Matching in Stochastically Evolving Graphs

TL;DR

This paper investigates algorithms for maximum matching in stochastically evolving graphs, introducing the concept of the price of stochasticity, and provides bounds and approximation schemes for the problem.

Contribution

It introduces the price of stochasticity, proves the existence of a deterministic optimal algorithm, and develops an efficient approximation scheme for expected maximum matching size.

Findings

The price of stochasticity is at most 2/3.

A fully randomized approximation scheme (FPRAS) is developed.

A deterministic optimal algorithm exists for the problem.

Abstract

This paper studies the maximum cardinality matching problem in stochastically evolving graphs. We formally define the arrival-departure model with stochastic departures. There, a graph is sampled from a specific probability distribution and it is revealed as a series of snapshots. Our goal is to study algorithms that create a large matching in the sampled graphs. We define the price of stochasticity for this problem which intuitively captures the loss of any algorithm in the worst case in the size of the matching due to the uncertainty of the model. Furthermore, we prove the existence of a deterministic optimal algorithm for the problem. In our second set of results we show that we can efficiently approximate the expected size of a maximum cardinality matching by deriving a fully randomized approximation scheme (FPRAS) for it. The FPRAS is the backbone of a probabilistic algorithm that is optimal when the model is defined over two timesteps. Our last result is an upper bound of $\frac{2}{3}$ on the price of stochasticity. This means that there is no algorithm that can match more than $\frac{2}{3}$ of the edges of an optimal matching in hindsight.