Abstract
It is common to classify satisfiability problems by their time complexity. We consider another complexity measure, namely the length of certificates (witnesses). Our results show that there is a similarity between these two types of complexity if we deal with certificates verifiable in subexponential time. In particular, the well-known result by Impagliazzo and Paturi [IP01] on the dependence of the time complexity of \(k\mbox{\tt-}\mathtt{SAT}\) on k has its counterpart for the certificate complexity: we show that, assuming the exponential time hypothesis (ETH), the certificate complexity of \(k\mbox{\tt-}\mathtt{SAT}\) increases infinitely often as k grows. Another example of time-complexity results that can be translated into the certificate-complexity setting is the results of [CIP06] on the relationship between the complexity of \(k\mbox{\tt-}\mathtt{SAT}\) and the complexity of SAT restricted to formulas of constant clause density. We also consider the certificate complexity of CircuitSAT and observe that if CircuitSAT has subexponential-time verifiable certificates of length cn, where cā<ā1 is a constant and n is the number of inputs, then an unlikely collapse happens (in particular, ETH fails).
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Dantsin, E., Hirsch, E.A. (2011). Satisfiability Certificates Verifiable in Subexponential Time. In: Sakallah, K.A., Simon, L. (eds) Theory and Applications of Satisfiability Testing - SAT 2011. SAT 2011. Lecture Notes in Computer Science, vol 6695. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21581-0_4
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DOI: https://doi.org/10.1007/978-3-642-21581-0_4
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