Maintaining Near-Popular Matchings

We study dynamic matching problems in graphs among agents with preferences. Agents and/or edges of the graph arrive and depart iteratively over time. The goal is to maintain matchings that are favorable to the agent population and stable over time. More formally, we strive to keep a small unpopularity factor by making only a small amortized number of changes to the matching per round. Our main result is an algorithm to maintain matchings with unpopularity factor

$$(\Delta +k)$$

by making an amortized number of

$$O(\Delta + \Delta ^2/k)$$

changes per round, for any

$$k > 0$$

. Here

$$\Delta $$

denotes the maximum degree of any agent in any round. We complement this result by a variety of lower bounds indicating that matchings with smaller factor do not exist or cannot be maintained using our algorithm.

As a byproduct, we obtain several additional results that might be of independent interest. First, our algorithm implies existence of matchings with small unpopularity factors in graphs with bounded degree. Second, given any matching

M

and any value

$$\alpha \ge 1$$

, we provide an efficient algorithm to compute a matching

$$M'$$

with unpopularity factor

$$\alpha $$

over

M

if it exists. Finally, our results show the absence of voting paths in two-sided instances, even if we restrict to sequences of matchings with larger unpopularity factors (below

$$\Delta )$$

.