Following on from the last post on maximum weighted matchings in the stream model, I wanted to briefly mention a result that appeared in STACS last week. The set-up is the same as before: there’s a stream of undirected weighted edges $\langle (e_1,w_{e_i}), \ldots, (e_m,w_{e_m})\rangle$ where $w_{e}$ is the weight of edge $e$ and we want to find an approximate maximum weighted matching. Last time, we showed a simple algorithm that gave a 5.828 approximation that was “preemptive online”, i.e., at any point in time the algorithm remembered no information other than the feasible matching found so far. The follow-up work achieved a 5.585 approximation but didn’t have this property. This begs numerous questions including if 5-ish is the “right answer” for this problem and whether there is a separation between preemptive-online algorithms and stream algorithms that use roughly the same amount of memory.

Which brings us to [Epstein, Levin, Mestre, Segev]. They showed a lower bound of 4.967 for any deterministic preemptive online algorithm. Furthermore, they show that there exists a deterministic stream algorithm that breaks through this bound and achieves a 4.911 approximation. As a commenter pointed out, the algorithm is nice and simple. Furthermore, it makes a good teaching example of how algorithms don’t necessary appear from thin air, but rather can be developed in a series of less mysterious steps. Here’s what they do in the paper:

• Design a simple deterministic 8-approximation: Geometrically group edge weights by rounding edge weight to the nearest power of 2 and ignore the “very light” edges. Greedily find a maximal matching among each group of edges. At the end of the stream, find a maximum matching among the stored edges.
• Randomize the algorithm to get a 4.911-approximation: Instead of grouping to the nearest power of 2, do randomized geometric grouping with the appropriate parameters.
• Derandomize! Thought… It might be interesting to investigate which other problems exhibit a separation between the preemptive online model and the stream model. You may need to be flexible with your definition of “preemptive online.”