Distributed Interactive Proofs for the Recognition of Some Geometric Intersection Graph Classes

TL;DR

This paper develops efficient distributed protocols for recognizing certain geometric intersection graph classes, providing proof labeling schemes and interactive protocols with logarithmic certificates, and establishing lower bounds on certificate sizes.

Contribution

It introduces novel distributed recognition protocols for permutation, trapezoid, circle, and polygon-circle graphs, with matching lower bounds for certificate sizes.

Findings

Logarithmic-sized certificates for permutation and trapezoid graphs

Three-round interactive protocols for circle and polygon-circle graphs

Logarithmic lower bounds on certificate sizes for recognition protocols

Abstract

A graph $G=(V,E)$ is a geometric intersection graph if every node $v \in V$ is identified with a geometric object of some particular type, and two nodes are adjacent if the corresponding objects intersect. Geometric intersection graph classes have been studied from both the theoretical and practical point of view. On the one hand, many hard problems can be efficiently solved or approximated when the input graph is restricted to a geometric intersection class of graphs. On the other hand, these graphs appear naturally in many applications such as sensor networks, scheduling problems, and others. Recently, in the context of distributed certification and distributed interactive proofs, the recognition of graph classes has started to be intensively studied. Different results related to the recognition of trees, bipartite graphs, bounded diameter graphs, triangle-free graphs, planar graphs, bounded genus graphs, $H$-minor free graphs, etc., have been obtained. The goal of the present work is to design efficient distributed protocols for the recognition of relevant geometric intersection graph classes, namely permutation graphs, trapezoid graphs, circle graphs, and polygon-circle graphs. More precisely, for the two first classes, we give proof labeling schemes recognizing them with logarithmic-sized certificates. For the other two classes, we give three-round distributed interactive protocols that use messages and certificates of size $\mathcal{O}(\log n)$. Finally, we provide logarithmic lower-bounds on the size of the certificates on the proof labeling schemes for the recognition of any of the aforementioned geometric intersection graph classes.