*Theoretical Computer Science* · volume 605, pages 51–61, 2015 · doi:10.1016/j.tcs.2015.08.027

We study the minimum backlog problem (MBP). This online problem arises, e.g., in the context of sensor networks. We focus on two main variants of MBP.

The *discrete MBP* is a 2-person game played on a graph $G=(V,E)$. The *player* is initially located at a vertex of the graph. In each time step, the *adversary* pours a total of one unit of water into *cups* that are located on the vertices of the graph, arbitrarily distributing the water among the cups. The player then moves from her current vertex to an adjacent vertex and empties the cup at that vertex. The player's objective is to minimize the *backlog*, i.e., the maximum amount of water in any cup at any time.

The *geometric MBP* is a continuous-time version of the MBP: the cups are points in the two-dimensional plane, the adversary pours water continuously at a constant rate, and the player moves in the plane with unit speed. Again, the player's objective is to minimize the backlog.

We show that the *competitive ratio* of any algorithm for the MBP has a lower bound of $\Omega(D)$, where $D$ is the diameter of the graph (for the discrete MBP) or the diameter of the point set (for the geometric MBP). Therefore we focus on determining a strategy for the player that guarantees a uniform upper bound on the absolute value of the backlog.

For the absolute value of the backlog there is a trivial lower bound of $\Omega(D)$, and the deamortization analysis of Dietz and Sleator gives an upper bound of $O(D\log N)$ for $N$ cups. Our main result is a tight upper bound for the geometric MBP: we show that there is a strategy for the player that guarantees a backlog of $O(D)$, independently of the number of cups.

We also study a *localized* version of the discrete MBP: the adversary has a location within the graph and must act locally (filling cups) with respect to his position, just as the player acts locally (emptying cups) with respect to her position. We prove that deciding the value of this game is PSPACE-hard.

- Michael A. Bender, Sándor P. Fekete, Alexander Kröller, Vincenzo Liberatore, Joseph S. B. Mitchell, Valentin Polishchuk, and Jukka Suomela: The minimum-backlog problem · MACIS 2007
- Valentin Polishchuk and Jukka Suomela: Optimal backlog in the plane · Algosensors 2008