New Theoretical Insights and Algorithmic Solutions for Reconstructing Score Sequences from Tournament Score Sets

TL;DR

The paper tackles the challenge of reconstructing tournament score sequences from score sets by deriving a Landau-based theoretical framework that yields a necessary and sufficient condition and a separate necessary condition. It then introduces three algorithms: a polynomial-time dynamic-programming method to reconstruct a score sequence, a scalable algorithm for large instances, and a polynomial-time network-building approach to enumerate all possible sequences; a fast, number-theoretic reconstruction algorithm further accelerates the process. The combination of a structured, group-theoretic viewpoint with net-based and modular techniques enables verifiable reconstruction of score sequences and even verification of Reid's conjecture, with demonstrated practicality in experimental evaluations. The work offers meaningful implications for ranking, scheduling, and data-imputation tasks where partial rankings must be completed or validated.

Abstract

The score set of a tournament is defined as the set of its distinct out-degrees. In 1978, Reid proposed the conjecture that for any set of nonnegative integers $D$, there exists a tournament $T$ with a degree set $D$. In 1989, Yao presented an arithmetical proof of the conjecture, but a general polynomial-time construction algorithm is not known. This paper proposes a necessary and sufficient condition and a separate necessary condition, based on the existing Landau's theorem for the problem of reconstructing score sequences from score sets of tournament graphs. The necessary condition introduces a structured set that enables the use of group-theoretic techniques, offering not only a framework for solving the reconstruction problem but also a new perspective for approaching similar problems. In particular, the same theoretical approach can be extended to reconstruct valid score sets given constraints on the frequency of distinct scores in tournaments. Based on these conditions, we have developed three algorithms that demonstrate the practical utility of our framework: a polynomial-time algorithm and a scalable algorithm for reconstructing score sequences, and a polynomial-time network-building method that finds all possible score sequences for a given score set. Moreover, the polynomial-time algorithm for reconstructing the score sequence of a tournament for a given score set can be used to verify Reid's conjecture. These algorithms have practical applications in sports analysis, ranking prediction, and machine learning tasks such as learning-to-rank models and data imputation, where the reconstruction of partial rankings or sequences is essential for recommendation systems and anomaly detection.

Figures (3)

• Figure 1: An example of a tournament graph.
• Figure 2: Comparison of time measurements as the maximum element in a fixed set of scores (with a size of 7) varies, under experimental and control algorithms.
• Figure 3: Comparison of execution time as the number of elements in a score set varies while keeping the maximum element $\geq 28$, under experimental and control algorithms.

Theorems & Definitions (28)

• Theorem 1: Landau Landau53_III
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof