Equal Sum Sequences and Imbalance Sets of Tournaments

TL;DR

This paper completely solves the tournament imbalance set problem by providing constructive proofs and algorithms, introduces the NP-complete equal sum sequences problem, and establishes a reduction linking the two problems.

Contribution

It offers a complete solution to the TIS problem with a pseudo-polynomial algorithm and generalizes the ESS problem to ESSeq, proving its NP-completeness.

Findings

TIS is weakly NP-complete due to reduction from ESSeq.

Constructive proofs and algorithms for realizing imbalance sets.

Generalization of ESS to ESSeq and NP-completeness proof.

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.