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Approximation algorithms for satisfiable and nearly satisfiable ordering CSPs

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.

Approximation algorithms for satisfiable and nearly satisfiable ordering CSPs

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 and the desired precision . As a consequence, for any finite ordering constraint language, we can compute a strong IDU transformation whose guarantee is within of the best guarantee achievable by the framework, in time depending only on and . The framework applies broadly and yields nontrivial approximation guarantees for a wide class of ordering predicates.

Paper Structure

This paper contains 47 sections, 39 theorems, 186 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Ordering CSPs
  3. Overview of our framework
  4. From weak to strong IDU transformations
  5. Relaxation for completely satisfiable instances
  6. Relaxation for nearly satisfiable instances
  7. Related work
  8. Technical overview of IDU transformations
  9. Applying the framework to satisfiable and nearly satisfiable ordering CSPs
  10. Results for single-predicate ordering CSPs of arity 3 and 4
  11. Dataset and verification code
  12. Completely satisfiable instances
  13. Nearly satisfiable instances
  14. Preliminaries
  15. Permutons
  16. ...and 32 more sections

Key Result

Theorem 2.4

Consider $\mathop{\mathrm{CSP}}\nolimits(\Pi)$ and a relaxation $\Pi'$ of $\Pi$. Assume that there is a polynomial-time algorithm that, given a $c$-satisfiable instance of $\mathop{\mathrm{CSP}}\nolimits(\Pi)$, finds a solution satisfying at least an $s$ fraction of constraints in the corresponding

Figures (10)

  • Figure 1: The figure shows the transformation from Example \ref{['example:algorithm']}. Each $x_i$ is randomly mapped to either $\pi(i)$ or $\pi(i)+n$.
  • Figure 2: Summary of the classification results for single-predicate ordering CSPs of arity 3 and 4. The diagram shows the number of predicates of different types. The boxes labeled "With a nontrivial $L$- or $R$-relaxation" count predicates for which at least one of the canonical tractable relaxations $\varphi_L$ and $\varphi_R$ is nontrivial. The boxes labeled "With a nontrivial $\varepsilon$-relaxation" are defined analogously.
  • Figure 3: The left panel shows five points sampled from a permuton (whose measure is supported on two segments), defining the pattern $(\mathsf{1\,3\,4\,2\,5})$. The right panel displays the permuton $\mu_\pi$ associated with the permutation $\pi=(\mathsf{3\,2\,5\,1\,4})\in {\mathbb{S}}_5$, whose support consists of five disjoint squares with side length $1/5$.
  • Figure 4: The increasing $I$, decreasing $D$, and uniform $U$ permutons.
  • Figure 5: The figure shows up-combinations of permutons:
  • ...and 5 more figures

Theorems & Definitions (108)

  • definition 1.1: Ordering CSP
  • definition 1.2
  • definition 1.3
  • definition 1.4
  • example 1.5
  • definition 2.1
  • definition 2.2
  • definition 2.3
  • Theorem 2.4
  • definition 2.5
  • ...and 98 more