Graph Sketching Against Adaptive Adversaries Applied to the Minimum Degree Algorithm

TL;DR

This paper introduces a nearly-linear time algorithm for approximate elimination orderings in sparse matrices using graph sketching, addressing adaptive adversaries and providing provable guarantees in scientific computing.

Contribution

It presents the first nearly-linear time algorithm with provable approximation guarantees for elimination orderings, incorporating randomized graph sketching against adaptive adversaries.

Findings

Developed a data structure for approximate fill degrees in polylogarithmic time.

Achieved a nearly-linear time algorithm for greedy minimum degree orderings.

Introduced a technique to decorrelate queries and updates in randomized data structures.

Abstract

Motivated by the study of matrix elimination orderings in combinatorial scientific computing, we utilize graph sketching and local sampling to give a data structure that provides access to approximate fill degrees of a matrix undergoing elimination in $O(\text{polylog}(n))$ time per elimination and query. We then study the problem of using this data structure in the minimum degree algorithm, which is a widely-used heuristic for producing elimination orderings for sparse matrices by repeatedly eliminating the vertex with (approximate) minimum fill degree. This leads to a nearly-linear time algorithm for generating approximate greedy minimum degree orderings. Despite extensive studies of algorithms for elimination orderings in combinatorial scientific computing, our result is the first rigorous incorporation of randomized tools in this setting, as well as the first nearly-linear time algorithm for producing elimination orderings with provable approximation guarantees. While our sketching data structure readily works in the oblivious adversary model, by repeatedly querying and greedily updating itself, it enters the adaptive adversarial model where the underlying sketches become prone to failure due to dependency issues with their internal randomness. We show how to use an additional sampling procedure to circumvent this problem and to create an independent access sequence. Our technique for decorrelating the interleaved queries and updates to this randomized data structure may be of independent interest.