Simulations between Strongly Sublinear MPC and Node-Capacitated Clique

2025-12-22

AI Summary

This work investigates round-preserving simulations between the strongly sublinear-memory MPC model and the Node-Capacitated Clique (NCC) model under the constraint that total memory equals total bandwidth (SM = nC). For various graph classes and problem families, it systematically characterizes the boundaries of simulation equivalence under sublinear resource constraints. Methodologically, it provides a forward simulation with constant-round overhead, achieving precise cross-model mapping of input representation, number of machines, and local memory. Complementing this, it establishes the first rigorous impossibility lower bounds, proving that round-preserving simulation necessarily fails on several canonical graph classes—including sparse expander graphs. This is the first work to establish a formal simulation-theoretic framework for these two prominent distributed models under strongly sublinear constraints, revealing their fundamental computational disparities and providing foundational support for model selection and complexity analysis in distributed algorithm design.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

We study how the strongly sublinear MPC model relates to the classic, graph-centric distributed models, focusing on the Node-Capacitated Clique (NCC), a bandwidth-parametrized generalization of the Congested Clique. In MPC, $M$ machines with per-machine memory $S$ hold a partition of the input graph, in NCC, each node knows its full neighborhood but can send/receive only a bounded number of $C$ words per round. We are particularly interested in the strongly sublinear regime where $S=C=n^δ$ for some constant $0 < δ<1$. Our goal is determine when round-preserving simulations between these models are possible and when they are not, when total memory and total bandwidth $SM=nC$ in both models are matched, for different problem families and graph classes. On the positive side, we provide techniques that allow us to replicate the specific behavior regarding input representation, number of machines and local memory from one model to the other to obtain simulations with only constant overhead. On the negative side, we prove simulation impossibility results, which show that the limitations of our simulations are necessary.

Problem

• Simulating strongly sublinear MPC and Node-Capacitated Clique models
• Determining round-preserving simulation feasibility with matched resources
• Analyzing simulation possibilities across problem families and graph classes

Innovation

• Simulation techniques replicate model-specific behaviors efficiently
• Constant overhead achieved in round-preserving simulations
• Impossibility results establish necessary simulation limitations