Dell, Lapinskas and Meeks [DLM SICOMP 2022] presented a general reduction from approximate counting to decision for a class of fine-grained problems that can be viewed as hyperedge counting or detection problems in an implicit hypergraph, thus obtaining tight equivalences between approximate counting and decision for many key problems such as $k$-clique, $k$-sum and more. Their result is a reduction from approximately counting the number of hyperedges in an implicit $k$-partite hypergraph to a polylogarithmic number of calls to a hyperedge oracle that returns whether a given subhypergraph contains an edge. The main result of this paper is a generalization of the DLM result for {\em output-sensitive} approximate counting, where the running time of the desired counting algorithm is inversely proportional to the number of witnesses. Our theorem is a reduction from approximately counting the (unknown) number of hyperedges in an implicit $k$-partite hypergraph to a polylogarithmic number of calls to a hyperedge oracle called only on subhypergraphs with a small ``measure''. If a subhypergraph has $u_i$ nodes in the $i$th node partition of the $k$-partite hypergraph, then its measure is $\prod_i u_i$. Using the new general reduction and by efficiently implementing measure-bounded colorful independence oracles, we obtain new improved output-sensitive approximate counting algorithms for $k$-clique, $k$-dominating set and $k$-sum. In graphs with $n^t$ $k$-cliques, for instance, our algorithm $(1\pm \epsilon)$-approximates the $k$-clique count in time $$\tilde{O}_\epsilon(n^{\omega(\frac{k-t-1}{3},\frac{k-t}{3},\frac{k-t+2}{3}) }+n^2),$$ where $\omega(a,b,c)$ is the exponent of $n^a\times n^b$ by $n^b\times n^c$ matrix multiplication. For large $k$ and $t>2$, this is a substantial improvement over prior work, even if $\omega=2$.

翻译：Dell、Lapinskas和Meeks [DLM SICOMP 2022] 针对一类可视为隐式超图中超边计数或检测问题的细粒度问题，提出了从近似计数到判定的一般性归约，从而为$k$-团、$k$-和等许多关键问题建立了近似计数与判定之间的紧等价关系。他们的结果将隐式$k$-部超图中的超边近似计数问题，归约到对数多项式次调用超边预言机（该预言机返回给定子超图是否包含边）。本文的主要结果是对DLM结果的推广，旨在实现**输出敏感**的近似计数，即期望计数算法的运行时间与见证数量成反比。我们的定理将隐式$k$-部超图中（未知）超边数量的近似计数，归约到对数多项式次调用一种仅作用于具有小“测度”的子超图的超边预言机。若一个子超图在$k$-部超图的第$i$个节点划分中有$u_i$个节点，则其测度为$\prod_i u_i$。利用这一新的通用归约，并通过高效实现测度有界的彩色独立集预言机，我们为$k$-团、$k$-支配集和$k$-和问题获得了新的改进的输出敏感近似计数算法。例如，在含有$n^t$个$k$-团的图中，我们的算法以时间$$\tilde{O}_\epsilon(n^{\omega(\frac{k-t-1}{3},\frac{k-t}{3},\frac{k-t+2}{3}) }+n^2)$$对$k$-团数量进行$(1\pm \epsilon)$-近似，其中$\omega(a,b,c)$表示$n^a\times n^b$矩阵与$n^b\times n^c$矩阵相乘的指数。对于较大的$k$和$t>2$，即使假设$\omega=2$，这也是对先前工作的显著改进。