Equal Sum Sequences and Imbalance Sets of Tournaments
Abstract
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.
Keywords
Cite
@article{arxiv.1402.2456,
title = {Equal Sum Sequences and Imbalance Sets of Tournaments},
author = {Muhammad Ali Khan},
journal= {arXiv preprint arXiv:1402.2456},
year = {2014}
}
Comments
Presented at the Retrospective Workshop on Discrete Geometry, Optimization and Symmetry, 25-29 Nov 2013, The Fields Institute, Toronto, Canada