The firefighter problem models the situation where an infection, a computer virus, an idea or fire etc. is spreading through a network and the goal is to save as many as possible nodes of the network through targeted ...
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The firefighter problem models the situation where an infection, a computer virus, an idea or fire etc. is spreading through a network and the goal is to save as many as possible nodes of the network through targeted vaccinations. The number of nodes that can be vaccinated at a single time-step is typically one, or more generally O(1). In a nonstandard model, the so called spreading model, the vaccinations also spread in contrast to the standard model. Our main results are concerned with general graphs in the spreading model. We provide a very simple exact 2(O(root n log n))-time algorithm. In the special case of trees, where the standard and spreading model are equivalent, our algorithm is substantially simpler than that exact subexponential algorithm for trees presented in Ref. 2. On the other hand, we show that the firefighter problem on weighted directed graphs in the spreading model cannot be approximated within a constant factor better than 1 - 1/e unless NP subset of DTIME (n(O(log log n))) We also present several results in the standard model. We provide approximation algorithms for planar graphs in case when at least two vaccinations can be performed at a time-step. We also derive trade-offs between approximation factors for polynomial-time solutions and the time complexity of exact or nearly exact solutions for instances of the fireifighter problem for the so called directed layered graphs.
We study exact algorithms for EUCLIDEAN TSP in R-d. In the early 1990s algorithms with n(O)(root n) running time were presented for the planar case, and some years later an algorithm with n(O)(n(1-1/d)) running time w...
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ISBN:
(纸本)9781538642306
We study exact algorithms for EUCLIDEAN TSP in R-d. In the early 1990s algorithms with n(O)(root n) running time were presented for the planar case, and some years later an algorithm with n(O)(n(1-1/d)) running time was presented for any d >= 2. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on EUCLIDEAN TSP, except for a lower bound stating that the problem admits no 2(O)(n(1-1/d-epsilon)) algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of EUCLIDEAN TSP by giving a 2(O(n1-1/d)) algorithm and by showing that a 2(o(n1-1/d)) algorithm does not exist unless ETH fails.
We introduce a general notion of miniaturization of a problem that comprises the different miniaturizations of concrete problems considered so far. We develop parts of the basic theory of miniaturizations. Using the a...
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ISBN:
(纸本)9783540230717
We introduce a general notion of miniaturization of a problem that comprises the different miniaturizations of concrete problems considered so far. We develop parts of the basic theory of miniaturizations. Using the appropriate logical formalism, we show that the miniaturization of a definable problem in W[t] lies in W[t], too. In particular, the miniaturization of the dominating set problem is in W[2]. Furthermore, we investigate the relation between f (k) center dot n(o(k)) time and subexponential time algorithms for the dominating set problem and for the clique problem. (c) 2005 Elsevier B.V. All rights reserved.
We give an algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to a wide range of geometric intersection g...
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ISBN:
(纸本)9781450355599
We give an algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to a wide range of geometric intersection graphs (intersections of similarly sized fat objects), yielding algorithms with running time 2(O(n1-1/d)) for any fixed dimension d >= 2 for many well known graph problems, including Independent Set, r -Dominating Set for constant r, and Steiner Tree. For most problems, we get improved running times compared to prior work;in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms work on the graph itself, i. e., do not require any geometric information. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques. The lower bound framework is based on a constructive embedding of graphs into d-dimensional grids, and it allows us to derive matching 2(Omega(n1-1/d)) lower bounds under the Exponential Time Hypothesis even in the much more restricted class of d-dimensional induced grid graphs.
The firefighter problem models the situation where an infection, a computer virus, an idea or fire etc. is spreading through a network and the goal is to save as many as possible nodes of the network through targeted ...
详细信息
ISBN:
(纸本)9781920682989
The firefighter problem models the situation where an infection, a computer virus, an idea or fire etc. is spreading through a network and the goal is to save as many as possible nodes of the network through targeted vaccinations. The number of nodes that can be vaccinated at a single time-step is typically one, or more generally O(1). In a nonstandard model, the so called spreading model, the vaccinations also spread in contrast to the standard model. Our main results are concerned with general graphs in the spreading model. We provide a very simple exact 2(O(root n log n))-time algorithm. In the special case of trees, where the standard and spreading model are equivalent, our algorithm is substantially simpler than that exact subexponential algorithm for trees presented in Ref. 2. On the other hand, we show that the firefighter problem on weighted directed graphs in the spreading model cannot be approximated within a constant factor better than 1 - 1/e unless NP subset of DTIME (n(O(log log n))) We also present several results in the standard model. We provide approximation algorithms for planar graphs in case when at least two vaccinations can be performed at a time-step. We also derive trade-offs between approximation factors for polynomial-time solutions and the time complexity of exact or nearly exact solutions for instances of the fireifighter problem for the so called directed layered graphs.
In the INTERVAL COMPLETION problem we are given an n-vertex graph G and an integer k, and the task is to transform G by making use of at most k edge additions into an interval graph. This is a fundamental graph modifi...
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In the INTERVAL COMPLETION problem we are given an n-vertex graph G and an integer k, and the task is to transform G by making use of at most k edge additions into an interval graph. This is a fundamental graph modification problem with applications in sparse matrix multiplication and molecular biology. The question about fixed-parameter tractability of INTERVAL COMPLETION was asked by Kaplan et al. [FOCS 1994;SIAM J. Comput. 1999] and was answered affirmatively more than a decade later by Villanger et al. [STOC 2007;SIAM J. Comput. 2009], who presented an algorithm with running time O(k(2k)n(3)m). We give the first subexponential parameterized algorithm solving INTERVAL COMPLETION in time k(O)((root)(k))n(O(1)). This adds INTERVAL COMPLETION to a very small list of parameterized graph modification problems solvable in subexponential time.
The firefighter problem models the situation where an infection, a computer virus, an idea or fire etc. is spreading through a network and the goal is to save as many as possible nodes of the network through targeted ...
详细信息
ISBN:
(纸本)9781920682989
The firefighter problem models the situation where an infection, a computer virus, an idea or fire etc. is spreading through a network and the goal is to save as many as possible nodes of the network through targeted vaccinations. The number of nodes that can be vaccinated at a single time-step is typically one, or more generally O(1). In a non-standard model, the so called spreading model, the vaccinations also spread in contrast to the standard *** main results are concerned with general graphs in the spreading *** provide a very simple exact 2O(√n log n)-time algorithm. In the special case of trees, where the standard and spreading model are equivalent, our algorithm is substantially simpler than that exact subexponential algorithm for trees presented in (Cai et al. 2008). On the other hand, we show that the firefighter problem on weighted directed graphs in the spreading model cannot be approximated within a constant factor better than 1−1/e unless NP ⊆ DTIME(nO(log log n)).We also present several results in the standard model. We provide approximation algorithms for planar graphs in case when at least two vaccinations can be performed at a time-step. We also derive trade-offs between approximation factors for polynomial-time solutions and the time complexity of exact or nearly exact solutions for instances of the firefighter problem for the so called directed layered graphs.
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