[Paper Review] Circuit Lower Bounds for the p-Spin Optimization Problem

David Gamarnik, Aukosh Jagannath+1 more|arXiv (Cornell University)|Sep 3, 2021

Complexity and Algorithms in Graphs4 citations

TL;DR

This paper establishes a circuit depth lower bound of $\log n / (2\log\log n)$ for polynomial-size Boolean circuits that produce near-optimal solutions to the $p$-spin optimization problem with Rademacher couplings. The result leverages the overlap gap property (OGP) and Fourier analysis via the Linial-Mansour-Nisan Theorem, marking the strongest known depth lower bound for combinatorial optimization problems requiring full solution output rather than decision answers.

ABSTRACT

We consider the problem of finding a near ground state of a $p$-spin model with Rademacher couplings by means of a low-depth circuit. As a direct extension of the authors' recent work [Gamarnik, Jagannath, Wein 2020], we establish that any poly-size $n$-output circuit that produces a spin assignment with objective value within a certain constant factor of optimality, must have depth at least $\log n/(2\log\log n)$ as $n$ grows. This is stronger than the known state of the art bounds of the form $Ω(\log n/(k(n)\log\log n))$ for similar combinatorial optimization problems, where $k(n)$ depends on the optimality value. For example, for the largest clique problem $k(n)$ corresponds to the square of the size of the clique [Rossman 2010]. At the same time our results are not quite comparable since in our case the circuits are required to produce a solution itself rather than solving the associated decision problem. As in our earlier work, the approach is based on the overlap gap property (OGP) exhibited by random $p$-spin models, but the derivation of the circuit lower bound relies further on standard facts from Fourier analysis on the Boolean cube, in particular the Linial-Mansour-Nisan Theorem. To the best of our knowledge, this is the first instance when methods from spin glass theory have ramifications for circuit complexity.

Motivation & Objective

• To establish tight circuit depth lower bounds for polynomial-size Boolean circuits solving the $p$-spin optimization problem with Rademacher couplings.
• To extend prior results on low-degree polynomial algorithms to general Boolean circuits by leveraging the overlap gap property (OGP).
• To demonstrate that circuits producing full spin assignments within a constant factor of optimality must have depth at least $\log n / (2\log\log n)$.
• To show that methods from spin glass theory, particularly the ensemble overlap gap property (e-OGP), have direct implications for circuit complexity.
• To provide a framework applicable to other models exhibiting OGP, advancing the frontier of circuit complexity in optimization.

Proposed method

• Utilizes the ensemble overlap gap property (e-OGP) to characterize the structure of near-optimal spin assignments in $p$-spin models with Rademacher couplings.
• Applies the Linial-Mansour-Nisan Theorem to show that low-depth circuits can only produce outputs with Fourier expansions concentrated on low-degree terms.
• Employs a continuity argument and symmetrization to bound the maximum deviation of Hamiltonian differences across correlated instances.
• Uses McDiarmid's inequality and Talagrand's comparison inequality to control the sub-Gaussian behavior of the maximum Hamiltonian difference over spin configurations.
• Applies a union bound over $O(n^{2p})$ pairs of time indices to derive a uniform bound on the maximum Hamiltonian fluctuation across correlated instances.
• Relies on universality arguments to extend results from Gaussian to Rademacher-distributed couplings, ensuring robustness of the OGP under distributional shifts.

Experimental results

Research questions

• RQ1 What is the minimum depth required for a polynomial-size Boolean circuit to produce a spin assignment within a constant factor of the optimal objective value in the $p$-spin model?
• RQ2 Can the overlap gap property (OGP) be used to derive circuit complexity lower bounds for optimization problems that require full solution output?
• RQ3 How does the depth requirement for solution-producing circuits compare to that of decision-only circuits in combinatorial optimization?
• RQ4 To what extent can Fourier analysis on the Boolean cube, particularly the Linial-Mansour-Nisan Theorem, be used to rule out low-depth circuits for optimization tasks?
• RQ5 Can OGP-based methods be extended to sparse random graph models, and what are the limitations of existing circuit insensitivity theorems in such settings?

Key findings

• Any polynomial-size $n$-output Boolean circuit that produces a spin assignment within a constant factor of the optimal objective value for the $p$-spin model must have depth at least $\log n / (2\log\log n)$.
• The lower bound is independent of the optimality gap, unlike previous results such as those for the clique problem, which depend on $k(n)^2$ where $k(n)$ is the clique size.
• The result is stronger than state-of-the-art bounds for similar problems because it applies to solution-finding circuits rather than decision problems.
• The proof relies on the ensemble overlap gap property (e-OGP), which ensures that near-optimal solutions either have high or low overlap, ruling out intermediate overlaps.
• The Linial-Mansour-Nisan Theorem is used to show that low-depth circuits cannot produce outputs with Fourier expansions that contradict the e-OGP structure.
• This is the first known application of spin glass theory—specifically OGP—to circuit complexity, establishing a novel bridge between statistical physics and computational complexity.