Conditional Hardness Results for Massively Parallel Computation from Distributed Lower Bounds

Abstract

We present the first conditional hardness results for massively parallel algorithms for some central graph problems including (approximating) maximum matching, vertex cover, maximal independent set, and coloring. In some cases, these hardness results match or get close to the state of the art algorithms. Our hardness results are conditioned on a widely believed conjecture in massively parallel computation about the complexity of the connectivity problem. We also note that it is known that an unconditional variant of such hardness results might be somewhat out of reach for now, as it would lead to considerably improved circuit complexity lower bounds and would concretely imply that NC 1 ⊊ P. We obtain our conditional hardness result via a general method that lifts unconditional lower bounds from the well-studied LOCAL model of distributed computing to the massively parallel computation setting.

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

“…In the super-linear regime of local memory where S = n 1+ε for some constant ε > 0, many graph problems -particularly, including minimum cut [LMSV11] -can be solved in O(1) rounds, using a relatively simple filtering idea. Much of the recent activities in the area has been on achieving similarly fast algorithms for various problems in the much harder memory regimes where S in nearly linear or even sublinear in n [CLM + 18, GGK + 18, BEG + 18, ASS + 18, ABB + 19, GU19, BBD + 19, BFU19, ASW19, GKMS19, CFG + 19, BHH19,GKU19].…”

Graph Sparsification for Derandomizing Massively Parallel Computation with Low Space

“…The additive (log log ) term in the bound is most likely necessary: Ghaffari et al [21] provided an Ω(log log ) conditional hardness result for maximal matching and MIS, even for randomized fully scalable MPC algorithms. They proved that unless there is an (log )-round (randomized) MPC algorithm for connectivity with local space = for a constant 0 < < 1 and poly( ) global space (see [39] for strong arguments about the hardness of that problem), there is no component-stable randomized MPC algorithm with local space = and poly( ) global space that computes a maximal matching or an MIS in (log log ) rounds.…”

Simple, Deterministic, Constant-Round Coloring in the Congested Clique

“…however, remains open. A corresponding constant-round algorithm there seems unlikely, due to the Ω(log log log ) conditional lower bound [10], but there is no reason to believe one could not improve the dependency on Δ to sub-logarithmic. Another area for possible improvement is our ( + 1+ ) global space bound in MPC, which is slightly weaker than the optimal ( + ).…”

“…This new partitioning idea was useful to support the more general (Δ + 1)-list coloring problem, as well as to provide efficient implementation low-space MPC, where (upon incorporating the subsequent polylogarithmic-round network decomposition algorithm of Rozhoň and Ghaffari [20]), the algorithm of Chang et al [6] works in (log log log ) rounds. This round complexity is matched by a conditional lower bound due to Ghaffari et al [10], which states that an (log log log )-round randomized component-stable coloring algorithm in low-space MPC would imply a 2 log (1) log -round deterministic LOCAL coloring algorithm, and a log (1) log -round randomized one. (Note, though, that our algorithms are not component-stable, since they involve global agreement on hash functions.…”