In this paper we consider a unified (polynomial time) approximation method for node-deletion problems with nontrivial and hereditary graph properties. One generic algorithm scheme is presented, which can be applied to...
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In this paper we consider a unified (polynomial time) approximation method for node-deletion problems with nontrivial and hereditary graph properties. One generic algorithm scheme is presented, which can be applied to any node-deletion problem for finding approximate solutions. It will be shown then that the quality of solutions found by this algorithm is determined by the quality of any minimal solution in any graph in which nodes are weighted according to a certain scheme chosen by the algorithm. For various node-deletion problems simple and natural schemes for weight assignment are considered. It will be proven that the weight of any minimal solution is a good approximation to the optimal weight when graphs are weighted according to them, implying that our generic algorithm indeed computes good approximate solutions for those node-deletion problems. (C) 1998 Elsevier Science B.V. All rights reserved.
This paper focuses on the scheduling problem on two parallel machines with delivery coordination. In particular, given a set of n jobs, we aim to find a schedule with a minimal makespan such that all jobs are first ex...
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This paper focuses on the scheduling problem on two parallel machines with delivery coordination. In particular, given a set of n jobs, we aim to find a schedule with a minimal makespan such that all jobs are first executed on two parallel machines then delivered at the destination with a transporter. This problem is known to be NP-hard Chang and Lee (Eur J Oper Res 158(2):470-487, 2004), cannot be solved with an approximation ratio strictly less than 3/2 unless P=NP. We close the gap by proposing a polynomial time algorithm whose approximation ratio is 3/2+epsilon with epsilon > 0, improve the previous best ratio 14/9 + epsilon.
The aim of this paper is to establish a new approximation algorithm for fixed points of nonexpansive mappings in general Banach spaces and to illustrate some numerical results. The approximation algorithm we shall dis...
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The aim of this paper is to establish a new approximation algorithm for fixed points of nonexpansive mappings in general Banach spaces and to illustrate some numerical results. The approximation algorithm we shall discuss is x(t,n) = (tT)(x0)(n), where x(0) is an element of D(T) is arbitrary, n is a natural number, and t is an element of (0, 1). We shall also provide some numerical error estimates.
We are concerned with the problem of scheduling monotonic moldable tasks on identical processors to minimize the makespan. We focus on the natural case where the number m of processors as resources is fixed or relativ...
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We are concerned with the problem of scheduling monotonic moldable tasks on identical processors to minimize the makespan. We focus on the natural case where the number m of processors as resources is fixed or relatively small compared with the number n of tasks. We present an efficient (3/2)-approximation algorithm with time complexity O(nmlog(nm)) (for m > n) and O(n(2) log n) (for m <= n). To the best of our knowledge, the best relevant known results are: (a) a (3/2 + epsilon)-approximation algorithm with time complexity O(nm log(n/epsilon)), (b) a fully polynomial-time approximation scheme for the case of m >= 16n/epsilon, and (c) a polynomial-time approximation scheme with time complexity O(n(g(1/epsilon))) when m is bounded by a polynomial in n , where g(center dot) is a super-exponential function. On the other hand, the novel general technique developed in this paper for removing the epsilon-term in the worst-case performance ratio can be applied to improving the performance guarantee of certain dual algorithms for other combinatorial optimization problems. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://***/licenses/by/4.0/)
In the k-level uncapacitated facility location problem, we have a set of demand points where clients are located. The demand of each client is known. Facilities have to be located at given sites in order to service th...
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In the k-level uncapacitated facility location problem, we have a set of demand points where clients are located. The demand of each client is known. Facilities have to be located at given sites in order to service the clients, and each client is to be serviced by a sequence of k different facilities, each of which belongs to a distinct level. There are no capacity restrictions on the facilities. There is a positive fixed cost of setting up a facility, and a per unit cost of shipping goods between each pair of locations. We assume that these distances are all nonnegative and satisfy the triangle inequality. The problem is to find an assignment of each client to a sequence of k facilities, one at each level, so that the demand of each client is satisfied, for which the sum of the setup costs and the service costs is minimized. We develop a randomized algorithm for the k-level facility location problem that is guaranteed to find a feasible solution of expected cost within a factor of 3 of the optimum cost. The algorithm is a randomized rounding procedure that uses an optimal solution of a linear programming relaxation and its dual to make a random choice of facilities to be opened. We show how this algorithm can be derandomized to yield a 3-approximation algorithm. (C) 1999 Elsevier Science B.V. All rights reserved.
In this paper, we introduce a squared metric k-facility location problem (SM-k-FLP) which is a generalization of the squared metric facility location problem and k-facility location problem (k-FLP). In the SM-k-FLP, w...
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In this paper, we introduce a squared metric k-facility location problem (SM-k-FLP) which is a generalization of the squared metric facility location problem and k-facility location problem (k-FLP). In the SM-k-FLP, we are given a client set and a facility set from a metric space, a facility opening cost for each , and an integer k. The goal is to open a facility subset with and to connect each client to the nearest open facility such that the total cost (including facility opening cost and the sum of squares of distances) is minimized. Using local search and scaling techniques, we offer a constant approximation algorithm for the SM-k-FLP.
We give a (1-1/e)-approximation algorithm for the may-profit generalized assignment problem (Max-GAP) with fixed profits when the profit (but not necessarily the size) of every item is independent from the bin it is a...
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We give a (1-1/e)-approximation algorithm for the may-profit generalized assignment problem (Max-GAP) with fixed profits when the profit (but not necessarily the size) of every item is independent from the bin it is assigned to. The previously best-known approximation ratio for this problem was 1/2. (c) 2005 Elsevier B.V. All rights reserved.
We derive a 3/2-approximation algorithm for the NP-hard parallel machine total weighted completion time problem with controllable processing times by the technique of convex quadratic programming relaxation. (C) 2001 ...
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We derive a 3/2-approximation algorithm for the NP-hard parallel machine total weighted completion time problem with controllable processing times by the technique of convex quadratic programming relaxation. (C) 2001 Elsevier Science B.V. All rights reserved.
Although, the Hamiltonicity of solid grid graphs are polynomial-time decidable, the complexity of the longest cycle problem in these graphs is still open. In this paper, by presenting a linear-time constant-factor app...
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Although, the Hamiltonicity of solid grid graphs are polynomial-time decidable, the complexity of the longest cycle problem in these graphs is still open. In this paper, by presenting a linear-time constant-factor approximation algorithm, we show that the longest cycle problem in solid grid graphs is in APX. More precisely, our algorithm finds a cycle of length at least 2n/3 + 1 in 2-connected n-node solid grid graphs. (C) 2015 Elsevier B.V. All rights reserved.
A subset F of vertices of a graph G is called a vertex cover P-k set if every path of order k in G contains at least one vertex from F. Denote by psi(k)(G) the minimum cardinality of a vertex cover P-k set in G. The v...
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A subset F of vertices of a graph G is called a vertex cover P-k set if every path of order k in G contains at least one vertex from F. Denote by psi(k)(G) the minimum cardinality of a vertex cover P-k set in G. The vertex cover P-k (VCPk) problem is to find a minimum vertex cover P-k set. It is easy to see that the VCP2 problem corresponds to the well-known vertex cover problem. In this paper, we restrict our attention to the VCP4 problem in cubic graphs. The paper proves that the VCP4 problem is NP-hard for cubic graphs. Further, we give sharp lower and upper bounds on psi(4)(G) for cubic graphs and propose a 2-approximation algorithm for the VCP4 problem in cubic graphs.
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