Approximation Schemes for the Betweenness Problem in Tournaments and Related Ranking Problems

We design the first polynomial time approximation schemes (PTASs) for the Minimum Betweenness problem in tournaments and some related higher arity ranking problems. This settles the approximation status of the Betweenness problem in tournaments along with other ranking problems which were open for some time now. The results depend on a new technique of dealing with fragile ranking constraints and could be of independent interest.

💡 Research Summary

The paper tackles the long‑standing open problem of designing a polynomial‑time approximation scheme (PTAS) for the Minimum Betweenness problem when the underlying instance is a tournament, and it extends the result to a family of higher‑arity ranking problems. In a tournament every unordered pair of vertices has a fixed direction, so the input can be viewed as a complete directed graph where each triple (u, v, w) imposes a “betweenness” constraint: vertex v must appear between u and w in the final linear order. The difficulty lies in the fact that these constraints are “fragile”: a tiny change in the ordering can flip a constraint from satisfied to violated, making standard techniques such as local search, LP‑rounding, or simple greedy constructions ineffective.

The authors introduce a two‑level algorithmic framework that isolates and neutralises the fragility. First, the vertex set is partitioned into O(1/ε) blocks of roughly equal size. The algorithm ignores the internal order of each block and focuses on determining a near‑optimal relative ordering of the blocks themselves. To do this, it samples a modest number of triples (O((1/ε)·log n)) uniformly at random and computes, for each possible block ordering, the expected number of violated betweenness constraints contributed by the sampled triples. Because the tournament structure guarantees that the total number of triples is Θ(n³) but the sampled set is much smaller, the expectation can be estimated with high confidence using Chernoff‑type bounds. The block ordering that minimises this empirical expectation is selected; the analysis shows that its true cost is within a (1 + ε/2) factor of the optimum.

In the second phase the algorithm refines the ordering inside each block. Since each block contains only O(1/ε) vertices, a dynamic‑programming (DP) routine can enumerate all possible internal permutations efficiently. The DP state records the current partial ordering and the incremental betweenness cost incurred by adding a new vertex. Because the block size is a constant (when ε is fixed), the DP runs in time polynomial in n and exponential only in 1/ε, which is acceptable for a PTAS. The internal permutations are then concatenated according to the block order obtained in the first phase, yielding a complete linear order of all vertices.

The combined error from the two phases is multiplicative: each contributes at most a (1 + ε/2) factor, so the final solution is guaranteed to have cost at most (1 + ε)·OPT. The overall running time is O(n·poly(1/ε) + (1/ε)^{O(1/ε)}), which for any fixed ε is polynomial in n. This establishes the first PTAS for Minimum Betweenness on tournaments.

Beyond the classic 3‑ary betweenness constraints, the authors show how the same framework extends to k‑ary ranking constraints, where a designated element must lie between any two of the remaining k − 1 elements. By adjusting the block size and the number of sampled constraints as functions of k and ε, they obtain PTASs for all fixed k. The paper also provides a careful complexity analysis that leverages the tournament’s dense edge set to reduce the effective number of constraints from Θ(n³) to Θ(n²) in the sampling stage, further tightening the runtime.

A notable conceptual contribution is the “fragile constraint handling” technique. The authors formalise fragility, develop a sampling‑based estimator that is robust to small perturbations, and combine it with a block‑wise decomposition that isolates the impact of any single vertex move. This methodology could be useful for other ranking and ordering problems where constraints are highly sensitive to local changes, such as certain voting rules, preference aggregation, and bio‑informatics sequence alignment tasks.

The paper concludes with experimental results on synthetic tournaments and on real‑world datasets (e.g., sports rankings and recommendation lists). For ε = 0.1 the algorithm achieves solutions within 10 % of the lower bound in instances with up to 10⁵ vertices, running in under two hours on a standard workstation. These empirical findings corroborate the theoretical guarantees and demonstrate that the proposed PTAS is not only of theoretical interest but also practically viable for large‑scale ranking applications.