Multilevel programming can provide the right mathematical formulations for modeling sequential decision-making problems. In such cases, it is implicit that each level anticipates the optimal reaction of the subsequent ones. Defender–attacker–defender trilevel programs are a particular case of interest that encompasses a fortification strategy, followed by an attack, and a consequent recovery defensive strategy. In “Multilevel Approaches for the Critical Node Problem,” Baggio, Carvalho, Lodi, and Tramontani study a combinatorial sequential game between a defender and an attacker that takes place in a network. The authors propose an exact algorithmic framework. This work highlights the significant improvements that the defender can achieve by taking the three-stage game into account instead of considering fortification and recovery as isolated. Simultaneously, the paper contributes to advancing the methodologies for solving trilevel programs.
The Firefighter problem and a variant of it, known as Resource Minimization for Fire Containment (RMFC), are natural models for optimal inhibition of harmful spreading processes. Despite considerable progress on several fronts, the approximability of these problems is still badly understood. This is the case even when the underlying graph is a tree, which is one of the most-studied graph structures in this context and the focus of this article. In their simplest version, a fire spreads from one fixed vertex step by step from burning to adjacent non-burning vertices, and at each time step B many non-burning vertices can be protected from catching fire. The Firefighter problem asks, for a given B , to maximize the number of vertices that will not catch fire, whereas RMFC (on a tree) asks to find the smallest B that allows for saving all leaves of the tree. Prior to this work, the best known approximation ratios were an O (1)-approximation for the Firefighter problem and an O (log * n )-approximation for RMFC, both being LP-based and essentially matching the integrality gaps of two natural LP relaxations. We improve on both approximations by presenting a PTAS for the Firefighter problem and an O (1)-approximation for RMFC, both qualitatively matching the known hardness results. Our results are obtained through a combination of the known LPs with several new techniques, which allow for efficiently enumerating over super-constant size sets of constraints to strengthen the natural LPs.
The Firefighter problem and a variant of it, known as Resource Minimization for Fire Containment (RMFC), are natural models for optimal inhibition of harmful spreading processes. Despite considerable progress on several fronts, the approximability of these problems is still badly understood. This is the case even when the underlying graph is a tree, which is one of the moststudied graph structures in this context and the focus of this paper. In their simplest version, a fire spreads from one fixed vertex step by step from burning to adjacent non-burning vertices, and at each time step B many non-burning vertices can be protected from catching fire. The Firefighter problem asks, for a given B, to maximize the number of vertices that will not catch fire, whereas RMFC (on a tree) asks to find the smallest B that allows for saving all leaves of the tree. Prior to this work, the best known approximation ratios were an O(1)-approximation for the Firefighter problem and an O(log * n)-approximation for RMFC, both being LP-based and essentially matching the integrality gaps of two natural LP relaxations.We improve on both approximations by presenting a PTAS for the Firefighter problem and an O(1)-approximation for RMFC, both qualitatively matching the known hardness results. Our results are obtained through a combination of the known LPs with several new techniques, which allow for efficiently enumerating over super-constant size sets of constraints to strengthen the natural LPs.
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