Graph Burning Problem (GBP)

• Graph Burning Problem is a discrete-time combinatorial model that abstracts contagion spread in networks by sequentially igniting sources to cover all vertices.
• It features key theoretical insights such as NP-completeness, tight bounds on the burning number, and specialized approximations for various graph classes.
• Methodologies include exact integer programming formulations and heuristic algorithms that ensure scalable performance on massive networks.

The graph burning problem (GBP) is a discrete-time combinatorial model that abstracts the spread of contagion or information through a network, focusing on the minimal number of steps required to fully “burn” (i.e., infect or cover) all vertices by repeatedly activating sources of fire. The core parameter of interest, the burning number, quantifies the minimum contagion speed under constrained parallelism. GBP has become a central subject in graph algorithms, complexity, and approximate optimization, yielding a wide spectrum of theoretical, algorithmic, and empirical insights.

1. Formal Model and Core Definitions

Given a simple, undirected graph $G = (V, E)$ of order %%%%1%%%%, the burning process proceeds in discrete rounds. At time $t = 0$, all vertices are unburned. At each integer time $t \geq 1$:

• Ignition (external activation): A single unburned vertex is selected as a new burning source.
• Contagion spread: Each vertex burned at time $t-1$ ignites all unburned neighbors at time $t$.

The set of burned vertices $B^t$ is recursively defined as: $$B^{0} = \emptyset; \quad B^{t} = B^{t-1} \cup \{ \text{new burning source at } t \} \cup \Big( \bigcup_{v \in B^{t-1}} N(v) \Big)$$ where $N(v)$ denotes the set of neighbors of $v$. The process continues until $B^T = V$.

A burning sequence of length $k$ is an ordered tuple $(x_1, x_2, ..., x_k)$ where $x_i \in V$, specifying the sequence of ignitions. The burning number $b(G)$ is the minimal $k$ for which a burning sequence exists, i.e.,

$$b(G)\;=\;\min\bigl\{\,k\;:\;\bigcup_{i=1}^{k} N_{k-i}[x_i] = V\,\bigr\}$$

where $N_{r}[v]=\{u\in V : \mathrm{dist}_G(u, v) \le r\}$.

Burning Number Conjecture: For every connected graph $G$ of order $n$, $b(G) \le \lceil \sqrt{n} \rceil$ (Antony et al., 2024). This conjecture is tight for paths and remains open for general graphs.

2. Hardness and Structural Results

2.1 NP-Completeness and Parameterized Complexity

The decision variant—given $(G, k)$, is $b(G) \le k$—is NP-complete for general graphs and remains so even when restricted to:

• Trees of maximum degree 3 (Bonato et al., 2015)
• Path-forests and spider graphs (Bonato et al., 2015)
• Interval graphs (Gupta, 2020, Antony et al., 2024)
• Connected cubic graphs (Antony et al., 2024)
• Proper interval graphs (Antony et al., 2024)
• Caterpillars of degree 3 (Kobayashi et al., 2020)

From the parameterized viewpoint:

• GBP is W[2]-complete when parameterized by $k$ (the burning number itself), via a reduction from SET COVER (Kobayashi et al., 2020).
• Neither vertex cover nor treewidth kernels allow polynomial-size reductions unless standard complexity collapses occur.
• GBP is fixed-parameter tractable (FPT) parameterized by clique-width plus maximum component diameter, and consequently by modular-width, treedepth, and distance to cographs. For distance to split graphs, bespoke branching and set-cover arguments yield FPT algorithms (Kobayashi et al., 2020).

2.2 Classical and Improved Bounds

Burning number admits the following upper and lower bounds:

• For general connected $G$ of order $n$ and diameter $D$, $b(G) \leq D + 1$ (trivial), and $b(G) \leq 2\lceil\sqrt{n}\rceil - 1$ (Antony et al., 2024).
• For paths and cycles, $b(P_n) = b(C_n) = \lceil\sqrt{n}\rceil$.
• For connected $P_k$-free graphs, $b(G) \le \lceil\frac{k+1}{2}\rceil$ (Antony et al., 2024).
• For minimum degree $\delta \ge 1$, $b(G) \le (1+o(1))\sqrt{3n/(\delta+1)}$ (Martinsson, 2023), and for $\delta \ge 23$, $b(G) \leq \lceil\sqrt{n}\rceil$ (Kamali et al., 2019).

Variants such as edge burning ($b_L(G) = b(L(G))$) and total burning ($b_T(G) = b(T(G))$) satisfy: $$b(G) - 1 \leq b_L(G) \leq b(G) + 1, \quad b(G) \leq b_T(G) \leq b(G) + 1$$ with NP-completeness persisting on their respective graph classes (Antony et al., 2024).

3. Exact and Approximate Algorithms

3.1 Exact Algorithms and Mathematical Programming

Recent advances have produced exact algorithms capable of scaling to graphs with millions of vertices:

• A set covering integer-programming (IP) model forms the basis: for given $B$, introduce binary variables $x_{v,i}$ indicating if $v$ is ignited as source in round $i$, imposing one-per-round, no-repeats, and coverage constraints (Pereira et al., 2024, Cajica-Maceda et al., 23 Nov 2025).
• Row generation (cutting-plane) techniques add coverage constraints “on demand.” This reduces the number of actively enforced constraints to $O(|V|)$ for practical instances, dramatically improving memory and runtime (Pereira et al., 2024).
• Empirically, PRYM can optimally burn graphs with up to $200\,000$ vertices in $\le 35$ seconds, and up to $4$ million vertices in several minutes (Pereira et al., 2024, Cajica-Maceda et al., 23 Nov 2025).

