CIAC 2017: Algorithms and Complexity pp 55-66

# Completeness Results for Counting Problems with Easy Decision

• Eleni Bakali
• Aggeliki Chalki
• Aris Pagourtzis
• Petros Pantavos
• Stathis Zachos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)

## Abstract

Counting problems with easy decision are the only ones among problems in complexity class $$\#\textsc {P}$$ that are likely to be (randomly) approximable, under the assumption $$\textsc {RP}\ne \textsc {NP}$$. $$\text {TotP}$$ is a subclass of $$\#\textsc {P}$$ that contains many of these problems. $$\text {TotP}$$ and $$\#\textsc {P}$$ share some complete problems under Cook reductions, the approximability of which does not extend to all problems in these classes (if $$\textsc {RP}\ne \textsc {NP}$$); the reason is that such reductions do not preserve the function value. Therefore Cook reductions do not seem useful in obtaining (in)approximability results for counting problems in $$\text {TotP}$$ and $$\#\textsc {P}$$.

On the other hand, the existence of $$\text {TotP}$$-complete problems (apart from the generic one) under stronger reductions that preserve the function value has remained an open question thus far. In this paper we present the first such problems, the definitions of which are related to satisfiability of Boolean circuits and formulas. We also discuss implications of our results to the complexity and approximability of counting problems in general.

## Notes

### Acknowledgments

We would like to thank Antonis Antonopoulos for many useful discussions as well as the anonymous reviewers for their observations and corrections.

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