Keren Censor-Hillel

Keren Censor-Hillel, Aditi Dudeja, George Giakkoupis

We study stochastic graph optimization problems in a novel distributed setting. As in the standard centralized setting, a random subgraph $G^*$ of a known base graph $G$ is realized by including each edge $e$ independently with a known probability $p_e$, and we must solve an optimization problem on $G^*$ despite uncertainty about its edges. In the standard setting, to cope with this uncertainty, the algorithm can query any edge of $G$ to learn if the edge exists in $G^*$, and its complexity is the number of queried edges. The distributed setting incorporates uncertainty in a natural manner, by having each vertex know only about its own edges in $G^*$ (and only communicate over them), and the complexity is measured by the number of synchronous communication rounds. We establish that distributed stochastic algorithms can be drastically faster than their non-stochastic counterparts and overcome known lower bounds, by showing fast distributed approximation algorithms for maximum matching, minimum vertex cover, and minimum dominating set.

Keren Censor-Hillel, Tomer Even, Virginia Vasillevska Williams et al.

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.