Some Inapproximability Results of MAP Inference and Exponentiated Determinantal Point Processes

TL;DR

This paper establishes strong computational hardness results for MAP inference and probabilistic inference in exponentiated DPPs, showing they are NP-hard to approximate within certain exponential factors, thus highlighting their computational intractability.

Contribution

It provides the first exponential inapproximability bounds for MAP inference and normalizing constants in E-DPPs, significantly advancing understanding of their computational complexity.

Findings

Unconstrained MAP inference is NP-hard to approximate within 2^{βn}

Log-determinant maximization is NP-hard to approximate within 5/4

Normalizing constant for high exponent E-DPPs is NP-hard to approximate within 2^{βpn}

Abstract

We study the computational complexity of two hard problems on determinantal point processes (DPPs). One is maximum a posteriori (MAP) inference, i.e., to find a principal submatrix having the maximum determinant. The other is probabilistic inference on exponentiated DPPs (E-DPPs), which can sharpen or weaken the diversity preference of DPPs with an exponent parameter $p$. We present several complexity-theoretic hardness results that explain the difficulty in approximating MAP inference and the normalizing constant for E-DPPs. We first prove that unconstrained MAP inference for an $n \times n$ matrix is $\textsf{NP}$-hard to approximate within a factor of $2^{\beta n}$, where $\beta = 10^{-10^{13}} $. This result improves upon the best-known inapproximability factor of $(\frac{9}{8}-\epsilon)$, and rules out the existence of any polynomial-factor approximation algorithm assuming $\textsf{P} \neq \textsf{NP}$. We then show that log-determinant maximization is $\textsf{NP}$-hard to approximate within a factor of $\frac{5}{4}$ for the unconstrained case and within a factor of $1+10^{-10^{13}}$ for the size-constrained monotone case. In particular, log-determinant maximization does not admit a polynomial-time approximation scheme unless $\textsf{P} = \textsf{NP}$. As a corollary of the first result, we demonstrate that the normalizing constant for E-DPPs of any (fixed) constant exponent $p \geq \beta^{-1} = 10^{10^{13}}$ is $\textsf{NP}$-hard to approximate within a factor of $2^{\beta pn}$, which is in contrast to the case of $p \leq 1$ admitting a fully polynomial-time randomized approximation scheme.