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An equivalent formulation of PCP theorem is: For Max 3-SAT it is $NP$-hard to distinguish between satisfiable formulas and formulas where at most $r$-fraction of the clauses are satisfiable (for some $r\lt 1$).

Is there any known dichotomy theorem that classifies all Max CSP based on whether they have hard gaps or not?

Edit Dec 16, 2010: MAX CSP with hard gap means that the problem has optimal inapproximability factor. For instance, 3SAT has hard gap at location one since it is polynomial time approximable to a factor $7/8$ but it is $NP$-hard to obtain approximation factor $7/8+ \epsilon$ even when all clauses are satisfiable.

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3 Answers 3

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Prasad Raghavendra in the STOC'08 best paper proved a dichotomy conjecture for approximating Max-CSP assuming the Unique Games Conjecture. This is not how he presented it originally, but he did give talks presenting things this way a couple of years later, e.g., at the IAS, where it was videotaped: http://www.math.ias.edu/seminars/abstract?event=36669

The difference from showing SNP-hardness is that here we talk about quantitatively optimal results.

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    $\begingroup$ what does 'quantitatively optimal' mean ? $\endgroup$
    – Suresh Venkat
    Oct 6, 2010 at 2:35
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    $\begingroup$ hardness factor that matches the best known approximation algorithm $\endgroup$
    – Dana Moshkovitz
    Oct 6, 2010 at 17:13
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Theorem 5.14 of Khanna, Sudan, Trevisan and Williamson [KSTW01] gives a dichotomy theorem for the gap versions with perfect completeness for the boolean MaxCSP problems.

[KSTW01] Sanjeev Khanna, Madhu Sudan, Luca Trevisan and David P. Williamson. The approximability of constrant satisfaction problems. SIAM Journal on Computing, 30(6):1863–1920, 2001. http://dx.doi.org/10.1137/S0097539799349948

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  • $\begingroup$ Interesting paper. How is this dichotomy theorem related to Raghavendra's result in Dana's answer?. $\endgroup$
    – Mohammad Al-Turkistany
    Oct 23, 2010 at 13:24
  • $\begingroup$ I think that the results are fairly different. The theorem in [KSTW01] which I mentioned in this answer is about perfect completeness version whereas Raghavendra’s result is not. The theorem in [KSTW01] is about boolean CSP, whereas Raghavendra’s is about CSP over any domain. But you should check by yourself, because I do not know the paper by Raghavendra well. $\endgroup$
    – Tsuyoshi Ito
    Oct 23, 2010 at 13:35
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If I'm not mistaken, the definitive result on this front is by Nadia Creignou, who showed that every problem in MAX CSP is either polytime solvable, or is MAX SNP-hard.

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  • $\begingroup$ MAX 2-SAT is MAX SNP-hard but very easily solvable for satisfiable instances (2-SAT $\in P$ while 3-SAT is not known to be in $P$). It is $NP$-hard to have $7/8+\epsilon$ -approximation for Max 3-SAT even for satisfiable instances (for any $\epsilon \gt 0$). $\endgroup$
    – Mohammad Al-Turkistany
    Oct 6, 2010 at 8:04

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