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.
Modeling a spreading process in a network is a widely studied topic and often relies on a graph theoretical approach (see [4,6,8,12,15,16]). Such processes occur for instance in epidemiology and social sciences. Indeed, the spreading process could be the spread of an infectious disease in a population or the spread of opinions through a social network. Different objectives may then be of interest, for instance minimizing the total number of infected persons by vaccinating at each time step some particular individuals, or making sure that some specific subset of individuals does not get infected at all, etc... The spreading process may also represent the spread of a fire. The associated firefighter problem, introduced in [9], has been studied intensively in the literature (see for instance [2,3,5,7,8,9,10,11,13,14,15]). In this paper, we consider some generalizations and variants of this problem which is defined as follows. Initially, a fire breaks out at some special vertex s of a graph. At each time step, we have to choose one vertex which will be protected by a firefighter. Then the fire spreads to all unprotected neighbors of the vertices on fire. The process ends when the fire can no longer spread, and then all vertices that are not on fire are considered as saved. The objective consists of choosing, at each time step, a vertex which will be protected by a firefighter such that a maximum number of vertices in the graph is saved at the end of the process.
The firefighter problem was proved to be NP-hard for bipartite graphs [14]. Much stronger results were proved later [7] implying a dichotomy: the firefighter problem is NP-hard even for trees of maximum degree three and it is solvable in polynomial-time for graphs with maximum degree three, provided that the fire breaks out at a vertex of degree at most two. Moreover, the firefighter problem is NP-hard for cubic graphs [13]. From the approximation point of view, the firefighter problem is e e-1 -approximable on trees [3] and it is not n 1-ε -approximable on general graphs for any ǫ ∈ (0, 1) [2], if P = NP. Moreover for trees where vertices have at most three children, the firefighter problem is 1.3997-approximable [11]. Finally, the firefighter problem is polynomial-time solvable for caterpillars and P-trees [14].
A problem related to the firefighter problem, denoted by S-Fire, was introduced in [13]. It consists of deciding if there is a strategy of choosing a vertex to be protected at each time step such that all vertices of a given set S are saved. S-Fire was proved to be NP-complete for trees of maximum degree three in which every leaf is at the same distance from the vertex where the fire starts and S is the set of leaves.
In this paper, we consider a generalized version of S-Fire. We denote by b-Save, where b ≥ 1 is an integer, the problem which consists of deciding if we can choose at most b vertices to be protected by firefighters at each time step and save all the vertices from a given set S. Thus, S-Fire is equivalent to 1-Save. The optimization version of b-Save will be denoted by Max b-Save. This problem consists of choosing at most b vertices to be protected at each time step and saving as many vertices as possible from a given set S. Hence, Max 1-Save corresponds to the firefighter problem when S is the set of all vertices of the graph. Max b-Save is known to be 2-approximable for trees when S is the set of all vertices [10].
A survey on the firefighter problem and related problems can be found in [8]. In this survey, the authors presented a list of open problems. Here, we will answer three of these open questions (questions 2, 4, and 8).
The first question asks for finding algorithms and complexity results of b-Save when b ≥ 2. We show that for any fixed b ≥ 2, b-Save is NP-complete for trees of maximum degree b + 2 when S is the set of all leaves. Moreover, we show that for any fixed b ≥ 2, Max b-Save is NP-hard for trees of maximum degree b + 3 when S is the set of all vertices. Finally, we show that for any b ≥ 1, b-Save is polynomial-time solvable for trees of maximum degree b + 2 when the fire breaks out at a vertex of degree at most b + 1.
The second question asks if there exists a constant c > 1 such that the greedy strategy of protecting, at each time step, a vertex of highest degree adjacent to a burning vertex gives a polynomial-time c-approximation for the firefighter problem on trees. We give a negative answer to this question.
Finally, the third question asks for finding classes of trees for which the firefighter problem can be solved in polynomial time. We present a polynomial-time algorithm to solve Max b-Save, b ≥ 1, in k-caterpillars a subclass of trees.
Our paper is organized as follows. Definitions, terminology and preliminaries are given in Section 2. In Section 3 we establish a dichotomy on the complexity of b-Save and show that the greedy strategy mentioned above gives no approximation guarantee. In Section 4
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