Constructing Near Spanning Trees with Few Local Inspections

TL;DR

This paper investigates the problem of constructing near spanning trees in bounded-degree graphs using only a constant number of local inspections, focusing on expansion properties and their impact on the feasibility of such constructions.

Contribution

It introduces conditions on graph expansion that enable deterministic construction of sparse spanning subgraphs with minimal inspections, and proves these conditions are tight.

Findings

Graphs with certain expansion properties allow near spanning trees with few inspections.

The expansion condition is shown to be optimal even with randomized algorithms.

Constructs high-girth, 3-regular graphs demonstrating the tightness of the expansion requirement.

Abstract

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge $e$ in $G$ we would like to decide whether $e$ belongs to a connected subgraph $G'$ consisting of $(1+\epsilon)n$ edges (for a prespecified constant $\epsilon >0$), where the decision for different edges should be consistent with the same subgraph $G'$. Can this task be performed by inspecting only a {\em constant} number of edges in $G$? Our main results are: (1) We show that if every $t$-vertex subgraph of $G$ has expansion $1/(\log t)^{1+o(1)}$ then one can (deterministically) construct a sparse spanning subgraph $G'$ of $G$ using few inspections. To this end we analyze a "local" version of a famous minimum-weight spanning tree algorithm. (2) We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of $3$-regular graphs of high girth, in which every $t$-vertex subgraph has expansion $1/(\log t)^{1-o(1)}$.