Conic Formulations of Graph Homomorphisms

TL;DR

This paper introduces conic programming formulations for graph homomorphisms, unifying classical and quantum variants, and explores their properties and implications for graph theory and quantum information.

Contribution

It develops conic feasibility programs characterizing classical and quantum graph homomorphisms, and studies their properties and interactions with generalized Lovász theta functions.

Findings

Conic programs characterize classical and quantum homomorphisms.

Generalized theta functions are multiplicative on certain graph products.

Several classical homomorphism results are extended to the conic setting.

Abstract

Given graphs $X$ and $Y$, we define two conic feasibility programs which we show have a solution over the completely positive cone if and only if there exists a homomorphism from $X$ to $Y$. By varying the cone, we obtain similar characterizations of quantum/entanglement-assisted homomorphisms and three previously studied relaxations of these relations. Motivated by this, we investigate the properties of these "conic homomorphisms" for general (suitable) cones. We also consider two generalized versions of the Lov\'asz theta function, and how they interact with these conic homomorphisms. We prove analogs of several results on classical graph homomorphisms as well as some monotonicity theorems. We also show that one of the generalized theta functions is multiplicative on lexicographic and disjunctive graph products.