Parametric Graph Templates: Properties and Algorithms

TL;DR

This paper introduces parametric graph templates to efficiently represent and analyze graphs with hierarchical and repetitive structures, enabling new polynomial and fixed-parameter tractable algorithms for key graph problems.

Contribution

It defines a new class of parametric graph templates and develops structurally-parametric algorithms for maximum flow, minimum cut, and tree subgraph isomorphism.

Findings

Algorithms are polynomial time for maximum flow and minimum cut.

Tree subgraph isomorphism is fixed-parameter tractable.

Templates can be recovered from instantiated graphs in quasi-polynomial time.

Abstract

Hierarchical structure and repetition are prevalent in graphs originating from nature or engineering. These patterns can be represented by a class of parametric-structure graphs, which are defined by templates that generate structure by way of repeated instantiation. We propose a class of parametric graph templates that can succinctly represent a wide variety of graphs. Using parametric graph templates, we develop structurally-parametric algorithm variants of maximum flow, minimum cut, and tree subgraph isomorphism. Our algorithms are polynomial time for maximum flow and minimum cut and are fixed-parameter tractable for tree subgraph isomorphism when parameterized by the size of the tree subgraph. By reasoning about the structure of the repeating subgraphs, we avoid explicit construction of the instantiation. Furthermore, we show how parametric graph templates can be recovered from an instantiated graph in quasi-polynomial time when certain parameters of the graph are bounded. Parametric graph templates and the presented algorithmic techniques thus create opportunities for reasoning about the generating structure of a graph, rather than an instance of it.