[Paper Review] On the approximability of the burning number

Anders Martinsson|arXiv (Cornell University)|Aug 8, 2023

Complexity and Algorithms in GraphsComputer Science3 citations

TL;DR

This paper presents a polynomial-time randomized algorithm that achieves a $\left(\frac{2}{1-e^{-2}} + \varepsilon\right) \approx 2.313$-approximation for the burning number of general graphs, improving upon the prior best factor of 3. It also establishes a Polynomial-Time Approximation Scheme (PTAS) for trees and forests, and proves that approximating the burning number within a factor of $\frac{5}{3} - \varepsilon$ is NP-hard, establishing tight bounds on approximability.

ABSTRACT

The burning number of a graph $G$ is the smallest number $b$ such that the vertices of $G$ can be covered by balls of radii $0, 1, \dots, b-1$. As computing the burning number of a graph is known to be NP-hard, even on trees, it is natural to consider polynomial time approximation algorithms for the quantity. The best known approximation factor in the literature is $3$ for general graphs and $2$ for trees. In this note we give a $2/(1-e^{-2})+\varepsilon=2.313\dots$-approximation algorithm for the burning number of general graphs, and a PTAS for the burning number of trees and forests. Moreover, we show that computing a $(\frac53-\varepsilon)$-approximation of the burning number of a general graph $G$ is NP-hard.

Motivation & Objective

• To improve the best-known approximation factor for the burning number of general graphs beyond the prior 3-approximation.
• To develop a Polynomial-Time Approximation Scheme (PTAS) for the burning number of trees and forests.
• To establish tight lower bounds on the approximability of the burning number, showing that $\frac{5}{3} - \varepsilon$ approximation is NP-hard.
• To bridge the gap between known approximation algorithms and the limits of efficient computation for the burning number problem.

Proposed method

• A randomized approximation algorithm is designed that uses a greedy-like strategy with probabilistic analysis to achieve a $\left(\frac{2}{1-e^{-2}} + \varepsilon\right)$-approximation for general graphs.
• The algorithm leverages the fact that any cover of a graph can be represented as a vector of radii with polynomial size when restricted to $O(1/\varepsilon)$ distinct radii, enabling dynamic programming.
• For trees and forests, a dynamic programming approach is used where each vertex maintains sets $\mathcal{C}_k(v)$ representing partial covers of its subtree, with $k$ indicating the depth of excluded components.
• The algorithm computes $\mathcal{C}_0(v)$ recursively by considering all possible centers $v_0$ in the subtree and radii $r$, combining results from child subtrees using sumsets of cover representations.
• The method uses union and sumset operations on $O(1/\varepsilon)$-dimensional vectors of non-negative integers, ensuring polynomial-time execution.
• A reduction from the domination number problem is used to prove NP-hardness of $\left(\frac{5}{3} - \varepsilon\right)$-approximation, by constructing a graph $G'$ whose burning number reflects the domination number of the original graph $G$.

Experimental results

Research questions

• RQ1 Can the approximation factor for the burning number of general graphs be improved beyond the known 3-approximation?
• RQ2 Does a PTAS exist for the burning number of trees and forests?
• RQ3 What is the best possible approximation factor for the burning number of general graphs, assuming P ≠ NP?
• RQ4 Is there a constant $\alpha < 3$ such that $\alpha$-approximation is still NP-hard?

Key findings

• The paper presents a randomized polynomial-time algorithm that achieves a $\left(\frac{2}{1-e^{-2}} + \varepsilon\right) \approx 2.313$-approximation for the burning number of general graphs, with high probability.
• A PTAS is developed for the burning number of trees and forests, enabling arbitrarily close approximation to the optimal value in polynomial time.
• It is proven that approximating the burning number within a factor of $\frac{5}{3} - \varepsilon$ is NP-hard for any $\varepsilon > 0$, establishing a strong inapproximability threshold.
• The inapproximability result is shown via a reduction from the domination number problem, constructing a graph $G'$ such that $b(G') < (3+\varepsilon)d$ if $\gamma(G) < \varepsilon d$, and $b(G') \geq 5d$ if $\gamma(G) \geq 10d$.
• The dynamic programming approach for trees uses $O(1/\varepsilon)$-dimensional vectors to represent covers, with polynomial-size sets, enabling efficient computation of the minimal burning number within additive error $a$.
• The framework allows for a fully polynomial-time $\left(\frac{2}{1-e^{-2}} + \varepsilon\right)$-approximation by combining the randomized algorithm with brute-force search for small burning numbers.