# On the Power of Symmetric Linear Programs

## TL;DR

This paper explores the power of symmetric linear programs in graph property decision problems, establishing equivalences with symmetric Boolean circuits and fixed-point logic with counting, and providing bounds for specific graph classes.

## Contribution

It characterizes symmetric LPs in terms of symmetric Boolean circuits and fixed-point logic with counting, and derives bounds for graph classes like perfect matchings and Hamiltonian graphs.

## Key findings

- Polynomial-size symmetric LPs for perfect matchings
- Exponential lower bounds for Hamiltonian graphs
- Equivalence of symmetric LPs and symmetric Boolean circuits

## Abstract

We consider families of symmetric linear programs (LPs) that decide a property of graphs (or other relational structures) in the sense that, for each size of graph, there is an LP defining a polyhedral lift that separates the integer points corresponding to graphs with the property from those corresponding to graphs without the property. We show that this is equivalent, with at most polynomial blow-up in size, to families of symmetric Boolean circuits with threshold gates. In particular, when we consider polynomial-size LPs, the model is equivalent to definability in a non-uniform version of fixed-point logic with counting (FPC). Known upper and lower bounds for FPC apply to the non-uniform version. In particular, this implies that the class of graphs with perfect matchings has polynomial-size symmetric LPs while we obtain an exponential lower bound for symmetric LPs for the class of Hamiltonian graphs. We compare and contrast this with previous results (Yannakakis 1991) showing that any symmetric LPs for the matching and TSP polytopes have exponential size. As an application, we establish that for random, uniformly distributed graphs, polynomial-size symmetric LPs are as powerful as general Boolean circuits. We illustrate the effect of this on the well-studied planted-clique problem.