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I am quoting a phrase of Martin Dyer in his paper Approximate Counting by Dynamic Programming:

Since 0-1 knapsack is self-reducible, existence of an fpras for the problem now follows indirectly from a general result of Sinclair and Jerrum[19]

In that paper of Sinclair and Jerrum it is stated that:

It follows that, for self-reducible structures, polynomial time randomised algorithms for counting to within factors of the form (1 +$n^{-\beta}$) are available either for all $\beta \in R$ or for no $\beta \in R$.

Question 1: Does statement 2 mean that counting self reducible structures might have an FPRAS?

Question 2: What is the indirect method Dyer is talking about? Is it some folklore method considered well known?

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This theorem from that paper of Sinclair and Jerrum is slightly stronger than the sentence quoted in the question, and gives the fpras mentioned by Dyer:

THEOREM 4.6. Let $R\subseteq \Sigma^*\times\Sigma^*$ be self-reducible. If there exists a polynomially time-bounded randomised approximate counter for $R$ within ratio $1 + O(n^\alpha)$ for some $\alpha\in\mathbb R$, then there exists a fully polynomial randomised approximation scheme for $\#R$.

Note the quantification over $R$. Your first question asks if "counting self-reducible structures might have an FPRAS", but this theorem is looking at particular $R$'s and showing that if $R$ has a weak kind of approximation algorithm, then $R$ also has a stronger kind of approximation algorithm.

Jerrum and Sinclair explain it like this:

The chief significance of Theorem 4.6 is that it establishes a notion of approximate counting which is robust with respect to polynomial time computation, at least for the large class of self-reducible relations [...] We suggest that this notion will be useful in the future classification of hard counting problems [...]

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  • $\begingroup$ Two small questions. What type of self-reducibility are we talking about here? Randomised self-reducibility? And second, any more specific ideas for question 2? Thanks in advance $\endgroup$ – Paramar Feb 19 '15 at 3:13
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    $\begingroup$ Check out Sinclair and Jerrum's paper (cs.berkeley.edu/~sinclair/approx.pdf) for the definition of self-reducibility, informal (p.94) and formal (p.99). It's quite different from randomised self-reducibility. For question 2 - I believe the theorem I quoted is the "indirect" general result being referred to. $\endgroup$ – Colin McQuillan Feb 20 '15 at 7:32

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