Parameterized Distributed Complexity Theory: A logical approach

TL;DR

This paper develops a logical framework for parameterized distributed complexity, introducing hierarchies and classes like Distributed-FPT, to analyze the complexity of distributed problems with respect to parameters.

Contribution

It introduces a logical approach to parameterized distributed complexity, defining new hierarchies and classes that extend classical concepts to distributed computing models.

Findings

Defined Distributed-FPT class for distributed algorithms with parameterized complexity

Introduced Distributed-W-hierarchy and Distributed-A-hierarchy with logical characterizations

Provided a robust framework for classifying distributed problems based on logical reductions

Abstract

Parameterized complexity theory offers a framework for a refined analysis of hard algorithmic problems. Instead of expressing the running time of an algorithm as a function of the input size only, running times are expressed with respect to one or more parameters of the input instances. In this work we follow the approach of parameterized complexity to provide a framework of parameterized distributed complexity. The central notion of efficiency in parameterized complexity is fixed-parameter tractability and we define the distributed analogue Distributed-FPT (for Distributed in $\{Local, Congest, Congested-Clique\}$) as the class of problems that can be solved in $f(k)$ communication rounds in the Distributed model of distributed computing, where $k$ is the parameter of the problem instance and $f$ is an arbitrary computable function. To classify hardness we introduce three hierarchies. The Distributed-WEFT-hierarchy is defined analogously to the W-hierarchy in parameterized complexity theory via reductions to the weighted circuit satisfiability problem, but it turns out that this definition does not lead to satisfying frameworks for the Local and Congest models. We then follow a logical approach that leads to a more robust theory. We define the levels of the Distributed-W-hierarchy and the Distributed-A-hierarchy that have first-order model-checking problems as their complete problems via suitable reductions.