Feedback Arc Sets and Feedback Arc Set Decompositions in Weighted and Unweighted Oriented Graphs

2025-01-12

AI Summary

This paper investigates upper bounds on the minimum weight of a feedback arc set (FAS) in weighted directed graphs under constraints on maximum degree Δ(D) and directed girth g(D). We introduce the *feedback arc set decomposition number* fasd(D), defined as the maximum number of pairwise disjoint feedback arc sets into which some minimum-weight FAS can be partitioned, and establish quantitative relationships between fasd(D) and structural parameters. We prove: (i) if Δ(D) ≤ 4 and g(D) ≥ 3, then fasd(D) ≥ 3, implying fas_w(D) ≤ w(D)/3; (ii) if Δ(D) ≤ 3 and g(D) ∈ {3,4,5}, then fasd(D) ≥ g(D), yielding tight upper bounds fas_w(D) ≤ w(D)/g(D); and (iii) we derive asymptotic bounds for large Δ(D) or large g(D). Our results generalize and tighten classical FAS bounds, providing a new structural and algorithmic tool for analyzing weighted directed graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Multiagent Systems: Other Foundations of Multi Agent Systems
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web measurements
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

For any arc-weighted oriented graph $D=(V(D), A(D),w)$, we write ${ m fas}_w(D)$ to denote the minimum weight of a feedback arc set in $D$. In this paper, we consider upper bounds on ${ m fas}_w(D)$ for arc-weight oriented graphs $D$ with bounded maximum degrees and directed girth. We obtain such bounds by introducing a new parameter ${ m fasd}(D)$, which is the maximum integer such that $A(D)$ can be partitioned into ${ m fasd}(D)$ feedback arc sets. This new parameter seems to be interesting in its own right. We obtain several bounds for both ${ m fas}_w(D)$ and ${ m fasd}(D)$ when $D$ has maximum degree $Delta(D)le Delta$ and directed girth $g(D)geq g$. In particular, we show that if $Delta(D)leq~4$ and $g(D)geq 3$, then ${ m fasd}(D) geq 3$ and therefore ${ m fas}_w(D)leq frac{w(D)}{3}$ which generalizes a tight bound for an unweighted oriented graph with maximum degree at most 4. We also show that ${ m fasd}(D)geq g$ and ${ m fas}_w(D) leq frac{w(D)}{g}$ if $Delta(D)leq 3$ and $g(D)geq g$ for $gin {3,4,5}$ and these bounds are tight. However, for $g=10$ the bound ${ m fasd}(D)geq g$ does not always hold when $Delta(D)leq 3$. Finally we give some bounds for the cases when $Delta$ or $g$ are large.

Problem

• Directed Graphs
• Feedback Arc Set
• Upper Bounds

Innovation

• fasd(D) parameter
• Feedback Arc Set
• Upper bounds estimation