We provide a fast \emph{witness-sensitive} algorithm for detecting an induced diamond (a $K_4$ minus an edge) in an $n$-vertex graph containing $t$ induced diamonds. Our algorithm runs in time $\tilde{O}(\min(n^{2.425}/t^{0.25}+n^2, n^ω))$ with high probability, improving upon the prior state of the art (witness-oblivious) algorithm that runs in time $O(n^ω\log{n})$ [Vassilevska Williams, Wang, Williams, Yu, SODA 2014] whenever $t \geq n^{(3-ω)/3}$, where $ω< 2.372$ is the matrix multiplication exponent. Our key insight is that the size of a clique containing one of the triangles of an induced diamond plays a crucial role in detecting such a diamond. We say that a diamond is $r$-heavy if this size is at least $r$, and we provide a fast detection algorithm for $r$-heavy diamonds in $\tilde{O}(r \cdot (n/r)^ω+ (n/r)^3+ nr)$ time. When there are no $r$-heavy diamonds, we provide a different fast detection algorithm in $\tilde{O}(\mathsf{MM}(n,n,n\sqrt{r/t}))$ time, where $\mathsf{MM}(a,b,c)$ denotes the time to multiply an $a \times b$ matrix by a $b \times c$ matrix, which is conditionally optimal for $r=\tilde{O}(1)$. Our main technical contribution is in designing a refinement framework for sampling vectors, which allows sampling vertices for detecting diamonds in a manner that is adaptive to the structure of graphs with no $r$-heavy diamonds. We establish that our technique is of a wide applicability, by showing how it also allows for faster witness-sensitive algorithms for $4$-SUM and for a special case of $4$-cycles.

翻译：我们提出一种快速的\textit{见证敏感}算法，用于在包含$t$个诱导菱形的$n$顶点图中检测诱导菱形（$K_4$减一条边）。该算法以高概率在$\tilde{O}(\min(n^{2.425}/t^{0.25}+n^2, n^ω))$时间内运行，改进了此前运行时间为$O(n^ω\log{n})$的最优（见证无关）算法[Vassilevska Williams, Wang, Williams, Yu, SODA 2014]（当$t \geq n^{(3-ω)/3}$时），其中$ω< 2.372$为矩阵乘法指数。我们的关键洞察在于：包含诱导菱形中一个三角形的团的大小对检测此类菱形起到决定性作用。我们将满足该大小至少为$r$的菱形称为$r$-重菱形，并针对$r$-重菱形提出$\tilde{O}(r \cdot (n/r)^ω+ (n/r)^3+ nr)$时间内的快速检测算法。当不存在$r$-重菱形时，我们提出另一种$\tilde{O}(\mathsf{MM}(n,n,n\sqrt{r/t}))$时间内的快速检测算法，其中$\mathsf{MM}(a,b,c)$表示$a \times b$矩阵与$b \times c$矩阵相乘所需时间，该算法对于$r=\tilde{O}(1)$条件最优。我们的主要技术贡献在于设计了一种采样向量的细化框架，能够以自适应于无$r$-重菱形图结构的方式对采样顶点进行检测。我们通过展示该技术同时适用于$4$-SUM问题以及$4$-环特殊情形的更快速见证敏感算法，证明了其具有广泛适用性。