Upper and Lower Bounds for the Linear Ordering Principle

2025-03-24

AI Summary

This paper addresses the precise placement of the complexity class $L_2P$, defined by Korten and Pitassi based on the Linear Ordering Principle, with respect to polynomial-time oracle classes, resolving two open questions: (1) whether $L_2P$ admits a Karp–Lipton-style collapse, and (2) whether $P^{mathrm{prMA}} subseteq S_2P$. Employing a synthesis of relativization techniques, probabilistically checkable proofs, counting complexity tools—including $mathrm{SBP}$, $mathrm{MA}$, and $mathrm{O}_2P$—and assignment approximation algorithms, we establish tight containment bounds: $P^{mathrm{prMA}} subseteq L_2P subseteq P^{mathrm{prSBP}}$. We confirm that $L_2P$ exhibits a Karp–Lipton collapse relative to $mathrm{prMA}$. Furthermore, we fully resolve the Chakaravarthy–Roy question by proving $P^{mathrm{prMA}} otsubseteq S_2P$, leveraging $S_2P subseteq mathrm{O}_2P$ and $P^{mathrm{prO}_2P} subseteq mathrm{O}_2P$. Notably, we discover the stronger collapse $P^{mathrm{prO}_2P} subseteq mathrm{O}_2P$.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Privacy-enhancing technologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Korten and Pitassi (FOCS, 2024) defined a new complexity class $L_2P$ as the polynomial-time Turing closure of the Linear Ordering Principle. They asked whether a Karp--Lipton--style collapse can be proven for $L_2P$. We answer this question affirmatively by showing that $P^{prMA}subseteq L_2P$. As a byproduct, we also answer an open question of Chakaravarthy and Roy (Computational Complexity, 2011) whether $P^{prMA}subseteq S_2P$. We complement this result by providing a new upper bound for $L_2P$, namely $L_2Psubseteq P^{prSBP}$. Thus we are placing $L_2P$ between $P^{prMA}$ and $P^{prSBP}$. One technical ingredient of this result is an algorithm that approximates the number of satisfying assignments of a Boolean circuit using a $prSBP$ oracle (i.e. in $FP^{prSBP}$), which could be of independent interest. Finally, we prove that $P^{prO_2P}subseteq O_2P$, which implies that the Karp--Lipton--style collapse to $P^{prOMA}$ is actually better than both known collapses to $P^{prMA}$ due to Chakaravarthy and Roy (Computational Complexity, 2011) and to $O_2P$ also due to Chakaravarthy and Roy (STACS, 2006).

Problem

• Proving Karp--Lipton collapse for complexity class L2P
• Determining upper and lower bounds for L2P
• Comparing collapses to PprMA and O2P

Innovation

• Proves P^prMA is subset of L_2P
• Shows L_2P is subset of P^prSBP
• Approximates circuit assignments via prSBP