Approximating observables is as hard as counting

TL;DR

This paper demonstrates that approximating local observables in Gibbs distributions is computationally as hard as counting solutions, even in cases where the underlying optimization problem is easy, by establishing a generic reduction from counting to estimating observables.

Contribution

It introduces a new reduction technique that relates approximate counting to local observable estimation, extending hardness results to bipartite graphs and various spin systems.

Findings

Hardness of estimating the average size of independent sets in bipartite graphs of max degree 6.

Tight hardness results for vertex-edge observables in antiferromagnetic 2-spin systems on bipartite graphs.

Hardness of approximating monochromatic edges in ferromagnetic Potts models.

Abstract

We study the computational complexity of estimating local observables for Gibbs distributions. A simple combinatorial example is the average size of an independent set in a graph. In a recent work, we established NP-hardness of approximating the average size of an independent set utilizing hardness of the corresponding optimization problem and the related phase transition behavior. Here, we instead consider settings where the underlying optimization problem is easily solvable. Our main contribution is to classify the complexity of approximating a wide class of observables via a generic reduction from approximate counting to the problem of estimating local observables. The key idea is to use the observables to interpolate the counting problem. Using this new approach, we are able to study observables on bipartite graphs where the underlying optimization problem is easy but the counting problem is believed to be hard. The most-well studied class of graphs that was excluded from previous hardness results were bipartite graphs. We establish hardness for estimating the average size of the independent set in bipartite graphs of maximum degree 6; more generally, we show tight hardness results for general vertex-edge observables for antiferromagnetic 2-spin systems on bipartite graphs. Our techniques go beyond 2-spin systems, and for the ferromagnetic Potts model we establish hardness of approximating the number of monochromatic edges in the same region as known hardness of approximate counting results.