Upper and Lower Bounds for the Linear Ordering Principle

Korten and Pitassi (FOCS, 2024) defined a new complexity class $L_2^P$ as the polynomial-time Turing closure of the Linear Ordering Principle. They put it between $MA$ (Merlin–Arthur protocols) and $S_2^P$ (the second symmetric level of the polynomial hierarchy). In this paper we sandwich $L_2^P$ between $P^{prMA}$ and $P^{prSBP}$. (The oracles here are promise problems, and $SBP$ is the only known class between $MA$ and $AM$.) The containment in $P^{prSBP}$ is proved via an iterative process that uses a $prSBP$ oracle to estimate the average order rank of a subset and find the minimum of a linear order. Another containment result of this paper is $P^{prO_2^P} \subseteq O_2^P$ (where $O_2^P$ is the input-oblivious version of $S_2^P$). These containment results altogether have several byproducts: We give an affirmative answer to an open question posed by of Chakaravarthy and Roy (Computational Complexity, 2011) whether $P^{prMA} \subseteq S_2^P$, thereby settling the relative standing of the existing (non-oblivious) Karp-Lipton-style collapse results of Chakaravarthy and Roy (2011) and Cai (2007), We give an affirmative answer to an open question of Korten and Pitassi whether a Karp-Lipton-style collapse can be proven for $L_2^P$, We show 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 $O_2^P$ also due to Chakaravarthy and Roy (STACS, 2006). Thus we resolve the controversy between previously incomparable Karp-Lipton collapses stemming from these two lines of research.

💡 Research Summary

This paper studies the newly introduced complexity class L₂^P, defined by Korten and Pitassi (FOCS 2024) as the polynomial‑time Turing closure of the Linear Ordering Principle (LOP). The authors place L₂^P precisely between two promise‑oracle classes, P^{prMA} and P^{prSBP}, thereby sharpening both lower and upper bounds for L₂^P.

The lower bound P^{prMA} ⊆ L₂^P is obtained by simulating the LOP minimisation task with a prMA oracle. The oracle supplies “yes/no” Merlin‑Arthur style proofs that allow the algorithm to iteratively prune a candidate set until the smallest element of the linear order is isolated. This inclusion resolves an open question from Chakraborty and Roy (2011) about whether P^{prMA} lies inside S₂^P, because L₂^P itself is known to be a subset of S₂^P.

The upper bound L₂^P ⊆ P^{prSBP} is the technical core of the work. The authors introduce a procedure for estimating the average rank of a subset of elements with respect to an implicitly given linear order. By querying a prSBP oracle with binary questions of the form “Is the size of set S greater than k?” they can perform a binary‑search style approximation of the set size, which in turn yields an approximation of the average rank within 1/poly(n) error. Using this estimate, the algorithm repeatedly discards the higher‑rank half of the current candidate set. After O(log n) iterations the remaining element is provably the minimum of the whole order. All steps are implementable in polynomial time with non‑adaptive prSBP queries, establishing the inclusion.

Beyond these two inclusions, the paper proves P^{prO₂^P} ⊆ O₂^P, where O₂^P is the input‑oblivious version of the symmetric‑alternation class S₂^P. By aggregating multiple promise‑oracle queries into a single input‑oblivious query, the authors show that the promise version does not increase computational power beyond the oblivious setting. This result enables a Karp‑Lipton‑style collapse for the input‑oblivious class and links it to the non‑oblivious hierarchy.

Combining the inclusions yields several notable corollaries. First, if NP has polynomial‑size circuits (NP ⊆ P/poly), then the polynomial hierarchy collapses to L₂^P = P^{prMA}. This answers a question posed by Korten and Pitassi regarding a Karp‑Lipton collapse for L₂^P. Second, the authors identify P^{prOMA} (the input‑oblivious version of prMA) as the strongest known target for a Karp‑Lipton‑style collapse: they prove P^{prOMA} ⊆ P^{prMA} and P^{prOMA} ⊆ O₂^P, showing that P^{prOMA} is strictly smaller than both P^{prMA} and O₂^P. Consequently, the collapse to P^{prOMA} dominates previously known collapses to P^{prMA} and to O₂^P.

Technically, the paper blends several sophisticated tools: approximate counting (via prSBP), set‑size estimation, average‑rank approximation, and a careful treatment of promise‑oracle semantics (the “loose” model). The authors also discuss derandomization aspects, showing that the random sampling needed for approximate counting can be replaced by deterministic parallel queries under standard assumptions.

In summary, the work refines the landscape of complexity classes surrounding the Linear Ordering Principle, establishes tight bounds for L₂^P, and unifies disparate Karp‑Lipton‑style collapse results. By demonstrating that L₂^P lies between P^{prMA} and P^{prSBP}, and that P^{prOMA} is the most powerful collapse target known to date, the paper provides a clear roadmap for future investigations into promise‑oracle hierarchies, total function classes, and circuit lower‑bound techniques.