Matrix Scaling: a New Heuristic for the Feedback Vertex Set Problem

TL;DR

This work introduces a matrix-scaling based heuristic for finding a minimum feedback vertex set in unweighted digraphs by perturbing the adjacency structure with loops and applying the Sinkhorn–Knopp balancing procedure. At each step, the algorithm identifies a vertex to remove by selecting the minimal diagonal entry in the balanced matrix, which approximates a vertex present in many disjoint cycle unions, and repeats after reductions until the graph is acyclic. The method achieves a theoretical running time of $O(|F|\log(|V|)|V|^{2})$ and demonstrates, through extensive experiments on Erdős–Rényi and regular digraphs, that it often produces smaller FVS sets and runs faster than existing heuristics, with additional post-processing to ensure minimality. The approach also incorporates lower-bound techniques for assessing approximation quality and analyzes practical variants, including disabling strong component reductions to gain speed. Overall, the paper presents a practically efficient, deterministic heuristic with strong empirical performance for the Directed Feedback Vertex Set problem.

Abstract

For a digraph $G$, a set $F\subseteq V(G)$ is said to be a feedback vertex set (FVS) if $G-F$ is acyclic. The problem of finding a smallest FVS is NP-hard. We present a matrix scaling technique for finding feedback vertex sets in un-weighted directed graphs that runs in $O(|F|\log(|V|)|V|^{2})$ time. Our technique is empirically shown to produce smaller feedback vertex sets than other known heuristics and in a shorter amount of time.

Figures (12)

• Figure 1: For $t\geq2$ the vertex $z$ is not in a minimum FVS, but is in nearly every DCU and most cycles.
• Figure 2: FVS_SH_Del
• Figure 3: Sinkhorn_Selection
• Figure 4: FVS_SH_DEL_MOD
• Figure 5: MaxDeg

Theorems & Definitions (1)

• Definition 1