[Paper Review] On Triangle Counting Parameterized by Twin-Width

Stefan Kratsch, Florian Nelles+1 more|arXiv (Cornell University)|Feb 14, 2022

Advanced Graph Theory Research4 citations

TL;DR

This paper presents an efficient parameterized algorithm for triangle counting in graphs, running in time $\mathcal{O}(d^2n + m)$, where $d$ is the twin-width and $n$, $m$ are vertices and edges. The method uses dynamic programming over a compact $d$-contraction sequence, maintaining auxiliary values to count triangles across contraction steps, achieving a significant improvement over unparameterized combinatorial algorithms on graphs with bounded twin-width.

ABSTRACT

In this report we present an algorithm solving Triangle Counting in time $O(d^2n+m)$, where n and m, respectively, denote the number of vertices and edges of a graph G and d denotes its twin-width, a recently introduced graph parameter. We assume that a compact representation of a d-contraction sequence of G is given.

Motivation & Objective

• To design a fast parameterized algorithm for triangle counting in graphs, exploiting the recently introduced twin-width parameter.
• To achieve a running time that is adaptive to the twin-width $d$, improving upon the best known unparameterized combinatorial algorithms.
• To provide a dynamic programming approach that efficiently tracks triangle counts through a sequence of vertex contractions.
• To demonstrate that triangle counting can be solved efficiently on graph classes with bounded twin-width, such as cographs, unit interval, and minor-closed graphs.
• To lay the foundation for extending the approach to other problems parameterized by twin-width.

Proposed method

• The algorithm operates on a compact representation of a $d$-contraction sequence, where vertices are successively contracted in a way that maintains red degree at most $d$.
• It uses dynamic programming to maintain and update auxiliary values for each vertex in the current trigraph, tracking black and red neighbor counts across contraction steps.
• For each contraction step, the algorithm updates the adjacency structure and auxiliary data in $\mathcal{O}(d)$ time, leveraging the bounded red degree.
• The triangle count is accumulated by tracking transitions between cases based on edge colors (black or red) in the trigraph representation during contractions.
• The procedure TriCountOneNeighbor and TriCountTwoNeighbors are used to count triangles involving one or two black edges in the current trigraph, respectively, in constant or linear time per call.
• The overall running time is bounded by $\mathcal{O}(d^2n + m)$, derived from $n$ contractions, each taking $\mathcal{O}(d^2)$ time due to neighbor checks and updates.

Experimental results

Research questions

• RQ1 Can triangle counting be solved in time $\mathcal{O}(d^2n + m)$ when parameterized by twin-width $d$?
• RQ2 Is there a dynamic programming approach over a contraction sequence that efficiently tracks triangle counts while maintaining $\mathcal{O}(d^2n + m)$ running time?
• RQ3 How does the performance of this algorithm compare to unparameterized combinatorial algorithms on graphs with small twin-width?
• RQ4 Can the approach be generalized to other problems parameterized by twin-width?
• RQ5 Is it possible to achieve an $\mathcal{O}(\operatorname{tww}^{\omega-1}n + m)$-time algorithm by leveraging fast matrix multiplication?

Key findings

• The algorithm runs in $\mathcal{O}(d^2n + m)$ time, where $d$ is the twin-width and $n$, $m$ are the number of vertices and edges.
• The method assumes a compact $d$-contraction sequence is given, which is a common assumption in twin-width-based algorithms.
• The algorithm maintains auxiliary values for each vertex, updated in $\mathcal{O}(d)$ time per contraction, ensuring efficient overall performance.
• Triangle counting is correctly computed by tracking case transitions in the trigraph representation, with each triangle counted exactly once as the contraction sequence progresses.
• The algorithm is adaptive, outperforming the best unparameterized combinatorial algorithms on graphs with small twin-width.
• The approach opens the door to designing efficient algorithms for other problems parameterized by twin-width, such as independent set or shortest paths.