Easy problems for tree-decomposable graphs
Journal of Algorithms
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
Infeasibility of instance compression and succinct PCPs for NP
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
On problems without polynomial kernels
Journal of Computer and System Sciences
Parameterized complexity of the firefighter problem
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Towards more efficient infection and fire fighting
CATS '11 Proceedings of the Seventeenth Computing: The Australasian Theory Symposium - Volume 119
Parameterized Complexity
Parameterized complexity of the firefighter problem
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Making life easier for firefighters
FUN'12 Proceedings of the 6th international conference on Fun with Algorithms
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The Firefighter problem is to place firefighters on the vertices of a graph to prevent a fire with known starting point from lighting up the entire graph. In each time step, a firefighter may be permanently placed on an unburned vertex and the fire spreads to its neighborhood in the graph in so far no firefighters are protecting those vertices. The goal is to let as few vertices burn as possible. This problem is known to be NP-complete, even when restricted to bipartite graphs or to trees of maximum degree three. Initial study showed the Firefighter problem to be fixed-parameter tractable on trees in various parameterizations. We complete these results by showing that the problem is in FPT on general graphs when parameterized by the number of burned vertices, but has no polynomial kernel on trees, resolving an open problem. Conversely, we show that the problem is W[1]-hard when parameterized by the number of unburned vertices, even on bipartite graphs. For both parameterizations, we additionally give refined algorithms on trees, improving on the running times of the known algorithms.