Online Firefighting on Cactus Graphs

TL;DR

This paper extends online firefighting from trees to cactus-like graphs, establishing fundamental limits and algorithmic strategies. It proves an $\Omega(\sqrt{n})$ lower bound on tadpole graphs, highlighting the adversary’s advantage due to cycles, and then gives $O(\sqrt{n})$-competitive algorithms for $1$-almost-trees and for cactus graphs (ALG_A and ALG_C). It introduces a robust charging framework and graph-reduction techniques to align online decisions with the offline optimum, even when residual graphs diverge due to cycle-breaking. In a special case where firefighters arrive in even numbers, a simpler greedy strategy (ALG_E) attains a constant $3$-competitive ratio. Collectively, the results significantly deepen our understanding of online containment in cyclic graphs and provide near-optimal guarantees across graph classes of bounded cycle structure.

Abstract

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.

Figures (7)

• Figure 1: An $(\alpha +1, \beta + 1)$-tadpole graph.
• Figure 2: Examples of covered sets.
• Figure 3: Graph reduction. The left-hand figure shows the graph at the beginning of the round, with the orange vertex as the fire source. The middle figure illustrates the spread of the fire to unprotected vertices by the end of the round where the blue vertices $a$ and $b$ are protected. The right-hand figure shows the graph after the graph reduction.
• Figure 4: The distances from $v, v^\prime$ to $\tilde{r}$ in $T(\widetilde{C}\backslash u_h)$ compared to $T(\widetilde{C}\backslash u_1)$.
• Figure 5: Delay. The fire can be delayed by protecting a cycle vertex ($a$) on the unique shortest path from a vertex to the root. Protecting a vertex $b$ on a not (unique) shortest path does not affect the distance to the root.

Theorems & Definitions (23)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• Lemma 6
• Lemma 7: Exclusive weight as a lower bound
• Lemma 9: The advantage of breaking cycle on $N(\tilde{r}) \cap C$
• Lemma 10: Quality of Break-cycle
• Lemma 12: Cycle vertices in $\Psi^*$