Linear vertex-kernels for several dense ranking r-CSPs

A Ranking r-Constraint Satisfaction Problem (ranking r-CSP) consists of a ground set of vertices V, an arity r >= 2, a parameter k and a constraint system c, where c is a function which maps rankings of r-sized subsets of V to {0,1}. The objective is to decide if there exists a ranking of the vertices satisfying all but at most k constraints. Famous ranking r-CSP include the Feedback Arc Set in Tournaments and Betweenness in Tournaments problems. We consider these problems from the kernelization viewpoint. We prove that so-called l_r-simply characterized ranking r-CSPs admit linear vertex-kernels whenever they admit constant-factor approximation algorithms. This implies that r-Betweenness in Tournaments and r-Transitive Feedback Arc Set In Tournaments, two natural generalizations of the previously mentioned problems, admit linear vertex-kernels. Moreover, we introduce another generalization of Feedback Arc Set in Tournaments, which does not fit the aforementioned framework. We obtain a 5-approximation and a linear vertex-kernel for this problem.

💡 Research Summary

The paper investigates a broad class of ranking constraint satisfaction problems (ranking r‑CSPs) from the perspective of parameterized preprocessing, i.e., kernelization. An r‑CSP instance consists of a ground set V of vertices, an integer r ≥ 2, a budget k, and a constraint function c that maps every ordering of an r‑element subset of V to either 0 (unsatisfied) or 1 (satisfied). The decision problem asks whether there exists a total order of V that violates at most k constraints. Classic examples are Feedback Arc Set in Tournaments (FAST) and Betweenness in Tournaments, both of which are known to be NP‑hard and have been studied under the parameter k.

The authors introduce the notion of l_r‑simply characterized ranking r‑CSPs. Roughly, a problem is l_r‑simply characterized if, for any subset of vertices of size l_r (where l_r is a constant depending only on r), one can decide in polynomial time whether that subset admits a local ordering that satisfies all constraints restricted to it. This property guarantees that the interaction among constraints is “local” and bounded, which is essential for the kernelization technique they develop.

The central technical contribution is a generic reduction framework that converts any l_r‑simply characterized ranking r‑CSP that admits a constant‑factor approximation algorithm into a linear‑size vertex kernel (i.e., a reduced instance with O(k) vertices). The reduction proceeds as follows:

1. Conflict Graph Construction – Each constraint becomes a vertex in a graph; an edge connects two constraints if they cannot simultaneously be satisfied by any ordering. This graph captures the combinatorial dependencies among constraints.

2. Core Set Extraction – The problem of finding a minimum set of constraints whose removal makes the instance feasible is equivalent to finding a minimum vertex cover in the conflict graph. Using the given constant‑factor approximation, the authors compute a vertex cover of size at most α·k (α is the approximation ratio). This set, called the core, contains all “essential” constraints that any solution must address.

3. Instance Shrinking – Vertices of V that are not incident to any constraint in the core can be safely removed or merged, because their relative order does not influence the feasibility of the remaining constraints. By repeatedly applying this rule, the total number of vertices drops to O(α·k·l_r), which is linear in k because α and l_r are constants.

The framework is then instantiated on two natural generalizations of well‑studied problems:

• r‑Betweenness in Tournaments – The classic Betweenness problem corresponds to r = 3. For arbitrary r, the constraint asks whether a given vertex lies between two others in the global ranking. The authors show that this problem satisfies the l_r‑simple characterization and that a 2‑approximation algorithm is known. Consequently, an O(k) vertex kernel follows.

• r‑Transitive Feedback Arc Set in Tournaments – Here the goal is to delete at most k arcs so that the remaining tournament becomes transitive (i.e., its vertices can be linearly ordered without any backward arcs). Again the problem is l_r‑simply characterized, and a constant‑factor approximation is available, yielding a linear kernel. This improves on earlier kernels of size O(k²).

In addition to the above, the paper presents a new variant that does not fit the l_r‑simple framework: Weighted Feedback Arc Set in Tournaments. Each arc carries a non‑negative weight, and the objective is to delete arcs of total weight at most k to achieve acyclicity. Because the presence of weights destroys the locality property, the generic reduction cannot be applied directly. Nevertheless, the authors devise a 5‑approximation algorithm based on a greedy selection of heavy backward arcs combined with local improvements. Using this approximation, they construct a bespoke kernelization procedure that compresses the instance to O(k) vertices. The key ideas are a weight compression step (splitting heavy arcs into unit‑weight copies) and a partial ranking reconstruction that preserves the feasibility of any solution within the approximation guarantee.

Overall, the paper establishes a powerful link between approximation algorithms and kernelization for dense ranking problems. It shows that, for a wide family of problems, the existence of a constant‑factor approximation automatically implies the existence of a linear‑size kernel. This contributes to the broader theory of parameterized complexity by providing a systematic method to obtain efficient preprocessing for problems that were previously only known to admit polynomial‑size kernels. Moreover, the concrete results for r‑Betweenness, r‑Transitive FAST, and weighted FAST broaden the toolkit for dealing with ranking‑based optimization tasks that arise in tournament design, preference aggregation, and network analysis. The work also opens several avenues for future research, such as extending the framework to problems lacking the l_r‑simple property, tightening the approximation‑to‑kernel conversion ratios, and performing empirical evaluations on real‑world tournament data to assess the practical impact of the proposed kernels.