  Given a sequence of n numbers, the Maximum Consecutive Subsums Problem (MCSP)
asks for the maximum consecutive sum of lengths l for each l = 1,...,n. No
algorithm is known for this problem which is significantly better than the
naive quadratic solution. Nor a super linear lower bound is known. The best
known bound for the MCSP is based on the the computation of the
(min,+)-convolution, another problem for which neither an O(n^{2-{\epsilon}})
upper bound is known nor a super linear lower bound. We show that the two
problems are in fact computationally equivalent by providing linear reductions
between them. Then, we concentrate on the problem of finding super linear lower
bounds and provide empirical evidence for an {\Omega}(nlogn) lower bounds for
both problems in the decision tree model.
