We present the first super-linear lower bounds for natural graph problems in the CON-GEST model, answering a long-standing open question. Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires Ω(n 2 / log 2 n) rounds in the worst case in the CONGEST model, as well as any algorithm for χ-coloring a graph, where χ is the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and nearquadratic number of rounds. Finally, we address the problem of computing an exact solution to weighted all-pairsshortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple Ω(n) lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a complexity of Θ(n/ log n). We also formally prove that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question.
In REF , it is shown that computing an optimal solutions to the maximum independent set and the minimum vertex cover problem both require timeΩ(n 2 ) in the CONGEST model.
Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model
ArXiv
ArXiv
Mathematics Computer Science