Online Firefighting on Cactus Graphs

It is known that the online firefighting is 2-competitive on trees (Coupechoux et al. 2019), which suggests that the problem is relatively easy on trees. We extend the study to graphs containing cycles. We first show that the presence of cycles gives a strong advantage to the adversary: cycles create situations where the algorithm and the optimal solution operate on different game states, and the adversary can exploit the uncertainty in the firefighter sequence to trap the algorithm. Specifically, we prove that even on a tadpole graph (a cycle with a tail path), no deterministic online algorithm achieves a competitive ratio better than $Ω(\sqrt{n})$, where n is the number of vertices. We then propose an $O(\sqrt{n})$-competitive algorithm for 1-almost trees, which contain at most one cycle and generalize tadpole graphs. We further generalize this algorithm to cactus graphs, in which multiple cycles may appear, but no two share more than one vertex, and show that the online firefighting problem on cactus graphs remains $O(\sqrt{n})$-competitive. Finally, since cactus graphs have treewidth at most 2, we study a variant where firefighters are released in pairs, that is, each round an even number of firefighters is available. Surprisingly, in this setting the competitive complexity is significantly reduced, and we prove that the problem is at most 3-competitive. The main technical challenges lie in both algorithm design and analysis, since the algorithm and the optimal solution may break different cycles and thus operate on different residual graphs. To overcome this difficulty, we design a charging framework that carefully partitions the vertices saved by the optimal solution and charges them to the vertices saved by the algorithm. Namely, the charging scheme is carefully constructed to ensure that each vertex saved by the algorithm is charged at most a constant number of times.

💡 Research Summary

The paper investigates the competitive complexity of the online firefighting problem when the underlying graph contains cycles. While it is known that on trees the problem admits a deterministic 2‑competitive algorithm (Coup­echoux et al., 2019), the authors show that the presence of even a single cycle dramatically increases the difficulty.

First, they prove a lower bound: on a tadpole graph (a cycle with a pendant path) any deterministic online algorithm has competitive ratio at least Ω(√n), where n is the number of vertices. The proof constructs an (α+1, β+1) tadpole, balances the two adversarial cases (protecting a cycle vertex versus a path vertex in the first round), and chooses α≈β² to obtain the √n factor.

To match this bound, the authors design an O(√n)‑competitive algorithm for 1‑almost‑trees (graphs with at most one extra edge, i.e., a single cycle). The key technical tool is the notion of a “covered set” κ(v): the set of vertices that become safe if v is protected before the fire spreads. For an algorithm’s protected set Ψ_A and the optimal solution’s protected set Ψ*, the algorithm’s profit is w(Ψ_A)=|κ(Ψ_A)|. The analysis partitions κ(Ψ*) into independent parts and “charges” each part to a subset of Ψ_A, ensuring that each algorithmic vertex is charged only a constant number of times. By carefully choosing where to break the unique cycle (either to save a large portion of the cycle or to delay the fire along the tail), they achieve a (5√n+2)‑competitive guarantee, which is asymptotically optimal.

The framework is then extended to cactus graphs, where each edge belongs to at most one cycle, so cycles may coexist but intersect in at most a single vertex. The authors apply the same charging idea cycle‑by‑cycle, constructing a priority ordering among cycles to avoid conflicts. The resulting algorithm attains a (20√n+1)‑competitive ratio, again matching the lower bound up to constant factors.

Finally, the paper explores a variant motivated by the treewidth‑2 nature of cactus graphs: each round supplies an even number of firefighters (specifically, two). In this setting, a “pair‑protection” strategy can simultaneously block two edges of a cycle, effectively neutralizing the adversary’s advantage. The authors prove that this modification reduces the competitive ratio to a constant, presenting a 3‑competitive algorithm for cactus graphs with even firefighter arrivals.

Overall, the work demonstrates that cycles break the alignment between online and optimal strategies that exists on trees, necessitating a new charging analysis based on covered sets. It establishes tight Θ(√n) bounds for a broad class of graphs (tadpoles, 1‑almost‑trees, cactus graphs) and shows that additional structural constraints (even firefighter batches) can dramatically improve performance, reducing the ratio to a constant. The techniques introduced—covered‑set partitioning and constant‑charge analysis—are likely to be useful for other online containment problems on sparse graphs.