8 research outputs found
An approximation algorithm for #k-SAT
"Vegeu el resum a l'inici del document del fitxer adjunt"
An Approximation Algorithm for #k-SAT
We present a simple randomized algorithm that approximates the number of
satisfying assignments of Boolean formulas in conjunctive normal form. To the
best of our knowledge this is the first algorithm which approximates #k-SAT for
any k >= 3 within a running time that is not only non-trivial, but also
significantly better than that of the currently fastest exact algorithms for
the problem. More precisely, our algorithm is a randomized approximation scheme
whose running time depends polynomially on the error tolerance and is mildly
exponential in the number n of variables of the input formula. For example,
even stipulating sub-exponentially small error tolerance, the number of
solutions to 3-CNF input formulas can be approximated in time O(1.5366^n). For
4-CNF input the bound increases to O(1.6155^n).
We further show how to obtain upper and lower bounds on the number of
solutions to a CNF formula in a controllable way. Relaxing the requirements on
the quality of the approximation, on k-CNF input we obtain significantly
reduced running times in comparison to the above bounds
Fine-Grained Reductions from Approximate Counting to Decision
In this paper, we introduce a general framework for fine-grained reductions
of approximate counting problems to their decision versions. (Thus we use an
oracle that decides whether any witness exists to multiplicatively approximate
the number of witnesses with minimal overhead.) This mirrors a foundational
result of Sipser (STOC 1983) and Stockmeyer (SICOMP 1985) in the
polynomial-time setting, and a similar result of M\"uller (IWPEC 2006) in the
FPT setting. Using our framework, we obtain such reductions for some of the
most important problems in fine-grained complexity: the Orthogonal Vectors
problem, 3SUM, and the Negative-Weight Triangle problem (which is closely
related to All-Pairs Shortest Path).
We also provide a fine-grained reduction from approximate #SAT to SAT.
Suppose the Strong Exponential Time Hypothesis (SETH) is false, so that for
some and all there is an -time algorithm for k-SAT. Then we
prove that for all , there is an -time algorithm for
approximate #-SAT. In particular, our result implies that the Exponential
Time Hypothesis (ETH) is equivalent to the seemingly-weaker statement that
there is no algorithm to approximate #3-SAT to within a factor of
in time (taking as part of the input).Comment: An extended abstract was presented at STOC 201
Nearly optimal independence oracle algorithms for edge estimation in hypergraphs
We study a query model of computation in which an n-vertex k-hypergraph can
be accessed only via its independence oracle or via its colourful independence
oracle, and each oracle query may incur a cost depending on the size of the
query. In each of these models, we obtain oracle algorithms to approximately
count the hypergraph's edges, and we unconditionally prove that no oracle
algorithm for this problem can have significantly smaller worst-case oracle
cost than our algorithms
An approximation algorithm for #k-SAT
"Vegeu el resum a l'inici del document del fitxer adjunt"