[Paper Review] Dynamic Graph Algorithms and Graph Sparsification: New Techniques and Connections

Gramoz Goranci|arXiv (Cornell University)|Sep 13, 2019

Complexity and Algorithms in Graphs199 references5 citations

TL;DR

This paper introduces novel dynamic algorithms and sparsification techniques for large-scale graphs, achieving sub-linear update times for Laplacian systems, effective resistances, and low-stretch spanning trees. It presents the first dynamic algorithms for global minimum cut, approximate all-pair maximum flow, and sparsest cut, along with a new notion of Local Sparsifiers that reveal deep connections between dynamic graph algorithms and graph sparsification.

ABSTRACT

Graphs naturally appear in several real-world contexts including social networks, the web network, and telecommunication networks. While the analysis and the understanding of graph structures have been a central area of study in algorithm design, the rapid increase of data sets over the last decades has posed new challenges for designing efficient algorithms that process large-scale graphs. These challenges arise from two usual assumptions in classical algorithm design, namely that graphs are static and that they fit into a single machine. However, in many application domains, graphs are subject to frequent changes over time, and their massive size makes them infeasible to be stored in the memory of a single machine. Driven by the need to devise new tools for overcoming such challenges, this thesis focuses on two areas of modern algorithm design that directly deal with processing massive graphs, namely dynamic graph algorithms and graph sparsification. We develop new algorithmic techniques from both dynamic and sparsification perspective for a multitude of graph-based optimization problems which lie at the core of Spectral Graph Theory, Graph Partitioning, and Metric Embeddings. Our algorithms are faster than any previous one and design smaller sparsifiers with better (approximation) quality. More importantly, this work introduces novel reduction techniques that show unexpected connections between seemingly different areas such as dynamic graph algorithms and graph sparsification.

Motivation & Objective

• Address the challenge of processing massive, dynamically changing graphs that exceed single-machine memory.
• Develop efficient dynamic algorithms for core graph problems in spectral graph theory, graph partitioning, and metric embeddings.
• Design new sparsification techniques that preserve key graph properties like shortest paths, reachability, and effective resistances with high quality and small size.
• Establish unexpected theoretical connections between dynamic graph algorithms and graph sparsification through novel reduction techniques.
• Provide tight upper and lower bounds on the trade-off between sparsifier size and approximation quality for vertex sparsifiers.

Proposed method

• Introduce Local Sparsifiers, a stronger variant of vertex sparsifiers that preserve local graph structure and enable efficient dynamic maintenance.
• Develop a dynamic algorithm for maintaining approximate solutions to Laplacian systems with sub-linear update and query time in weighted undirected graphs.
• Extend dynamic techniques to maintain vertex spectral sparsifiers and all-pair effective resistances in undirected, weighted graphs.
• Design a dynamic algorithm for low-stretch spanning trees with sub-polynomial stretch and sub-linear update time in unweighted undirected graphs.
• Use a Why-Delta transformation adaptation to construct planarity-preserving cut and distance sparsifiers for Okamura-Seymour graphs.
• Prove conditional lower bounds showing that exact maintenance of effective resistances in dynamic graphs is infeasible under standard complexity assumptions.

Experimental results

Research questions

• RQ1 Can we maintain approximate solutions to Laplacian systems in sub-linear update and query time in dynamic, weighted undirected graphs?
• RQ2 What is the trade-off between sparsifier size and approximation quality for vertex sparsifiers that preserve shortest paths and reachability?
• RQ3 Can we design dynamic algorithms for maintaining low-stretch spanning trees and low-diameter clusterings with sub-linear update time?
• RQ4 What are the limits of dynamic algorithms for fundamental graph problems like global minimum cut and all-pair maximum flow?
• RQ5 How can we unify dynamic graph algorithms and graph sparsification through novel reduction techniques and structural insights?

Key findings

• The paper presents the first dynamic algorithm for maintaining approximate solutions to Laplacian systems in sub-linear update and query time in undirected, weighted graphs.
• It achieves the first dynamic maintenance of vertex spectral sparsifiers and all-pair effective resistances in undirected, weighted graphs with sub-linear update time.
• The authors prove conditional lower bounds showing that exact maintenance of effective resistances in dynamic graphs is impossible in sub-linear time under standard complexity assumptions.
• A new dynamic algorithm is developed for maintaining low-stretch spanning trees with sub-polynomial stretch and sub-linear update time in unweighted, undirected graphs.
• The paper provides the current best-known algorithms for incrementally maintaining global minimum cut, approximate all-pair maximum flow, and sparsest cut in unweighted, undirected graphs.
• The authors introduce Local Sparsifiers as a stronger variant of vertex sparsifiers and derive tight upper and lower bounds on the trade-off between size and approximation quality for such sparsifiers.