Reconstruction of complete interval tournaments

Let $a, b$ and $n$ be nonnegative integers $(b \geq a, \ b > 0, \ n \geq 1)$, $\mathcal{G}_n(a,b)$ be a multigraph on $n$ vertices in which any pair of vertices is connected with at least $a$ and at most $b$ edges and \textbf{v =} $(v_1, v_2, …, v_n)$ be a vector containing $n$ nonnegative integers. We give a necessary and sufficient condition for the existence of such orientation of the edges of $\mathcal{G}_n(a,b)$, that the resulted out-degree vector equals to \textbf{v}. We describe a reconstruction algorithm. In worst case checking of \textbf{v} requires $\Theta(n)$ time and the reconstruction algorithm works in $O(bn^3)$ time. Theorems of H. G. Landau (1953) and J. W. Moon (1963) on the score sequences of tournaments are special cases $b = a = 1$ resp. $b = a \geq 1$ of our result.

💡 Research Summary

The paper studies a generalized tournament model called a complete interval tournament. In an (a,b)-tournament on n vertices, every unordered pair of vertices is linked by at least a and at most b directed arcs (or “points”). The classical tournament corresponds to a = b = 1, while a = b ≥ 1 yields a multitournament. The authors address two fundamental questions: (1) given a non‑negative integer vector v = (v₁,…,vₙ) (or its sorted version s = (s₁ ≤ … ≤ sₙ)), does there exist an orientation of the underlying multigraph Gₙ(a,b) whose out‑degree sequence equals v? and (2) if the answer is affirmative, how can one actually construct such a digraph efficiently?

First, they derive necessary conditions that extend Landau’s and Moon’s classic theorems. The obvious constraints are Σ_{i=1}^{k}s_i ≥ a·C(k,2) for every k (a lower bound on the total points among the k weakest players) and Σ_{i=1}^{n}s_i ≤ b·C(n,2) (the global upper bound). However, when a ≠ b these two inequalities are not sufficient. To close the gap they introduce a “loss function” L_k defined recursively by L₀ = 0 and L_k = max{L_{k‑1}, b·C(k,2) – Σ_{i=1}^{k}s_i}. Intuitively, L_k is a lower bound on the number of points that must be “lost” among the first k players because the maximum possible total b·C(k,2) cannot be reached.

Using L_k they prove a strengthened set of inequalities (Lemma 3): for every k,  a·C(k,2) ≤ Σ_{i=1}^{k}s_i ≤ b·C(k,2) – L_k – (n‑k)s_k. These bounds are both necessary and sufficient: any non‑decreasing integer sequence satisfying them can be realized as the score sequence of an (a,b)-tournament.

The authors then present a linear‑time verification algorithm, ScoreCheck. It computes the cumulative sums S_i, the binomial coefficients B_i = C(i,2), and the loss values L_i in a single pass, checking the lower and upper bounds of Lemma 3 for each i. If a violation occurs, it reports the offending index and whether the score is too small or too large; otherwise it confirms that the sequence passes both necessary conditions. The algorithm runs in Θ(n) time and uses Θ(n) auxiliary space (or O(1) space if the scores are streamed).

For reconstruction, they design a two‑level procedure. The outer routine Main iterates k from n down to 3, repeatedly invoking ScoreSlicing to “slice off” the last column and row of the point table. ScoreSlicing is an extension of Ryser’s matrix construction technique. For the current player k it computes:

• the additional points A_i = Σ_{j≤i}p_j – a·C(i,2) (how many points the first i players can still win beyond the mandatory a per pair),
• the missing points M = (k‑1)·b – p_k (how many points player k still needs to reach its prescribed out‑degree), and then distributes points r_{i,k} (i < k) and r_{k,i} while respecting:  a ≤ r_{i,k} + r_{k,i} ≤ b,  0 ≤ r_{i,k} ≤ p_i,  the monotonicity of the provisional score vector, and the loss constraints derived earlier. The algorithm repeatedly selects the largest index x with r_{x,k} < b, determines how many points can be “sliced” from player x, and updates the provisional scores accordingly. The inner loops run at most b·k times, leading to an overall worst‑case running time of O(b·n³). The final point table R (an n×n matrix with zeros on the diagonal) represents a concrete (a,b)-tournament realizing the original score sequence.

The paper includes detailed examples. A tiny (2,10)-tournament with three players and score sequence (3,4,5) is worked out step by step, illustrating the computation of L_k, the choice of r_{i,k}, and the final reconstruction. A larger six‑player example (a=2, b=10, scores (9,9,19,20,32,34)) demonstrates how ScoreCheck validates the sequence and how Main together with ScoreSlicing gradually reduces the problem, updating the provisional scores and the point table until a full tournament is obtained.

Complexity analysis shows that ScoreCheck is optimal (Θ(n) time) and that the reconstruction algorithm, while polynomial, is dominated by the factor b·n³; for fixed b this is cubic in n, which is acceptable for moderate sizes. Memory usage is O(n²) for storing the point table, but the authors note that the verification stage can be performed with constant additional memory if the scores are streamed.

In conclusion, the authors provide a complete characterization of feasible score sequences for complete interval tournaments, generalizing the classical Landau and Moon theorems. Their linear‑time decision procedure and constructive algorithm fill a gap in the literature on multigraph tournaments and have potential applications in ranking systems, sports scheduling, and any domain where pairwise comparisons with bounded point allocations are required.