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# Conditional Dichotomy of Boolean Ordered Promise CSPs

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LIPIcs.ICALP.2021.37.pdf
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## Acknowledgements

We thank Libor Barto, whose talk [Libor Barto, 2018] and insightful discussions inspired our work.

## Cite As

Joshua Brakensiek, Venkatesan Guruswami, and Sai Sandeep. Conditional Dichotomy of Boolean Ordered Promise CSPs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.37

## Abstract

Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Satisfaction Problems (CSPs) where each predicate has a strong and a weak form and given a CSP instance, the objective is to distinguish if the strong form can be satisfied vs. even the weak form cannot be satisfied. Since their formal introduction by Austrin, Guruswami, and Håstad [Per Austrin et al., 2017], there has been a flurry of works on PCSPs, including recent breakthroughs in approximate graph coloring [Barto et al., 2018; Andrei A. Krokhin and Jakub Opršal, 2019; Marcin Wrochna and Stanislav Zivný, 2020]. The key tool in studying PCSPs is the algebraic framework developed in the context of CSPs where the closure properties of the satisfying solutions known as polymorphisms are analyzed. The polymorphisms of PCSPs are significantly richer than CSPs - even in the Boolean case, we still do not know if there exists a dichotomy result for PCSPs analogous to Schaefer’s dichotomy result [Thomas J. Schaefer, 1978] for CSPs. In this paper, we study a special case of Boolean PCSPs, namely Boolean Ordered PCSPs where the Boolean PCSPs have the predicate x ≤ y. In the algebraic framework, this is the special case of Boolean PCSPs when the polymorphisms are monotone functions. We prove that Boolean Ordered PCSPs exhibit a computational dichotomy assuming the Rich 2-to-1 Conjecture [Mark Braverman et al., 2021] which is a perfect completeness surrogate of the Unique Games Conjecture. In particular, assuming the Rich 2-to-1 Conjecture, we prove that a Boolean Ordered PCSP can be solved in polynomial time if for every ε > 0, it has polymorphisms where each coordinate has Shapley value at most ε, else it is NP-hard. The algorithmic part of our dichotomy result is based on a structural lemma showing that Boolean monotone functions with each coordinate having low Shapley value have arbitrarily large threshold functions as minors. The hardness part proceeds by showing that the Shapley value is consistent under a uniformly random 2-to-1 minor. As a structural result of independent interest, we construct an example to show that the Shapley value can be inconsistent under an adversarial 2-to-1 minor.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Constraint and logic programming
• Theory of computation → Approximation algorithms analysis
• Theory of computation → Problems, reductions and completeness
• Applied computing → Economics
##### Keywords
• promise constraint satisfaction
• Boolean ordered PCSP
• Shapley value
• rich 2-to-1 conjecture
• random minor

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