Mathematical Programs:

A recent taxonomy (Cajica-Maceda et al., 23 Nov 2025) distinguishes:

• PROP-MILP: Direct propagation-based MILP.
• COV-CSP and COV-ILP: Compact coverage-based CSP/ILP via clustered maximum coverage reduction.
• GBP-ILP: Coverage-based ILP with monotonicity constraints and direct minimization.
• sQUBO/uQUBO: Quadratic unconstrained binary optimization, facilitating application to classical/quantum annealing.

Each varies in variable count, conceptual simplicity, and suitability for large-scale or quantum optimization (see table below).

Formulation	Variables	Constraints	Advantage
PROP-MILP	$2Un+1$	$U n + U + n$	Explicit simulation
COV-CSP	$g n$	$g + n$	Smallest, compact
GBP-ILP	$U n$	$(U-1) + n$	No binary search
sQUBO	$g n + n \log_2 g$	—	No penalty tuning
uQUBO	$g n$	—	Quantum-ready

4. Approximation and Heuristic Methods

4.1 Approximation Algorithms

• General graphs: A $3$-approximation exists by repeatedly choosing farthest unburned vertices (BFF algorithm) (Díaz et al., 2020). For $b(G)>2$, the approximation ratio is $(3-2/b(G))$.
• Trees: The best provable factor is $2$ (Gautam et al., 2023), with a recently improved $1.75$-approximation for trees (Gautam et al., 2022).
• Cactus graphs: Admit a $2.75$-approximation (Gautam et al., 2023).
• Multi-rooted directed trees (polytrees): $3$-approximation; arborescences (single-rooted) achieve $1.905$-approximation (Gautam et al., 2023).

Conjecturally, it is NP-hard to approximate $b(G)$ to within any factor less than $3$ for general graphs (Gupta, 2020).

4.2 Heuristics and Metaheuristics

• Centrality-based heuristics: Eigenvector and betweenness centralities seed the selection of burning sources, with practical variants including the Backbone Based Greedy (BBGH), Improved Cutting Corners (ICCH), and the Component Based Recursive Heuristic (CBRH) (Gautam et al., 2020).
• Greedy Set-Cover Reductions: The problem reduces to a Clustered Maximum Coverage Problem (CMCP), with a simple greedy yielding a half-approximation for coverage and near-optimal burning numbers in real-world graphs (García-Díaz et al., 2024).
• Evolutionary methods: The Centrality-Based Genetic Algorithm for Graph Burning (CBAG) uses centrality-driven chromosome initialization and problem-specific crossover/mutation (Nazeri et al., 2022).
• Farthest-First Traversal (BFF): Generalizes the Gonzalez metric $k$-center algorithm, providing a practical, fast, and empirically near-optimal approximation in real graphs (Díaz et al., 2020). Empirical evaluations show that these heuristics often closely match the best-known or optimal solutions on benchmarks of up to tens of thousands of nodes—typically differing by one step or less on practical instances (Díaz et al., 2020, Nazeri et al., 2022, Gautam et al., 2020, García-Díaz et al., 2024).

5. GBP in Special Graph Classes and Variants

5.1 Trees

For trees, tight structural bounds and special algorithms exist:

• If $n_2$ is the count of degree-2 vertices, $b(T) \le \lceil \sqrt{n + n_2 + 8} \rceil - 1$ (Das et al., 2023).
• For perfect and complete binary trees, the burning number matches the height or height plus one.
• Recursive algorithms exist for full binary trees not perfect, achieving $b(T) \le \lceil\sqrt{n+9}\rceil - 1$ (Das et al., 2023).
• $2$-approximation algorithms are standard (Gautam et al., 2023) with improved factors for trees in (Gautam et al., 2022).

5.2 Geometric and Random Graphs

• Interval, permutation, and disk graphs: Burning number is NP-complete even with explicit representations (Gupta, 2020).
• Random graphs $\mathcal{G}(n,p)$: Burning number is typically the graph diameter; in the supercritical regime, $b(G) = D(G)$ or $D(G)+1$ with high probability (Mitsche et al., 2015).
• Random geometric graphs: $b(G) = \Theta(r^{-2/3})$ for connectivity radius $r \gg r_c$.
• Grids: For large $m \times n$ grids, $b(P_m \square P_n) = (3/2)^{1/3}(mn)^{1/3} + o(1)$ (Mitsche et al., 2015).

5.3 Probabilistic and Drunken Variants

Variants allowing probabilistic or randomized selection of burning sources demonstrate the cost of suboptimal strategies:

• “Drunken” random activations incur a $\sqrt{\log n}$–factor penalty on paths, but if selections are restricted to unburned vertices, the penalty becomes constant-factor (Mitsche et al., 2015).

6. Future Directions and Open Problems

Key unresolved topics and emerging problems:

• Tightening Approximability: Whether the $3$-approximation lower bound is tight for general graphs or can be formally improved remains unresolved (Gupta, 2020).
• Special Graph Families: Characterizing the exact burning number for planar, chordal, and minor-free graphs, or for graphs parameterized by deletion to special classes (e.g., cographs) (Antony et al., 2024, Kobayashi et al., 2020).
• Variants: Algorithmic and parameterized complexity for edge and total burning, both of which are only partially understood (Antony et al., 2024).
• Scaling to Massive Networks: Continued development of mathematical programming and decomposition algorithms for graphs with $n>10^6$.
• Quantum Optimization: Evaluation of QUBO-based encodings for quantum annealing and QAOA solvers (Cajica-Maceda et al., 23 Nov 2025).
• Probabilistic and temporal models: Burning with imperfect information, on time-varying or weighted graphs.

The burning number remains a focal graph invariant for diffusion processes, entangling combinatorial, geometric, and computational complexity aspects. Progress in its analysis and computation continues to inform understanding of worst-case contagion, resource allocation, and algorithmic containment in complex networks.