10. ^ W.F.de la Vega and M.Karpinski, 9/8-Approximation Algorithm for Random MAX-3SAT, ECCC TR 02-070 (2002);RAIRO-Operations Research 41(2007),pp.95-107]

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