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A Constant Factor Approximation for Directed Feedback Vertex Set in Graphs of Bounded Genus

Hao Sun

TL;DR

This work proves a constant-factor approximation for the directed feedback vertex set problem on graphs embedded on surfaces of bounded genus. The main approach combines a primal–dual scheme for hitting facial dicycles with a genus-reduction separator that removes a constant-factor of the LP optimum and recurses on smaller-genus instances, complemented by an inductive LP-gap analysis. The paper develops topological tools (left/right separation, tubular neighbourhoods, and surgery) to justify the genus-reducing decompositions and to bound the overall approximation factor as a function of the genus $g$. The results extend constant-factor approximations known for planar graphs to bounded-genus graphs, with implications for the related dicycle-packing LP gap and DFVS-size bounds in terms of $g$.

Abstract

The minimum directed feedback vertex set problem consists in finding the minimum set of vertices that should be removed in order to make a directed graph acyclic. This is a well-known NP-hard optimization problem with applications in various fields, such as VLSI chip design, bioinformatics and transaction processing deadlock prevention and node-weighted network design. We show a constant factor approximation for the directed feedback vertex set problem in graphs of bounded genus.

A Constant Factor Approximation for Directed Feedback Vertex Set in Graphs of Bounded Genus

TL;DR

This work proves a constant-factor approximation for the directed feedback vertex set problem on graphs embedded on surfaces of bounded genus. The main approach combines a primal–dual scheme for hitting facial dicycles with a genus-reduction separator that removes a constant-factor of the LP optimum and recurses on smaller-genus instances, complemented by an inductive LP-gap analysis. The paper develops topological tools (left/right separation, tubular neighbourhoods, and surgery) to justify the genus-reducing decompositions and to bound the overall approximation factor as a function of the genus . The results extend constant-factor approximations known for planar graphs to bounded-genus graphs, with implications for the related dicycle-packing LP gap and DFVS-size bounds in terms of .

Abstract

The minimum directed feedback vertex set problem consists in finding the minimum set of vertices that should be removed in order to make a directed graph acyclic. This is a well-known NP-hard optimization problem with applications in various fields, such as VLSI chip design, bioinformatics and transaction processing deadlock prevention and node-weighted network design. We show a constant factor approximation for the directed feedback vertex set problem in graphs of bounded genus.
Paper Structure (6 sections, 24 theorems, 9 equations, 3 figures, 1 algorithm)

This paper contains 6 sections, 24 theorems, 9 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our techniques
  3. Preliminaries
  4. Hitting the facial cycles of a digraph
  5. Solving the case of no facial cycles
  6. Statement and proofs of topological results we use

Key Result

Theorem 1

For any fixed genus $g$, there is a polynomial-time $O(g)$-approximation for DFVS for graphs of genus $g$. Moreover, the algorithm returns a DFVS with cost $O(g)$ times the optimum solution to DFVS LP.

Figures (3)

  • Figure 1: $L$ and $R$ from \ref{['LRside']} in yellow and red respectively curve $C'$ depicted in black. The curve $h$ leaving $C'$ from the left and entering from the right is depicted in dark green. The closed curve formed by $h$ and the subcurve of $C'$ between $h(0)$ and $h(1)$ depicted in light green forms a non-facial closed curve.
  • Figure 2: Nodes of $U_{\tau_+}$ and $V_{\tau_-}$ shown in blue.
  • Figure 3: On the left, there are $u_{1,1}$-${v_2}$ and $u_{2,1}$-${v_3}$ paths (green and blue vertices respectively) of weight at most $\frac{1}{12} N$ and $s$-$q$ path of length at most $\frac{1}{12} N$. The red cycle would then have weight at most $N$, which is a contradiction. On the right are the sets $R_i$, vertices at distance $i$ from $P_2 \cup C^1_{( v_{j+1}, v_b)}$.

Theorems & Definitions (50)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • Definition 1
  • Lemma 1
  • Definition 2
  • Proposition 4
  • proof
  • Lemma 2
  • ...and 40 more