The Complexity of Pattern Counting in Directed Graphs, Parameterised by the Outdegree

TL;DR

This paper investigates the fixed-parameter tractability of counting subgraph copies in directed graphs, introducing new structural parameters and characterizing when the problem is computationally feasible based on these parameters.

Contribution

It introduces the fractional cover number and source number as key parameters, providing algorithms and complexity characterizations for counting subgraphs in directed graphs.

Findings

Algorithms with specific runtime bounds based on new parameters.

Characterization of fixed-parameter tractability using fractional cover and source numbers.

Optimality of classic algorithms under ETH assumptions.

Abstract

We study the fixed-parameter tractability of the following fundamental problem: given two directed graphs $\vec H$ and $\vec G$, count the number of copies of $\vec H$ in $\vec G$. The standard setting, where the tractability is well understood, uses only $|\vec H|$ as a parameter. In this paper we take a step forward, and adopt as a parameter $|\vec H|+d(\vec G)$, where $d(\vec G)$ is the maximum outdegree of $|\vec G|$. Under this parameterization, we completely characterize the fixed-parameter tractability of the problem in both its non-induced and induced versions through two novel structural parameters, the fractional cover number $\rho^*$ and the source number $\alpha_s$. On the one hand we give algorithms with running time $f(|\vec H|,d(\vec G)) \cdot |\vec G|^{\rho^*\!(\vec H)+O(1)}$ and $f(|\vec H|,d(\vec G)) \cdot |\vec G|^{\alpha_s(\vec H)+O(1)}$ for counting respectively the copies and induced copies of $\vec H$ in $\vec G$; on the other hand we show that, unless the Exponential Time Hypothesis fails, for any class $\vec C$ of directed graphs the (induced) counting problem is fixed-parameter tractable if and only if $\rho^*(\vec C)$ ($\alpha_s(\vec C)$) is bounded. These results explain how the orientation of the pattern can make counting easy or hard, and prove that a classic algorithm by Chiba and Nishizeki and its extensions (Chiba, Nishizeki SICOMP 85; Bressan Algorithmica 21) are optimal unless ETH fails.