In this paper,we consider the P-prize-collecting set cover(P-PCSC)problem,which is a generalization of the set cover *** this problem,we are given a set system(U,S),where U is a ground set and S⊆2U is a collection of ...
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In this paper,we consider the P-prize-collecting set cover(P-PCSC)problem,which is a generalization of the set cover *** this problem,we are given a set system(U,S),where U is a ground set and S⊆2U is a collection of subsets satisfying∪S∈SS=*** subset in S has a nonnegative cost,and every element in U has a nonnegative penalty cost and a nonnegative *** goal is to find a subcollection C⊆S such that the total cost,consisting of the cost of subsets in C and the penalty cost of the elements not covered by C,is minimized and at the same time the combined profit of the elements covered by C is at least P,a specified profit *** main work is to obtain a 2f+ε-approximation algorithm for the P-PCSC problem by using the primal-dual and Lagrangian relaxation methods,where f is the maximum frequency of an element in the given set system(U,S)andεis a fixed positive number.
We present a primal-dual aOElog(n)aOE parts per thousand-approximation algorithm for the version of the asymmetric prize collecting traveling salesman problem, where the objective is to find a directed tour that visit...
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We present a primal-dual aOElog(n)aOE parts per thousand-approximation algorithm for the version of the asymmetric prize collecting traveling salesman problem, where the objective is to find a directed tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. The previous algorithm for the problem (V.H. Nguyen and T.T Nguyen in Int. J. Math. Oper. Res. 4(3):294-301, 2012) which is not combinatorial, is based on the Held-Karp relaxation and heuristic methods such as the Frieze et al.'s heuristic (Frieze et al. in Networks 12:23-39, 1982) or the recent Asadpour et al.'s heuristic for the ATSP (Asadpour et al. in 21st ACM-SIAM symposium on discrete algorithms, 2010). Depending on which of the two heuristics is used, it gives respectively 1+aOElog(n)aOE parts per thousand and as an approximation ratio. Our algorithm achieves an approximation ratio of aOElog(n)aOE parts per thousand which is weaker than but represents the first combinatorial approximation algorithm for the Asymmetric Prize-Collecting TSP.
Considerable attention has been paid to sweep coverage in Wireless Sensor Networks (WSNs) for the past few years. In sweep coverage, mobile sensor nodes are scheduled to visit Points of Interests (POIs) periodically. ...
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Considerable attention has been paid to sweep coverage in Wireless Sensor Networks (WSNs) for the past few years. In sweep coverage, mobile sensor nodes are scheduled to visit Points of Interests (POIs) periodically. Due to the limited energy of mobile sensors, energy consumption poses an important challenge in sweep coverage. In this paper, we extend the Energy Restricted Sweep Coverage to the General Energy Restricted Sweep Coverage (GERSC) by assuming the unit energy consumption of mobile sensors varies in different road sections. The goal of GERSC is also to minimize the required number of mobile sensors in sweep coverage under the energy constraint. In GERSC, the approximation ratio of the state-of-the-art algorithm named ERSweepCoverage is no longer guaranteed. Therefore, we devise a constant-factor approximation algorithm named IERSC for GERSC. The approximation ratio of IERSC is beta[3 alpha/min(theta,1-theta) + 1]. theta is the predefined parameter of IERSC ranging from 0 to 1. With theta being 12, IERSC has the best approximation ratio of (6 alpha + 1)beta. Finally, the promising experimental results demonstrate the effectiveness and efficiency of the proposed algorithm. (C) 2021 Elsevier B.V. All rights reserved.
The hitting set problem is a generalization of the vertex cover problem to hypergraphs. Xu et al. (Theor Comput Sci 630:117-125, 2016) presented a primal-dual algorithm for the submodular vertex cover problem with lin...
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The hitting set problem is a generalization of the vertex cover problem to hypergraphs. Xu et al. (Theor Comput Sci 630:117-125, 2016) presented a primal-dual algorithm for the submodular vertex cover problem with linear/submodular penalties. Motivated by their work, we study the submodular hitting set problem with linear penalties (SHSLP). The goal of the SHSLP is to select a vertex subset in the hypergraph to cover some hyperedges and penalize the uncovered ones such that the total cost of covering and penalty is minimized. Based on the primal-dual scheme, we obtain ak-approximation algorithm for the SHSLP, wherekis the maximum number of vertices in all hyperedges.
In this paper, we give an algorithm that finds an e-approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in polynomial time if the rank of the quadratic function and the num...
