Numerous approaches study the vulnerability of networks against social contagion. Graph burning studies how fast a contagion, modeled as a set of fires, spreads in a graph. The burning process takes place in synchrono...
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ISBN:
(数字)9783030148126
ISBN:
(纸本)9783030148119;9783030148126
Numerous approaches study the vulnerability of networks against social contagion. Graph burning studies how fast a contagion, modeled as a set of fires, spreads in a graph. The burning process takes place in synchronous, discrete rounds. In each round, a fire breaks out at a vertex, and the fire spreads to all vertices that are adjacent to a burning vertex. The selection of vertices where fires start defines a schedule that indicates the number of rounds required to burn all vertices. Given a graph, the objective of an algorithm is to find a schedule that minimizes the number of rounds to burn graph. Finding the optimal schedule is known to be NP-hard, and the problem remains NP-hard when the graph is a tree or a set of disjoint paths. The only known algorithm is an approximation algorithm for disjoint paths, which has an approximation ratio of 1.5. We present approximation algorithms for graph burning. For general graphs, we introduce an algorithm with an approximation ratio of 3. When the graph is a tree, we present another algorithm with approximation ratio 2. Moreover, we consider a setting where the graph is a forest of disjoint paths. In this setting, when the number of paths is constant, we provide an optimal algorithm which runs in polynomial time. When the number of paths is more than a constant, we provide two approximation schemes: first, under a regularity condition where paths have asymptotically equal lengths, we show the problem admits an approximation scheme which is fully polynomial. Second, for a general setting where the regularity condition does not necessarily hold, we provide another approximation scheme which runs in time polynomial in the size of the graph.
We provide improved approximation algorithms for the min-max generalization problems considered by Du, Eppstein, Goodrich, and Lueker [1]. In min-max generalization problems, the input consists of data. items with wei...
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ISBN:
(纸本)9783642153686
We provide improved approximation algorithms for the min-max generalization problems considered by Du, Eppstein, Goodrich, and Lueker [1]. In min-max generalization problems, the input consists of data. items with weights and a lower bound w(1b), and the goal is to partition individual items into groups of weight at least w(1b), while minimizing the maximum weight of a group. The rules of legal partitioning are specific to a problem. Du et al. consider several problems in this vein: (I) partitioning a graph into connected subgraphs, (2) partitioning unstructured data into arbitrary classes and (3) partitioning a 2-dimensional array into non-overlapping contiguous rectangles (subarrays) that satisfy the above size requirements. We significantly improve approximation ratios for all the problems considered by Du et al., and provide additional motivation for these problems. Moreover, for the first problem, while Du et al. give approximation algorithms for specific graph families, namely, 3-connected and 4-connected planar graphs, no approximation algorithm that works for all graphs was known prior to this work.
In this paper we investigate approximation algorithms for the multi-vehicle scheduling problem (MVSP). In MVSP we are given a graph G = (V, E), where each vertex u of V is associated with a job j(u), and each edge e h...
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ISBN:
(纸本)9783642175138
In this paper we investigate approximation algorithms for the multi-vehicle scheduling problem (MVSP). In MVSP we are given a graph G = (V, E), where each vertex u of V is associated with a job j(u), and each edge e has a non-negative weight w(e). There are m identical vehicles available to service the jobs. Each job j(u) has its own release time r(u) and handling time h(u). A job j(u) can only be serviced by one vehicle after its release time r(u), and the handling time h(u) represents the time needed to finish processing j(u). The objective is to find a schedule in which the maximum completion time of the jobs, i.e. the makespan, is minimized. In this paper we present a 3-approximation algorithm for MVSP on trees, and a (5 - 2/m)-approximation algorithm for MVSP on general graphs.
The edge dominating set problem is one of the fundamental covering problems in the field of combinatorial optimization. In this paper, we consider the b-edge dominating set problem, a generalized version of the edge d...
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ISBN:
(纸本)3540280618
The edge dominating set problem is one of the fundamental covering problems in the field of combinatorial optimization. In this paper, we consider the b-edge dominating set problem, a generalized version of the edge dominating set problem. In this version, we are given a simple undirected graph G = (V, E) and a demand vector is an element of Z(E)(+). A set F of edges in G is called a b-edge dominating set if each edge e is an element of E is adjacent to at least b(e) edges in F, where we allow F to contain multiple copies of edges in E. Given a cost vector W is an element of Q(+)(E), the problem asks to find a minimum cost of a b-edge dominating set. We first show that there is a 8/3-approximation algorithm for this problem. We then consider approximation algorithms for other related problems.
In this paper, we consider the restless bandit problem, which is one of the most well-studied generalizations of the celebrated stochastic multi-armed bandit problem in decision theory. In its ultimate generality, the...
