Rounding Dynamic Matchings Against an Adaptive Adversary

TL;DR

This paper introduces a new framework for dynamic matchings that achieves near-optimal approximation ratios against adaptive adversaries with improved update times, advancing the state of the art in dynamic graph algorithms.

Contribution

The paper develops a novel dynamic matching sparsification scheme and a framework for rounding fractional matchings, enabling the first randomized algorithms against adaptive adversaries with strong approximation guarantees.

Findings

Achieves (2+ε)-approximate algorithms with constant or polylog update time.

Provides (2-δ)-approximate algorithms in bipartite graphs with small polynomial update time.

Offers polynomially better update time to approximation tradeoffs than previous methods.

Abstract

We present a new dynamic matching sparsification scheme. From this scheme we derive a framework for dynamically rounding fractional matchings against \emph{adaptive adversaries}. Plugging in known dynamic fractional matching algorithms into our framework, we obtain numerous randomized dynamic matching algorithms which work against adaptive adversaries (the first such algorithms, as all previous randomized algorithms for this problem assumed an \emph{oblivious} adversary). In particular, for any constant $\epsilon>0$, our framework yields $(2+\epsilon)$-approximate algorithms with constant update time or polylog worst-case update time, as well as $(2-\delta)$-approximate algorithms in bipartite graphs with arbitrarily-small polynomial update time, with all these algorithms' guarantees holding against adaptive adversaries. All these results achieve \emph{polynomially} better update time to approximation tradeoffs than previously known to be achievable against adaptive adversaries.