In this article, the job shop scheduling problem with two batch-processing machines is considered. The machines have limited capacity and the jobs have non-identical job sizes. The jobs are processed in batches and th...
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In this article, the job shop scheduling problem with two batch-processing machines is considered. The machines have limited capacity and the jobs have non-identical job sizes. The jobs are processed in batches and the total size of each batch cannot exceed the machine capacity. The processing times of a job on the two machines are proportional. We show the problem of minimising makespan is NP-hard in the strong sense. Then we provide an approximation algorithm with worst-case ratio no more than 4, and the running time of the algorithm is O(n log n). Finally, the performance of the proposed algorithm is tested by different levels of instances. Computational results demonstrate the effectiveness of the algorithm for all the instances.
For continuous-time Markov chains, the model-checking problem with respect to continuous-time stochastic logic (CSL) has been introduced and shown to be decidable by Aziz, Sanwal, Singhal and Brayton in 1996 [ 1, 2]. ...
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For continuous-time Markov chains, the model-checking problem with respect to continuous-time stochastic logic (CSL) has been introduced and shown to be decidable by Aziz, Sanwal, Singhal and Brayton in 1996 [ 1, 2]. Their proof can be turned into an approximation algorithm with worse than exponential complexity. In 2000, Baier, Haverkort, Hermanns and Katoen [ 4, 5] presented an efficient polynomial-time approximation algorithm for the sublogic in which only binary until is allowed. In this paper, we propose such an efficient polynomial-time approximation algorithm for full CSL. The key to our method is the notion of stratified CTMCs with respect to the CSL property to be checked. On a stratified CTMC, the probability to satisfy a CSL path formula can be approximated by a transient analysis in polynomial time (using uniformization). We present a measure-preserving, linear-time and - space transformation of any CTMC into an equivalent, stratified one. This makes the present work the centerpiece of a broadly applicable full CSL model checker. Recently, the decision algorithm by Aziz et al. was shown to work only for stratified CTMCs. As an additional contribution, our measure-preserving transformation can be used to ensure the decidability for general CTMCs.
We study how to partition an interval graph with non-negative vertex weights into k connected subgraphs such that the minimum total weight of any part of the partition is maximized. For k = 2, it is shown that for any...
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We study how to partition an interval graph with non-negative vertex weights into k connected subgraphs such that the minimum total weight of any part of the partition is maximized. For k = 2, it is shown that for any epsilon > 0, a (1 + epsilon)-approximation can be found in O((1/epsilon) n(3)) time, i.e., it admits a fully polynomial-time approximation scheme (FPTAS). For any fixed k > 2, the problem also admits an FPTAS when restricted to k-connected interval graphs.
The firefighter problem is used to model the spread of fire, infectious diseases, and computer viruses. This paper deals with firefighter problem on rooted trees. It is known that the firefighter problem is NP-hard ev...
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The firefighter problem is used to model the spread of fire, infectious diseases, and computer viruses. This paper deals with firefighter problem on rooted trees. It is known that the firefighter problem is NP-hard even for rooted trees of maximum degree 3. We propose techniques to improve a given approximation algorithm. First, we introduce an implicit enumeration technique. By applying the technique to existing (I)approximation algorithm, we obtain (1) approximation algorithm when a root has k children. In case of ternary trees, k = 3 and thus the approximation ratio satisfies (1 - (k-1/(k-1)e+1) >= 0.6892, which improves the existing result 1 - 1/e >= 0.6321. Second technique is based on backward induction and improves an approximation algorithm for firefighter problem on ternary trees. If we apply the technique to existing (l-l/e)-approximation algorithm, we obtain 0.6976-approximation algorithm. Lastly, we combine the above two techniques and Obtain 0.7144-approximation algorithm for firefighter problem on ternary trees.
This paper describes an extremely fast polynomial time algorithm, the Near Optimal Vertex Cover algorithm (NOVCA) that produces an optimal or near optimal vertex cover for any known undirected graph G (V, E). NOVCA co...
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This paper describes an extremely fast polynomial time algorithm, the Near Optimal Vertex Cover algorithm (NOVCA) that produces an optimal or near optimal vertex cover for any known undirected graph G (V, E). NOVCA constructs the vertex cover by repeatedly adding, at each step, all vertices adjacent to the vertex of minimal degree; in the case of a tie, it selects the one having the maximum sum of degrees of its neighbors. The results identifying bounds on the size of the minimum vertex cover as well as polynomial complexity of algorithm are given with experimental verification. Future research efforts will be directed at tuning the algorithm and providing proof for better approximation ratio with NOVCA compared to any other available vertex cover algorithms.
We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of ...
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We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. In the all-neighbours knapsack problem, an item can be selected only if all its neighbours are also selected. We give approximation algorithms and hardness results when the vertices have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected. (C) 2012 Elsevier B. V. All rights reserved.
In this paper, we consider a problem of VLSI (very large scale integrated) design occurring in the routing phase. The problem is to determine the optimal size selection for the gates in a combinatorial circuit which u...
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In this paper, we consider a problem of VLSI (very large scale integrated) design occurring in the routing phase. The problem is to determine the optimal size selection for the gates in a combinatorial circuit which uses the problem of finding a shortest path in an oriented acyclic graph for making certain updates between any two successive iterations. For this NP-hard problem, we give an approximation algorithm.
We consider the parallel computing environment where m organizations provide machines and several jobs to be executed. While cooperation of organizations is required to minimize the global makespan, each organization ...
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We consider the parallel computing environment where m organizations provide machines and several jobs to be executed. While cooperation of organizations is required to minimize the global makespan, each organization also expects the faster completion of its own jobs primarily and thus it is not necessarily cooperative. To handle the situations, we formulate the a-cooperative multi-organization scheduling problem (alpha-MOSP), where alpha >= 1 is a parameter representing the degree of cooperativeness. alpha-MOSP minimizes the makespan under the multi-organization constraint that each organization does not allow the completion time of its own jobs to be delayed a times of that in the case where those jobs are executed by itself. First, we reveal the relation between the makespan and the degree of cooperativeness. We show that the multi-organization constraint may degrade the optimal makespan by m times for a = 1, while the degradation ratio is bounded by alpha/(alpha- 1) for a > 1. This implies that weak cooperation improves the makespan dramatically. Second, we study the complexity of a-MOSP. We show its strongly NP-hardness and inapproximability for the approximation factor less than max{(alpha + 1)/alpha, 3/2}. We also show the hardness of transformation from an optimal schedule under no multi-organization constraint to an optimal schedule for a-MOSP. This result is a witness for inexistence of a general polynomial-time transformation algorithm that preserves the approximation ratio.
Grammar-based compression is a well-studied technique to construct a context-free grammar (CFG) deriving a given text uniquely. In this work, we propose an online algorithm for grammar-based compression. Our algorithm...
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Grammar-based compression is a well-studied technique to construct a context-free grammar (CFG) deriving a given text uniquely. In this work, we propose an online algorithm for grammar-based compression. Our algorithm guarantees O (log(2) n) approximation ratio for the minimum grammar size, where n is an input size, and it runs in input linear time and output linear space. In addition, we propose a practical encoding, which transforms a restricted CFG into a more compact representation. Experimental results by comparison with standard compressors demonstrate that our algorithm is especially effective for highly repetitive text.
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