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ISBN:
(纸本)9780898716801
In this paper, we consider the restless bandit problem, which is one of the most well-studied generalizations of the celebrated stochastic multi-armed bandit problem in decision theory. In its ultimate generality, the restless bandit problem is known to be PSPACE-Hard to approximate to any non-trivial factor, and little progress has been made on this problem despite its significance in modeling activity allocation under uncertainty. We make progress on this problem by showing that for an interesting and general subclass that we term MONOTONE bandits, a surprisingly simple and intuitive greedy policy yields a factor 2 approximation. Such greedy policies are termed index policies, and are popular due to their simplicity and their optimality for the stochastic multi-armed bandit problem. The MONOTONE bandit problem strictly generalizes the stochastic multi-armed bandit problem, and naturally models multi-project scheduling where the state of a project becomes increasingly uncertain when the project is not scheduled. We develop several novel techniques in the design and analysis of the index policy. Our algorithm proceeds by introducing a novel "balance" constraint to the dual of a well-known LP relaxation to the restless bandit problem. This is followed by a structural characterization of the optimal solution by using both the exact primal as well as dual complementary slackness conditions. This yields an interpretation of the dual variables as potential functions from which we derive the index policy and the associated analysis.
The buy-at-bulk network design problem has been extensively studied in the general graph model. In this paper we consider the geometric version of the problem, where all points in a Euclidean space are candidates for ...
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ISBN:
(纸本)9783642033667
The buy-at-bulk network design problem has been extensively studied in the general graph model. In this paper we consider the geometric version of the problem, where all points in a Euclidean space are candidates for network nodes. We present the first general approach for geometric versions of basic variants of the buy-at-bulk network design problem. It enables us to obtain quasi-polynomial-time approximation schemes for basic variants of the buy-at-bulk geometric network design problem with polynomial total demand. Then, for instances with few sinks and low capacity links, we design very fast polynomial-time low-constant approximations algorithms.
We study the problem of scheduling n independent jobs on a system of m identical parallel machines in the presence of reservations. This constraint is practically important;for various reasons, some machines are not a...
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ISBN:
(纸本)9783540772194
We study the problem of scheduling n independent jobs on a system of m identical parallel machines in the presence of reservations. This constraint is practically important;for various reasons, some machines are not available during specified time intervals. The objective is to minimize the makespan. This problem is inapproximable in the general case unless P = NP which motivates the study of suitable restrictions. We use an approach based on algorithms for multiple subset sum problems;our technique yields a polynomial time approximation scheme (PTAS) which is best possible in the sense that the problem does not admit an FPTAS unless P = NP. The PTAS presented here is the first one for the problem under consideration;so far, not even for special cases approximation schemes have been proposed. We also derive a low cost algorithm with a constant approximation ratio and discuss additional FPTASes for special cases and complexity results.
Inspired by air traffic control and other applications where moving objects have to be labeled, we consider the following (static) point labeling problem: given a set P of n points in the plane and labels that are uni...
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ISBN:
(纸本)9783642137303
Inspired by air traffic control and other applications where moving objects have to be labeled, we consider the following (static) point labeling problem: given a set P of n points in the plane and labels that are unit squares, place a label with each point;in P in such a way that the number of free labels (labels not intersecting any other label) is maximized. We develop efficient constant-factor approximation algorithms for this problem, as well as PTASs, for various label-placement models.
In this paper, we consider a variant of knapsack problem. There are two knapsacks with probably different capacities, owned by two agents respectively. Given a set of items, each with a fixed size and a profit, the tw...
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ISBN:
(纸本)9783642226151
In this paper, we consider a variant of knapsack problem. There are two knapsacks with probably different capacities, owned by two agents respectively. Given a set of items, each with a fixed size and a profit, the two agents select items and pack them into their own knapsacks under the capacity constraint. Same items can be packed simultaneously to different knapsacks. However, in this case the profit of such items can vary. One agent packs items into his knapsack to maximize the total profit, while another agent can only pack items into his knapsack as well but he cares the total profits of items packed into two knapsacks. The latter agent is a leader while the former is a follower. We aim at designing an approximation algorithm for the leader assuming that the follower is selfish. For different settings we provide approximation results.
We study the following generalization of the maximum matching problem in general graphs: Given a simple non-directed graph G = (V, E) and a partition of the edges into k classes (i.e. E = E-1 boolean OR ... boolean OR...
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ISBN:
(纸本)9783662444658;9783662444641
We study the following generalization of the maximum matching problem in general graphs: Given a simple non-directed graph G = (V, E) and a partition of the edges into k classes (i.e. E = E-1 boolean OR ... boolean OR E-k), we would like to compute a matching M on G of maximum cardinality or profit, such that vertical bar M boolean AND E-j vertical bar <= w(j) for every class E-j. Such problems were first studied in the context of network design in [17]. We study the problem from a linear programming point of view: We provide a polynomial time 1/2-approximation algorithm for the weighted case, matching the integrality gap of the natural LP formulation of the problem. For this, we use and adapt the technique of approximate convex decompositions [19] together with a different analysis and a polyhedral characterization of the natural linear program to derive our result. This improves over the existing 1/2, but with additive violation of the color bounds, approximation algorithm [14].
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