  Reid conjectured that any finite set of non-negative integers is the score
set of some tournament and Yao gave a non-constructive proof of Reid's
conjecture using arithmetic arguments. No constructive proof has been found
since. In this paper, we investigate a related problem, namely, which sets of
integers are imbalance sets of tournaments. We completely solve the tournament
imbalance set problem (TIS) and also estimate the minimal order of a tournament
realizing an imbalance set. Our proofs are constructive and provide a
pseudo-polynomial time algorithm to realize any imbalance set. Along the way,
we generalize the well-known equal sum subsets problem (ESS) to define the
equal sum sequences problem (ESSeq) and show it to be NP-complete. We then
prove that ESSeq reduces to TIS and so, due to the pseudo-polynomial time
complexity, TIS is weakly NP-complete.

Combinatorics