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In this paper, we give an algorithm that finds an e-approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in polynomial time if the rank of the quadratic function and the number of integer variables are fixed. The running time of the algorithm is expected unless P=NP. In order to design this algorithm we introduce the novel concepts of spherical form MIQP and of aligned vectors, and we provide a number of results of independent interest. In particular, we give a strongly polynomial algorithm to find a symmetric decomposition of a matrix, and show a related result on simultaneous diagonalization of matrices.
A space-efficient approximation algorithm for the grammar-based compression problem, which requests for a given string to find a smallest context-free grammar deriving the string, is presented. For the input length it...
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A space-efficient approximation algorithm for the grammar-based compression problem, which requests for a given string to find a smallest context-free grammar deriving the string, is presented. For the input length it and an optimum CFG size g, the algorithm consumes only O(g log g) space and O(n log*n) time to achieve O((log*n) log n) approximation ratio to the optimum compression, where log*n is the maximum number of logarithms satisfying log log...log n > 1. This ratio is thus regarded to almost O(log n), which is the currently best approximation ratio. While g depends on the string, it is known that g = Omega(log n) and g = O(n/log(k)n) for strings from k-letter alphabet [12].
We give the first 2-approximation algorithm for the cluster vertex deletion problem. This approximation factor is tight, since approximating the problem within any constant factor smaller than 2 is UGC-hard. Our algor...
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We give the first 2-approximation algorithm for the cluster vertex deletion problem. This approximation factor is tight, since approximating the problem within any constant factor smaller than 2 is UGC-hard. Our algorithm combines previous approaches, based on the local ratio technique and the management of twins, with a novel construction of a "good" cost function on the vertices at distance at most 2 from any vertex of the input graph. As an additional contribution, we also study cluster vertex deletion from the polyhedral perspective, where we prove almost matching upper and lower bounds on how well linear programming relaxations can approximate the problem.
In packing steel products of coils into cassettes, we need to consider both the width and weight capacity of cassettes. Each coil has weight in (0, 1/3] and width in (1/6, 1/3] when scaling both the weight and width c...
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In packing steel products of coils into cassettes, we need to consider both the width and weight capacity of cassettes. Each coil has weight in (0, 1/3] and width in (1/6, 1/3] when scaling both the weight and width capacities to 1. With the objective of minimizing the number of cassettes to pack the coils, the problem is modeled by two-dimensional vector packing. To efficiently pack the coils having sizes specified by the ranges, we develop a 4/3-approximation algorithm.
One ultimate goal of wireless sensor networks is to collect the sensed data from a set of sensors and transmit them to some sink node via a data gathering tree. In this work, we are interested in data aggregation, whe...
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One ultimate goal of wireless sensor networks is to collect the sensed data from a set of sensors and transmit them to some sink node via a data gathering tree. In this work, we are interested in data aggregation, where the sink node wants to know the value for a certain function of all sensed data, such as minimum, maximum, average, and summation. Given a data aggregation tree, sensors receive messages from children periodically, merge them with its own packet, and send the new packet to its parent. The problem of finding an aggregation tree with the maximum lifetime has been proved to be NP-hard and can be generalized to finding a spanning tree with the minimum maximum vertex load, where the load of a vertex is a nondecreasing function of its degree in the tree. Although there is a rich body of research in those problems, they either fail to meet a theoretical bound or need high running time. In this paper, we develop a novel algorithm with provable performance bounds for the generalized problem. We show that the running time of our algorithm is in the order of O(mn alpha)(m, n)), where m is the number of edges, n is the number of sensors, and alpha is the inverse Ackerman function. Though our work is motivated by applications in sensor networks, the proposed algorithm is general enough to handle a wide range of degree-oriented spanning tree problems, including bounded degree spanning tree problem and minimum degree spanning tree problem. When applied to these problems, it incurs a lower computational cost in comparison to existing methods. Simulation results validate our theoretical analysis.
We consider the scheduling problem of n independent jobs on m identical parallel processors in order to minimize makespan, the completion time of the last job. We propose a new approximation algorithm that iteratively...
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We consider the scheduling problem of n independent jobs on m identical parallel processors in order to minimize makespan, the completion time of the last job. We propose a new approximation algorithm that iteratively combines partial solutions to the problem. The worst-case performance ratio of the algorithm is (z+ 1)/(z) - (1)/(mz), where z is the number of initial partial solutions that are obtained by partitioning the set of jobs into z families of subsets which satisfy suitable properties. The computational behavior of our worst-case performance ratio provided encouraging results on three families of instances taken from the literature.
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