Gopinath Mishra

Sam Coy, Artur Czumaj, Gopinath Mishra

We consider the classic $k$-center problem {in the constant dimensional Euclidean space} under a parallel setting, on the low-local-space Massively Parallel Computation (MPC) model, with local space per machine of ${O}(n^δ)$, where $δ\in (0,1)$ is an arbitrary constant. As a central clustering problem, the $k$-center problem has been studied extensively. Still, until very recently, all parallel MPC algorithms have been requiring $Ω(k)$ or even $Ω(k n^δ)$ local space per machine. While this setting covers the case of small values of $k$, for a large number of clusters these algorithms require large local memory, making them poorly scalable. The case of large $k$, $k \ge Ω(n^δ)$, has been considered recently for the low-local-space MPC model by Bateni et al.\ (2021), who gave an ${O}(\log \log n)$-round MPC algorithm that produces $k(1+o(1))$ centers whose cost has multiplicative approximation of ${O}(\log\log\log n)$. In this paper we extend the algorithm of Bateni et al. and design a low-local-space MPC algorithm that in ${O}(\log\log n)$ rounds returns a clustering with $k(1+o(1))$ clusters that is an ${O}(\log^*n)$-approximation for $k$-center.

Sam Coy, Artur Czumaj, Peter Davies et al.

We consider the distributed complexity of the (degree+1)-list coloring problem, in which each node $u$ of degree $d(u)$ is assigned a palette of $d(u)+1$ colors, and the goal is to find a proper coloring using these color palettes. The (degree+1)-list coloring problem is a natural generalization of the classical $(Δ+1)$-coloring and $(Δ+1)$-list coloring problems, both being benchmark problems extensively studied in distributed and parallel computing. In this paper we settle the complexity of the (degree+1)-list coloring problem in the Congested Clique model by showing that it can be solved deterministically in a constant number of rounds.

Yi-Jun Chang, Yanyu Chen, Dipan Dey et al.

We investigate the \emph{minimum weight cycle (MWC)} problem in the $\mathsf{CONGEST}$ model of distributed computing. For undirected weighted graphs, we design a randomized algorithm that achieves a $(k+1)$-approximation, for any \emph{real} number $k \ge 1$. The round complexity of algorithm is \[ \tilde{O}\!\Big( n^{\frac{k+1}{2k+1}} + n^{\frac{1}{k}} + D\, n^{\frac{1}{2(2k+1)}} + D^{\frac{2}{5}} n^{\frac{2}{5}+\frac{1}{2(2k+1)}} \Big). \] where $n$ denotes the number of nodes and $D$ is the unweighted diameter of the graph. This result yields a smooth trade-off between approximation ratio and round complexity. In particular, when $k \geq 2$ and $D = \tilde{O}(n^{1/4})$, the bound simplifies to \[ \tilde{O}\!\left( n^{\frac{k+1}{2k+1}} \right) \] On the lower bound side, assuming the Erdős girth conjecture, we prove that for every \emph{integer} $k \ge 1$, any randomized $(k+1-ε)$-approximation algorithm for MWC requires \[ \tildeΩ\!\left( n^{\frac{k+1}{2k+1}} \right) \] rounds. This lower bound holds for both directed unweighted and undirected weighted graphs, and applies even to graphs with small diameter $D = Θ(\log n)$. Taken together, our upper and lower bounds \emph{match up to polylogarithmic factors} for graphs of sufficiently small diameter $D = \tilde{O}(n^{1/4})$ (when $k \geq 2$), yielding a nearly tight bound on the distributed complexity of the problem. Our results improve upon the previous state of the art: Manoharan and Ramachandran (PODC~2024) demonstrated a $(2+ε)$-approximation algorithm for undirected weighted graphs with round complexity $\tilde{O}(n^{2/3}+D)$, and proved that for any arbitrarily large number $α$, any $α$-approximation algorithm for directed unweighted or undirected weighted graphs requires $Ω(\sqrt{n}/\log n)$ rounds.