Approximation algorithms for satisfiable and nearly satisfiable ordering CSPs
2026-03-31 • Data Structures and Algorithms
Data Structures and Algorithms
AI summaryⓘ
The authors study how to approximate solutions for problems where you need to arrange items in a specific order to satisfy some conditions. They create a general method that first simplifies the problem, solves this simpler version, and then uses a special random process to get a solution for the original problem. They show that this random process can always be chosen from a well-defined set called strong IDU transformations, which makes finding the best approximation easier. Their method works for many types of ordering problems and allows computing near-best solutions efficiently for any fixed problem complexity and desired accuracy.
ordering constraint satisfaction problemsapproximation algorithmsrandomized transformationsstrong IDU transformationspredicate arityrelaxationranking problemsschedulingoptimizationconstraint language
Authors
Yury Makarychev
Abstract
We study approximation algorithms for satisfiable and nearly satisfiable instances of ordering constraint satisfaction problems (ordering CSPs). Ordering CSPs arise naturally in ranking and scheduling, yet their approximability remains poorly understood beyond a few isolated cases. We introduce a general framework for designing approximation algorithms for ordering CSPs. The framework relaxes an input instance to an auxiliary ordering CSP, solves the relaxation, and then applies a randomized transformation to obtain an ordering for the original instance. This reduces the search for approximation algorithms to an optimization problem over randomized transformations. Our main technical contribution is to show that the power of this framework is captured by a structured class of transformations, which we call strong IDU transformations: every transformation used in the framework can be replaced by a strong IDU transformation without weakening the resulting approximation guarantee. We then classify strong IDU transformations and show that optimizing over them reduces to an explicit optimization problem whose dimension depends only on the maximum predicate arity $k$ and the desired precision $δ> 0$. As a consequence, for any finite ordering constraint language, we can compute a strong IDU transformation whose guarantee is within $δ$ of the best guarantee achievable by the framework, in time depending only on $k$ and $δ$. The framework applies broadly and yields nontrivial approximation guarantees for a wide class of ordering predicates.