Fully Polynomial-Time Distributed Computation in Low-Treewidth Graphs

TL;DR

This paper introduces efficient distributed algorithms for solving key graph problems in low-treewidth graphs, achieving polynomial dependence on treewidth, linear on diameter, and polylogarithmic on node count, using a unified framework with novel techniques.

Contribution

It presents a fully polynomial-time distributed framework for low-treewidth graphs, including a new tree decomposition algorithm and the concept of stateful walk constraints, enabling efficient solutions to complex problems.

Findings

Algorithms for shortest paths, maximum matching, and girth run in   ext{polylog}(n) rounds.

Tree decomposition can be computed in   ext{polylog}(n) rounds.

The framework generalizes to various constrained walk problems in distributed settings.

Abstract

We consider global problems, i.e. problems that take at least diameter time, even when the bandwidth is not restricted. We show that all problems considered admit efficient solutions in low-treewidth graphs. By ``efficient'' we mean that the running time has polynomial dependence on the treewidth, a linear dependence on the diameter (which is unavoidable), and only a polylogarithmic dependence on $n$, the number of nodes in the graph. We present the algorithms solving the following problems in the CONGEST model which all attain $\tilde{O(\tau^{O(1)}D)}$-round complexity (where $\tau$ and $D$ denote the treewidth and diameter of the graph, respectively): (1) Exact single-source shortest paths (actually, the more general problem of computing a distance labeling scheme) for weighted and directed graphs, (2) exact bipartite unweighted maximum matching, and (3) the weighted girth for both directed and undirected graphs. We derive all of our results using a single unified framework, which consists of two novel technical ingredients, The first is a fully polynomial-time distributed tree decomposition algorithm, which outputs a decomposition of width $O(\tau^2\log n)$ in $\tilde{O}(\tau^{O(1)}D)$ rounds (where $n$ is the number of nodes in the graph). The second ingredient, and the technical highlight of this paper, is the novel concept of a \emph{stateful walk constraint}, which naturally defines a set of feasible walks in the input graph based on their local properties (e.g., augmenting paths). Given a stateful walk constraint, the constrained version of the shortest paths problem (or distance labeling) requires the algorithm to output the shortest \emph{constrained} walk (or its distance) for a given source and sink vertices. We show that this problem can be efficiently solved in the CONGEST model by reducing it to an \emph{unconstrained} version of the problem.