New Structures and Algorithms for Length-Constrained Expander Decompositions

TL;DR

This paper introduces a near-linear time algorithm for length-constrained expander decompositions that balances decomposition size and routing path length, advancing graph algorithms for length-sensitive problems.

Contribution

It presents a novel algorithm for length-constrained expander decompositions with a trade-off between decomposition size and routing length, supported by new structural theorems and flow algorithms.

Findings

Algorithm computes decompositions in near-linear time.

Decomposition size can be controlled via a parameter.

Supports routing paths of exponential length in 1/epsilon.

Abstract

Expander decompositions form the basis of one of the most flexible paradigms for close-to-linear-time graph algorithms. Length-constrained expander decompositions generalize this paradigm to better work for problems with lengths, distances and costs. Roughly, an $(h,s)$-length $\phi$-expander decomposition is a small collection of length increases to a graph so that nodes within distance $h$ can route flow over paths of length $hs$ with congestion at most $1/\phi$. In this work, we give a close-to-linear time algorithm for computing length-constrained expander decompositions in graphs with general lengths and capacities. Notably, and unlike previous works, our algorithm allows for one to trade off off between the size of the decomposition and the length of routing paths: for any $\epsilon > 0$ not too small, our algorithm computes in close-to-linear time an $(h,s)$-length $\phi$-expander decomposition of size $m \cdot \phi \cdot n^\epsilon$ where $s = \exp(\text{poly}(1/\epsilon))$. The key foundations of our algorithm are: (1) a simple yet powerful structural theorem which states that the union of a sequence of sparse length-constrained cuts is itself sparse and (2) new algorithms for efficiently computing sparse length-constrained flows.