Determine whether a bound of 2^n suffices for optimal strategy length

Determine whether every water transport problem on an n-vertex graph admits an optimal strategy of length at most 2^n.

Background

The paper studies the water transport problem in the connected-set averaging model, where a strategy consists of averaging the weights on connected vertex sets and seeks to maximize the final weight at a designated target vertex. Prior work had established finite attainment of an optimum but had not supplied a uniform strategy-length bound depending only on the number of vertices.

The paper proves a substantially larger uniform bound of n^{(2+o(1))n}. Consequently, the specifically proposed bound 2^n remains unresolved in the text: the authors establish computable uniform boundedness but do not determine whether the sharper exponential bound 2^n always suffices.