[Paper Review] Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model

Keren Censor-Hillel, Seri Khoury+1 more|arXiv (Cornell University)|Jan 1, 2017

Numerical methods for differential equations13 citations

TL;DR

This paper establishes the first super-linear lower bounds for natural graph problems in the CONGEST model, proving that exact computation of minimum vertex cover, maximum independent set, and χ-coloring require Ω(n²/log²n) rounds. It further shows that even simple P problems like identical subgraph detection and weighted cycle detection require Ω(n²) and Ω(n²/log n) rounds, respectively, and formally proves that the standard Alice-Bob framework cannot yield super-linear lower bounds for weighted APSP, highlighting a fundamental barrier in current techniques.

ABSTRACT

We present the first super-linear lower bounds for natural graph problems in the CONGEST 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 a near-quadratic number of rounds in the CONGEST model, as well as any algorithm for computing 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 near-quadratic number of rounds. Finally, we address the problem of computing an exact solution to weighted all-pairs-shortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple linear lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a sub-linear complexity. 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.

Motivation & Objective

• To resolve the long-standing open question of whether natural graph problems in the CONGEST model require super-linear time.
• To demonstrate that problems in P, not just NP-hard ones, can have near-quadratic round complexity.
• To formally show that the standard Alice-Bob communication framework cannot yield super-linear lower bounds for weighted all-pairs shortest-paths (APSP).
• To establish a hierarchy of complexity classes for global problems in the CONGEST model, including near-quadratic hardness.

Proposed method

• Uses a refined reduction framework from 2-party communication complexity, leveraging a bit-gadget to enable logarithmic-size cuts.
• Constructs lower-bound graphs with small cuts (O(1) or O(log n)) but large input sizes (Θ(n²)) to amplify communication complexity.
• Applies the framework to NP-hard problems (vertex cover, independent set, 3-coloring) and P problems (identical subgraph detection, weighted cycle detection).
• Proves that in the Alice-Bob model, the number of bits exchanged is bounded by O(|C|n log n), leading to a round complexity lower bound of Ω(CC(f)/|C| log n).
• Extends the analysis to t-party shared-blackboard models, showing that even with multiple players, the upper bound on communication complexity limits the achievable lower bound to O(n).
• Formally proves that any reduction using the Alice-Bob framework cannot yield super-linear lower bounds for weighted APSP, regardless of function or graph construction.

Experimental results

Research questions

• RQ1 Do any natural graph problems in the CONGEST model require super-linear time, specifically Ω(n²/log²n) rounds?
• RQ2 Can problems in P, such as detecting identical subgraphs or weighted cycles, require near-quadratic time in the CONGEST model?
• RQ3 Is the standard Alice-Bob communication framework sufficient to prove super-linear lower bounds for weighted APSP in the CONGEST model?
• RQ4 Does a complexity hierarchy exist in the CONGEST model, with problems ranging from Θ(D) to Θ(n²/log²n)?
• RQ5 Can weighted APSP be solved in linear time, or is its complexity fundamentally higher?

Key findings

• Exact minimum vertex cover and maximum independent set require Ω(n²/log²n) rounds in the CONGEST model, even for randomized algorithms with high probability.
• χ-coloring and 3-coloring of 3-colorable graphs also require Ω(n²/log²n) rounds, even in constant-diameter networks.
• The problem of detecting identical subgraphs requires Ω(n²) rounds deterministically, and weighted cycle detection requires Ω(n²/log n) rounds.
• A randomized algorithm for identical subgraph detection completes in O(D) rounds, showing a strong separation between deterministic and randomized complexity for global problems.
• The standard Alice-Bob framework cannot yield super-linear lower bounds for weighted APSP, regardless of the function or graph construction used.
• Any lower bound for weighted APSP via this framework is at most Ω(n), which separates it from the unweighted case (Θ(n/log n)) but falls short of super-linear bounds.