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Tight Localizations of Feedback Sets

Michael Hecht, Krzysztof Gonciarz, Szabolcs Horvát

TL;DR

This work addresses the directed feedback arc set problem (FASP) and its dual, the feedback vertex set problem (FVSP), by introducing a practical $O(|V||E|^4)$-time heuristic, TIGHT-CUT, designed for large, sparse graphs typical in circuit design. Central to the approach are the building block ISO-CUT, which leverages isolated cycles to localize optimal arc cuts, and GOOD-GUESS, which selects promising cuts via minimum $s$-$t$-cuts; together they form the global algorithm TIGHT-CUT and its relaxed variant TIGHT-CUT*. Empirical evaluation across synthetic and real-world circuit graphs shows TIGHT-CUT* frequently achieves approximation ratios under 1.6 and substantially outperforms the Greedy Removal heuristic, while solving large instances that ILP-based methods struggle with. The results indicate meaningful practical gains for circuit testing and other applications requiring acyclic subgraph extraction, with potential for further speedups through parallelization and optimized min-cut implementations.

Abstract

The classical NP-hard feedback arc set problem (FASP) and feedback vertex set problem (FVSP) ask for a minimum set of arcs $\varepsilon \subseteq E$ or vertices $ν\subseteq V$ whose removal $G\setminus \varepsilon$, $G\setminus ν$ makes a given multi-digraph $G=(V,E)$ acyclic, respectively. Though both problems are known to be APX-hard, approximation algorithms or proofs of inapproximability are unknown. We propose a new $\mathcal{O}(|V||E|^4)$-heuristic for the directed FASP. While a ratio of $r \approx 1.3606$ is known to be a lower bound for the APX-hardness, at least by empirical validation we achieve an approximation of $r \leq 2$. The most relevant applications, such as circuit testing, ask for solving the FASP on large sparse graphs, which can be done efficiently within tight error bounds due to our approach.

Tight Localizations of Feedback Sets

TL;DR

This work addresses the directed feedback arc set problem (FASP) and its dual, the feedback vertex set problem (FVSP), by introducing a practical -time heuristic, TIGHT-CUT, designed for large, sparse graphs typical in circuit design. Central to the approach are the building block ISO-CUT, which leverages isolated cycles to localize optimal arc cuts, and GOOD-GUESS, which selects promising cuts via minimum --cuts; together they form the global algorithm TIGHT-CUT and its relaxed variant TIGHT-CUT*. Empirical evaluation across synthetic and real-world circuit graphs shows TIGHT-CUT* frequently achieves approximation ratios under 1.6 and substantially outperforms the Greedy Removal heuristic, while solving large instances that ILP-based methods struggle with. The results indicate meaningful practical gains for circuit testing and other applications requiring acyclic subgraph extraction, with potential for further speedups through parallelization and optimized min-cut implementations.

Abstract

The classical NP-hard feedback arc set problem (FASP) and feedback vertex set problem (FVSP) ask for a minimum set of arcs or vertices whose removal , makes a given multi-digraph acyclic, respectively. Though both problems are known to be APX-hard, approximation algorithms or proofs of inapproximability are unknown. We propose a new -heuristic for the directed FASP. While a ratio of is known to be a lower bound for the APX-hardness, at least by empirical validation we achieve an approximation of . The most relevant applications, such as circuit testing, ask for solving the FASP on large sparse graphs, which can be done efficiently within tight error bounds due to our approach.

Paper Structure

This paper contains 16 sections, 7 theorems, 13 equations, 9 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical Considerations
  3. Preliminaries
  4. Problem Formulation
  5. Isolated Cycles
  6. The Algorithm
  7. Building Block
  8. A Good Guess
  9. The Global Approach
  10. Validation & Benchmarking
  11. Synthetic Instances
  12. Real-world datasets
  13. Conclusion
  14. Relaxed Version of TIGHT--CUT
  15. The Dualism of the FVSP & FASP
  16. ...and 1 more sections

Key Result

theorem 1

Let $G=(V,E,\mathrm{head},\mathrm{tail})$ be a multi--digraph with arc weight $\omega : E \longrightarrow \mathbb{R}^+$.

Figures (9)

  • Figure 1: Elementary and simple cycles.
  • Figure 2: $G_1$ with colored isolated cycles $\{a,b,c\}$, $\{d,e,f\}$, $\{g,h,i,j,k\}$, $\{g,l,m,i,j,k\}$ and $G_2$.
  • Figure 3: Approximation ratios of TIGHT--CUT* (left) and GR (right) for random graphs.
  • Figure 4: Runtime ratios of TIGHT--CUT* / EM plotted against $|E|/|V|$ (left) and feedback length (right).
  • Figure 5: Runtime ratios of TIGHT--CUT* / EM (left) and distribution of TIGHT--CUT* and GR on weighted digraphs with logarithmic $y$--scale (right).
  • ...and 4 more figures

Theorems & Definitions (19)

  • definition 1
  • Remark 2.1
  • definition 2: cycle cover & isolated cycles
  • Remark 2.2
  • theorem 1
  • lemma 1
  • proposition 1
  • proposition 2
  • definition 3: almost isolated cycles
  • definition 4: directed line graph
  • ...and 9 more