Sampling with a Black Box: Faster Parameterized Approximation Algorithms for Vertex Deletion Problems

TL;DR

Sampling with a Black Box presents a modular framework for parameterized approximations to vertex deletion problems by coupling a polynomial-time sampling step with a fast parameterized algorithm. The core idea uses a probabilistic reduction of OPT via iterative vertex removals and a subsequent extension with a black-box algorithm, with a refined analysis based on tail bounds and KL-divergence to optimize parameters. The paper demonstrates improved running times for several problems (e.g., FVS, POVD, and finite-forbidden vertex deletions) over prior methods, and provides general tools to convert sampling steps into approximation guarantees. This technique broadens the applicability of parameterized approximations and offers practical speedups for a wide class of deletion problems, with potential extensions to weighted settings and directed variants.

Abstract

In this paper we introduce Sampling with a Black Box, a generic technique for the design of parameterized approximation algorithms for vertex deletion problems (e.g., Vertex Cover, Feedback Vertex Set, etc.). The technique relies on two components: $\bullet$ A Sampling Step. A polynomial time randomized algorithm which given a graph $G$ returns a random vertex $v$ such that the optimum of $G\setminus \{v\}$ is smaller by $1$ than the optimum of $G$ with some prescribed probability $q$. We show such algorithms exists for multiple vertex deletion problems. $\bullet$ A Black Box algorithm which is either an exact parameterized algorithm or a polynomial time approximation algorithm. Our technique combines these two components together. The sampling step is applied iteratively to remove vertices from the input graph, and then the solution is extended using the black box algorithm. The process is repeated sufficiently many times so that the target approximation ratio is attained with a constant probability. The main novelty of our work lies in the analysis of the framework and the optimization of the parameters it uses. We use the technique to derive parameterized approximation algorithm for several vertex deletion problems, including Feedback Vertex Set, $d$-Hitting Set and $\ell$-Path Vertex Cover. In particular, for every approximation ratio $1<β<2$, we attain a parameterized $β$-approximation for Feedback Vertex Set which is faster than the parameterized $β$-approximation of [Jana, Lokshtanov, Mandal, Rai and Saurabh, MFCS 23']. Furthermore, our algorithms are always faster than the algorithms attained using Fidelity Preserving Transformations [Fellows, Kulik, Rosamond, and Shachnai, JCSS 18'].

Figures (8)

• Figure 3: A plot of the running time of our algorithm for Pathwidth One Vertex Deletion. The $x$-axis corresponds to the approximation ratio, while the $y$-axis corresponds to the base of the exponent in the running time. A point $(\beta, d)$ in the plot describes a running time of the form $d^{k} \cdot n^{\mathcal{O}(1)}$ for a $\beta$-approximation.
• Figure 4: Comparison of the functions $m_{\delta, q}(x)$ with varying $\delta$ values (blue, green, and cyan), alongside the function $\ln\left( \frac{1}{\phi(x,q)}\right)$ (red). Recall that $m_{\delta, q}(x)$ is a linear function of $x$ and $m_{ \delta, q}(\beta) = h_{ q}(\delta)$. Also observe that the functions $\ln\left( \frac{1}{\phi(x,q)}\right)$ and $m_{ \delta, q}(x)$ meet at $\delta$.
• Figure 5: The plot of the function $\psi(x) = \mathcal{D}\left(\frac{1}{\alpha}\, \middle\|\,x \right)$ and $y = \mathcal{D}\left(\frac{1}{\alpha}\, \middle\|\,q \right) - \frac{\ln(c)}{\alpha}$ for $\alpha = 1.2, q = 0.5$ and $c = 1.1$. Note that $\psi(x)$ has a zero at $\frac{1}{\alpha}$ and it is monotone in intervals $(1,\frac{1}{\alpha}]$ and $[\frac{1}{\alpha},1)$.
• Figure 6: The graph $T_2$
• Figure 7: Comparison of the running times of various algorithms for $3$-Hitting Set. The $x$-axis corresponds to the approximation ratio, while the $y$-axis corresponds to the base of the exponent in the running time. A point $(\beta, d)$ in the plot describes a running time of the form $d^{k} \cdot n^{\mathcal{O}(1)}$ for a $\beta$-approximation. The red point corresponds to the 2-approximation algorithm from brankovicParameterizedApproximationAlgorithms2012, with a running time of $1.29^{k} \cdot n^{\mathcal{O}(1)}$. Even though our result outperforms brankovicParameterizedApproximationAlgorithms2012 and Fellows2018, it only improves upon KulikS2020 for values of $\beta$ such that $\beta \lessapprox 1.16$.

Theorems & Definitions (115)

• Definition 3.1: Parameterized Approximation
• Definition 3.2: Sampling Step
• Lemma 3.3
• Lemma 3.3
• Theorem 3.4: Sampling with a Black-Box
• Theorem 3.5: simple formula for $\alpha=1$
• Theorem 3.6: simple formula for $\alpha =2$ and $c=1$
• Definition 3.7
• Lemma 3.8: Fellows2018
• Lemma 3.9