A Dichotomy Theorem for Linear Time Homomorphism Orbit Counting in Bounded Degeneracy Graphs

11/16/2022
by   Daniel Paul-Pena, et al.
0

Counting the number of homomorphisms of a pattern graph H in a large input graph G is a fundamental problem in computer science. There are myriad applications of this problem in databases, graph algorithms, and network science. Often, we need more than just the total count. Especially in large network analysis, we wish to compute, for each vertex v of G, the number of H-homomorphisms that v participates in. This problem is referred to as homomorphism orbit counting, as it relates to the orbits of vertices of H under its automorphisms. Given the need for fast algorithms for this problem, we study when near-linear time algorithms are possible. A natural restriction is to assume that the input graph G has bounded degeneracy, a commonly observed property in modern massive networks. Can we characterize the patterns H for which homomorphism orbit counting can be done in linear time? We discover a dichotomy theorem that resolves this problem. For pattern H, let l be the length of the longest induced path between any two vertices of the same orbit (under the automorphisms of H). If l <= 5, then H-homomorphism orbit counting can be done in linear time for bounded degeneracy graphs. If l > 5, then (assuming fine-grained complexity conjectures) there is no near-linear time algorithm for this problem. We build on existing work on dichotomy theorems for counting the total H-homomorphism count. Somewhat surprisingly, there exist (and we characterize) patterns H for which the total homomorphism count can be computed in linear time, but the corresponding orbit counting problem cannot be done in near-linear time.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
10/16/2020

Near-Linear Time Homomorphism Counting in Bounded Degeneracy Graphs: The Barrier of Long Induced Cycles

Counting homomorphisms of a constant sized pattern graph H in an input g...
research
08/17/2023

Computing complexity measures of degenerate graphs

We show that the VC-dimension of a graph can be computed in time n^log d...
research
08/28/2023

Efficient Batch Dynamic Graphlet Counting

Graphlet counting is an important problem as it has numerous application...
research
11/04/2019

Counting Small Permutation Patterns

A sample of n generic points in the xy-plane defines a permutation that ...
research
11/14/2019

Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six

We consider the problem of counting all k-vertex subgraphs in an input g...
research
01/15/2020

A color-avoiding approach to subgraph counting in bounded expansion classes

We present an algorithm to count the number of occurrences of a pattern ...
research
07/24/2020

Tromino Tilings with Pegs via Flow Networks

A tromino tiling problem is a packing puzzle where we are given a region...

Please sign up or login with your details

Forgot password? Click here to reset