Witness-Sensitive Detection of Induced Diamonds

2026-05-09

AI Summary

This work addresses the problem of efficiently detecting at least one induced diamond (a $K_4$ minus an edge) in an $n$-vertex graph containing $t$ such induced subgraphs. The authors propose a witness-sensitive algorithm that introduces a novel “heavy/light” diamond classification based on clique size, combined with a structure-adaptive sampling scheme and a vector refinement framework. Their approach uniquely integrates witness sensitivity with conditionally optimal matrix multiplication scheduling $\text{MM}(a,b,c)$, achieving particularly high efficiency in graphs without $r$-heavy diamonds. The algorithm detects an induced diamond with high probability in $\widetilde{O}(\min(n^{2.425}/t^{0.25} + n^2, n^\omega))$ time, improving upon the previous $O(n^\omega \log n)$ witness-oblivious method—especially when $t \geq n^{(3-\omega)/3}$—and extends naturally to related problems such as 4-SUM and certain 4-cycle detection tasks.

Abstract

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.

Problem

• induced diamond
• witness-sensitive
• graph detection
• combinatorial algorithms
• matrix multiplication

Innovation

• witness-sensitive
• induced diamond
• r-heavy clique
• adaptive sampling
• matrix multiplication