6 research outputs found
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
A fixed-parameter perspective on #BIS
The problem of (approximately) counting the independent sets of a bipartite
graph (#BIS) is the canonical approximate counting problem that is complete in
the intermediate complexity class #RH\Pi_1. It is believed that #BIS does not
have an efficient approximation algorithm but also that it is not NP-hard. We
study the robustness of the intermediate complexity of #BIS by considering
variants of the problem parameterised by the size of the independent set. We
exhaustively map the complexity landscape for three problems, with respect to
exact computation and approximation and with respect to conventional and
parameterised complexity. The three problems are counting independent sets of a
given size, counting independent sets with a given number of vertices in one
vertex class and counting maximum independent sets amongst those with a given
number of vertices in one vertex class. Among other things, we show that all of
these problems are NP-hard to approximate within any polynomial ratio. (This is
surprising because the corresponding problems without the size parameter are
complete in #RH\Pi_1, and hence are not believed to be NP-hard.) We also show
that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS,
whereas the other two are W[1]-hard to approximate even within any polynomial
ratio. Finally, we show that, when restricted to graphs of bounded degree, all
three problems have efficient exact fixed-parameter algorithms.Comment: to appear in Algorithmic
Approximately counting and sampling small witnesses using a colourful decision oracle
In this paper, we prove "black box" results for turning algorithms which decide whether or not a witness exists into algorithms to approximately count the number of witnesses, or to sample from the set of witnesses approximately uniformly, with essentially the same running time. We do so by extending the framework of Dell and Lapinskas (STOC 2018), which covers decision problems that can be expressed as edge detection in bipartite graphs given limited oracle access; our framework covers problems which can be expressed as edge detection in arbitrary k-hypergraphs given limited oracle access. (Simulating this oracle generally corresponds to invoking a decision algorithm.) This includes many key problems in both the fine-grained setting (such as k-SUM, k-OV and weighted k-Clique) and the parameterised setting (such as induced subgraphs of size k or weight-k solutions to CSPs). From an algorithmic standpoint, our results will make the development of new approximate counting algorithms substantially easier; indeed, it already yields a new state-of-the-art algorithm for approximately counting graph motifs, improving on Jerrum and Meeks (JCSS 2015) unless the input graph is very dense and the desired motif very small. Our k-hypergraph reduction framework generalises and strengthens results in the graph oracle literature due to Beame et al. (ITCS 2018) and Bhattacharya et al. (CoRR abs/1808.00691)
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