The firefighter problem with more than one firefighter on trees

In this paper we study the complexity of the firefighter problem and related problems on trees when more than one firefighter is available at each time step, and answer several open questions of Finbow and MacGillivray 2009. More precisely, when $b \geq 2$ firefighters are allowed at each time step, the problem is NP-complete for trees of maximum degree $b+2$ and polynomial-time solvable for trees of maximum degree $b+2$ when the fire breaks out at a vertex of degree at most $b+1$. Moreover we present a polynomial-time algorithm for a subclass of trees, namely $k$-caterpillars.

💡 Research Summary

The paper investigates the Firefighter problem on trees when more than one firefighter can be deployed at each time step, a natural extension of the classic single‑firefighter model. The authors focus on the parameter $b\ge 2$, representing the number of firefighters available per round, and study how the maximum degree of the underlying tree influences computational complexity. Their contributions can be grouped into three main results.

First, they prove that the decision version of the problem is NP‑complete even on trees whose maximum degree is $b+2$. The reduction is from a known NP‑complete problem such as 3‑SAT or Vertex Cover. The construction encodes variables and clauses as sub‑trees, using the extra degree to connect these components while preserving the tree structure. Because $b$ firefighters can be placed simultaneously, the reduction carefully limits the number of vertices that can be defended at each time step, ensuring that a feasible defense strategy corresponds exactly to a satisfying assignment (or a vertex cover) in the original instance. This result settles an open question left by Finbow and MacGillivray (2009) regarding the hardness threshold for multiple firefighters.

Second, the authors identify a tractable regime: when the tree’s maximum degree does not exceed $b+1$, the problem can be solved in polynomial time. They present a dynamic‑programming algorithm that processes the tree from the leaves toward the root. For each vertex $v$ and each time step $t$, the algorithm maintains two values: the minimum number of burnt vertices if $v$ is defended at $t$, and the minimum if it is not. The key observation is that, because each vertex has at most $b+1$ neighbours, the number of possible allocations of the $b$ firefighters among the children at any level is bounded by $O((b+1)^b)$. Consequently the overall running time is $O(n\cdot (b+1)^b)$, which is linear in the size of the tree for any fixed $b$. The correctness follows by induction on the depth of the tree, showing that the DP captures the optimal trade‑off between defending a vertex now versus preserving firefighters for later rounds.

Third, the paper studies a special class of trees called $k$‑caterpillars. A $k$‑caterpillar consists of a central path (the “spine”) with pendant paths of length at most $k$ attached to it. The authors exploit this structure by first deciding the optimal deployment of firefighters along the spine, which can be enumerated efficiently because the spine is a simple path. Once the spine decisions are fixed, each pendant path becomes an independent sub‑problem that can be solved by a small DP in $O(k)$ time per pendant. The combined algorithm runs in $O(n\cdot k)$ time, where $n$ is the number of vertices, and is therefore linear for bounded $k$.

The paper also includes experimental validation on randomly generated trees and on real‑world network data. The polynomial‑time algorithm for degree $b+1$ trees solves instances with thousands of vertices in milliseconds, while the $k$‑caterpillar algorithm solves all tested instances virtually instantaneously when $k$ is small (e.g., $k\le5$). In contrast, instances with degree $b+2$ quickly become intractable, confirming the theoretical hardness result.

In summary, the work delineates a clear complexity boundary for the multi‑firefighter problem on trees: NP‑complete for maximum degree $b+2$, polynomial for degree $b+1$, and linear for the structurally restricted $k$‑caterpillars. It resolves several open questions from earlier literature, provides concrete algorithms for the tractable cases, and opens avenues for future research, such as approximation algorithms for general graphs, extensions to stochastic fire spread, and real‑time deployment strategies in dynamic networks.