Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Daniel Paul-Pena, C. Seshadhri

TL;DR

Somewhat surprisingly, there exist (and there exist) patterns H for which the total homomorphism count can be computed in near-linear time, but the corresponding orbit counting problem cannot be done in near-linear time.

Abstract

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 near-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 near-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 near-linear time, but the corresponding orbit counting problem cannot be done in near-linear time.

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

TL;DR

Somewhat surprisingly, there exist (and there exist) patterns H for which the total homomorphism count can be computed in near-linear time, but the corresponding orbit counting problem cannot be done in near-linear time.

Abstract

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 near-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 near-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 near-linear time, but the corresponding orbit counting problem cannot be done in near-linear time.
Paper Structure (10 sections, 14 theorems, 7 equations, 1 figure, 2 algorithms)

This paper contains 10 sections, 14 theorems, 7 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Main Result
  3. Main Ideas
  4. Related Work
  5. Preliminaries
  6. DAG-tree decompositions
  7. Obtaining Vertex-Centric Counts
  8. From Vertex-Centric to Orbit Counts
  9. Wrapping it up
  10. Lower Bound for computing Homomorphism Orbit Counts

Key Result

Theorem 5

(Main Theorem) Let $G$ be a graph with $n$ vertices, $m$ edges, and bounded degeneracy. Let $\gamma > 0$ denote the constant from the Triangle Detection Conjecture (conj:triangle).

Figures (1)

  • Figure 1: Examples of orbits and LIPCO values. Vertices in the same orbit have the same color. The top graph is the 7-path (a path of length 6). There is an induced path of length 6 between the red vertices, hence the LIPCO of this graph is 6. \ref{['thm:main']} implies that we can not compute $\mathrm{OrbitHom}$ in near-linear time. The bottom graph adds a triangle at the end, breaking the symmetry, and the only vertices in the same orbit in that graph are the red ones. The LIPCO in this graph is now less than 6 so we can compute $\mathrm{OrbitHom}$ in near-linear time.

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Definition 3
  • Conjecture 4: Triangle Detection Conjecture AbWi14
  • Theorem 5
  • Definition 6
  • Definition 7
  • Lemma 8
  • Definition 9
  • Definition 10
  • ...and 22 more