Optimization, Randomized Approximability, and Boolean Constraint Satisfaction Problems

TL;DR

This paper provides a comprehensive classification of optimization Boolean constraint satisfaction problems based on whether their optimal solutions are polynomial-time solvable, hard, or intermediate, considering both sum and product measures of constraints.

Contribution

It introduces a unified framework for analyzing weighted Boolean CSPs with product-based optimality and establishes a complete trichotomy theorem classifying problems into three complexity categories.

Findings

Classified problems into PO, NPO-hard, and intermediate categories.

Developed a new analysis method using T-constructibility for weighted constraints.

Extended previous sum-based classifications to product-based optimality measures.

Abstract

We give a unified treatment to optimization problems that can be expressed in the form of nonnegative-real-weighted Boolean constraint satisfaction problems. Creignou, Khanna, Sudan, Trevisan, and Williamson studied the complexity of approximating their optimal solutions whose optimality is measured by the sums of outcomes of constraints. To explore a wider range of optimization constraint satisfaction problems, following an early work of Marchetti-Spaccamela and Romano, we study the case where the optimality is measured by products of constraints' outcomes. We completely classify those problems into three categories: PO problems, NPO-hard problems, and intermediate problems that lie between the former two categories. To prove this trichotomy theorem, we analyze characteristics of nonnegative-real-weighted constraints using a variant of the notion of T-constructibility developed earlier for complex-weighted counting constraint satisfaction problems.