[Paper Review] Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers

Li Chen, Gramoz Goranci+3 more|arXiv (Cornell University)|May 5, 2020

Complexity and Algorithms in Graphs75 references4 citations

TL;DR

This paper introduces a general framework for designing efficient dynamic approximate algorithms on undirected graphs using vertex sparsifiers. By leveraging hierarchical constructions of j-trees and spectral sparsifiers, it achieves sublinear update and query times for dynamic all-pair maximum flow, shortest paths, and effective resistances, with near-optimal approximation ratios and improved dependencies on graph size and error parameters.

ABSTRACT

We present a general framework of designing efficient dynamic approximate algorithms for optimization on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers, gives data structures that maintain approximate solutions in sub-linear update and query time. We illustrate the applicability of our paradigm to the following problems. (1) A fully-dynamic algorithm that approximates all-pair maximum-flows/minimum-cuts up to a nearly logarithmic factor in $ ilde{O}(n^{2/3})$ amortized time against an oblivious adversary, and $ ilde{O}(m^{3/4})$ time against an adaptive adversary. (2) An incremental data structure that maintains $O(1)$-approximate shortest path in $n^{o(1)}$ time per operation, as well as fully dynamic approximate all-pair shortest path and transshipment in $ ilde{O}(n^{2/3+o(1)})$ amortized time per operation. (3) A fully-dynamic algorithm that approximates all-pair effective resistance up to an $(1+ε)$ factor in $ ilde{O}(n^{2/3+o(1)} ε^{-O(1)})$ amortized update time per operation. The key tool behind result (1) is the dynamic maintenance of an algorithmic construction due to Madry [FOCS' 10], which partitions a graph into a collection of simpler graph structures (known as j-trees) and approximately captures the cut-flow and metric structure of the graph. The $O(1)$-approximation guarantee of (2) is by adapting the distance oracles by [Thorup-Zwick JACM `05]. Result (3) is obtained by invoking the random-walk based spectral vertex sparsifier by [Durfee et al. STOC `19] in a hierarchical manner, while carefully keeping track of the recourse among levels in the hierarchy.

Motivation & Objective

• To close the performance gap between static and dynamic graph algorithms by enabling efficient dynamic maintenance of approximate solutions.
• To address the challenge of maintaining approximate flows, distances, and effective resistances in dynamic graphs with sublinear update and query times.
• To develop a unifying framework based on vertex sparsifiers that generalizes to multiple graph optimization problems.
• To achieve approximation ratios matching the best known static oracles while significantly improving dynamic update efficiency.

Proposed method

• The framework reduces dynamic graph problems to maintaining a data structure for vertex sparsifiers, specifically using j-tree decompositions to capture cut and metric structures.
• For max-flow and min-cut, it dynamically maintains a construction from Madry (FOCS'10) that decomposes the graph into j-trees with bounded stretch.
• For shortest paths, it adapts Thorup-Zwick distance oracles to an incremental setting, achieving O(1)-approximation with n^{o(1)} update time.
• For effective resistances, it uses a hierarchical application of Durfee et al.'s spectral vertex sparsifiers with careful recourse tracking across levels.
• It employs a potential function and iterative boosting via the lmax function to maintain convex combinations of sparsifiers with bounded stretch.
• The algorithm uses a greedy update rule based on partial derivatives of the lmax function to guide the selection of sparsifier components at each step.

Experimental results

Research questions

• RQ1 Can a general framework be developed to design dynamic approximate algorithms for undirected graph optimization problems using vertex sparsifiers?
• RQ2 What is the best achievable trade-off between approximation ratio and update time for dynamic all-pair max-flow/min-cut?
• RQ3 Can dynamic shortest path and transshipment problems be maintained with subpolynomial update time while preserving constant-factor approximation?
• RQ4 How can effective resistances be maintained dynamically with (1+ε)-approximation and near-optimal update time?
• RQ5 Can the framework be extended to work against adaptive adversaries while maintaining sublinear update times?

Key findings

• The paper achieves a fully dynamic all-pair max-flow/min-cut approximation with O(log n log log n) factor in Õ(n^{2/3+o(1)}) amortized update time against an oblivious adversary.
• For adaptive adversaries, the update time increases to Õ(m^{3/4}), still sublinear, while maintaining the same approximation ratio.
• An incremental all-pair shortest path data structure provides O(1)-approximation in n^{o(1)} time per operation, improving upon prior dynamic shortest path bounds.
• A fully dynamic effective resistance algorithm achieves (1+ε)-approximation in Õ(n^{2/3+o(1)}ε^{-O(1)}) amortized update time, improving over the previous Õ(n^{5/6}ε^{-6}) bound.
• The framework matches the best-known approximation ratios of static oracles—oblivious routing, distance oracles, and static effective resistance computation—while achieving sublinear dynamic update times.
• The method is general and applies to multiple problems by reducing them to the maintenance of appropriate vertex sparsifiers with bounded stretch and recourse.