Distributed Approximation on Power Graphs

TL;DR

This paper explores the complexity of solving graph problems on the square of a graph in distributed and centralized models, revealing surprising structural speedups and congestion-induced slowdowns for problems like MVC and MDS.

Contribution

It provides new distributed algorithms and hardness results for graph problems on G^2, highlighting the contrasting effects of congestion and structural properties.

Findings

O(n/ε)-round approximation for MVC on G^2 in CONGEST

Polynomial-time 5/3-approximation for MVC on G^2 in centralized model

Ω(n^2) lower bound for MDS on G^2 in CONGEST

Abstract

We investigate graph problems in the following setting: we are given a graph $G$ and we are required to solve a problem on $G^2$. While we focus mostly on exploring this theme in the distributed CONGEST model, we show new results and surprising connections to the centralized model of computation. In the CONGEST model, it is natural to expect that problems on $G^2$ would be quite difficult to solve efficiently on $G$, due to congestion. However, we show that the picture is both more complicated and more interesting. Specifically, we encounter two phenomena acting in opposing directions: (i) slowdown due to congestion and (ii) speedup due to structural properties of $G^2$. We demonstrate these two phenomena via two fundamental graph problems, namely, Minimum Vertex Cover (MVC) and Minimum Dominating Set (MDS). Among our many contributions, the highlights are the following. - In the CONGEST model, we show an $O(n/\epsilon)$-round $(1+\epsilon)$-approximation algorithm for MVC on $G^2$, while no $o(n^2)$-round algorithm is known for any better-than-2 approximation for MVC on $G$. - We show a centralized polynomial time $5/3$-approximation algorithm for MVC on $G^2$, whereas a better-than-2 approximation is UGC-hard for $G$. - In contrast, for MDS, in the CONGEST model, we show an $\tilde{\Omega}(n^2)$ lower bound for a constant approximation factor for MDS on $G^2$, whereas an $\Omega(n^2)$ lower bound for MDS on $G$ is known only for exact computation. In addition to these highlighted results, we prove a number of other results in the distributed CONGEST model including an $\tilde{\Omega}(n^2)$ lower bound for computing an exact solution to MVC on $G^2$, a conditional hardness result for obtaining a $(1+\epsilon)$-approximation to MVC on $G^2$, and an $O(\log \Delta)$-approximation to the MDS problem on $G^2$ in $\mbox{poly}\log n$ rounds.