Browsing by Subject "Polynomial approximation"

Now showing 1 - 9 of 9

Open Access
Batch scheduling to minimize the weighted number of tardy jobs
(Pergamon Press, 2007) Erel, E.; Ghosh, J. B.
In this paper, we address a single-machine scheduling problem with due dates and batch setup times to minimize the weighted number of tardy jobs. We give a pseudo-polynomial dynamic program and a fully-polynomial approximation scheme for the case where the due dates are uniform within a family.
Open Access
Double bound method for solving the p-center location problem
(Elsevier, 2013) Calik, H.; Tansel, B. C.
We give a review of existing methods for solving the absolute and vertex restricted p-center problems on networks and propose a new integer programming formulation, a tightened version of this formulation and a new method based on successive restrictions of the new formulation. A specialization of the new method with two-element restrictions obtains the optimal p-center solution by solving a series of simple structured integer programs in recognition form. This specialization is called the double bound method. A relaxation of the proposed formulation gives the tightest known lower bound in the literature (obtained earlier by Elloumi et al., [1]). A polynomial time algorithm is presented to compute this bound. New lower and upper bounds are proposed. Problems from the OR-Library [2] and TSPLIB [3] are solved by the proposed algorithms with up to 3038 nodes. Previous computational results were restricted to networks with at most 1817 nodes.
Open Access
FPTAS for half-products minimization with scheduling applications
(Elsevier, 2008) Erel, E.; Ghosh, J. B.
A special class of quadratic pseudo-boolean functions called "half-products" (HP) has recently been introduced. It has been shown that HP minimization, while NP-hard, admits a fully polynomial time approximation scheme (FPTAS). In this note, we provide a more efficient FPTAS. We further show how an FPTAS can also be derived for the general case where the HP function is augmented by a problem-dependent constant and can justifiably be assumed to be nonnegative. This leads to an FPTAS for certain partitioning type problems, including many from the field of scheduling.
Open Access
Joint mixability of some integer matrices
(Elsevier B.V., 2016) Bellini, F.; Karaşan, O. E.; Pınar, M. Ç.
We study the problem of permuting each column of a given matrix to achieve minimum maximal row sum or maximum minimal row sum, a problem of interest in probability theory and quantitative finance where quantiles of a random variable expressed as the sum of several random variables with unknown dependence structure are estimated. If the minimum maximal row sum is equal to the maximum minimal row sum the matrix has been termed jointly mixable (see e.g. Haus (2015), Wang and Wang (2015), Wang et al. (2013)). We show that the lack of joint mixability (the joint mixability gap) is not significant, i.e., the gap between the minimum maximal row sum and the maximum minimal row sum is either zero or one for a class of integer matrices including binary and complete consecutive integers matrices. For integer matrices where all entries are drawn from a given set of discrete values, we show that the gap can be as large as the difference between the maximal and minimal elements of the discrete set. The aforementioned result also leads to a polynomial-time approximation algorithm for matrices with restricted domain. Computing the gap for a {0,1,2}-matrix is proved to be equivalent to finding column permutations minimizing the difference between the maximum and minimum row sums. A polynomial procedure for computing the optimum difference by solving the maximum flow problem on an appropriate graph is given. © 2016 Elsevier B.V. All rights reserved.
Open Access
Minimizing weighted mean absolute deviation of job completion times from their weighted mean
(Elsevier, 2011) Erel, E.; Ghosh, J. B.
We address a single-machine scheduling problem where the objective is to minimize the weighted mean absolute deviation of job completion times from their weighted mean. This problem and its precursors aim to achieve the maximum admissible level of service equity. It has been shown earlier that the unweighted version of this problem is NP-hard in the ordinary sense. For that version, a pseudo-polynomial time dynamic program and a 2-approximate algorithm are available. However, not much (except for an important solution property) exists for the weighted version. In this paper, we establish the relationship between the optimal solution to the weighted problem and a related one in which the deviations are measured from the weighted median (rather than the mean) of the job completion times; this generalizes the 2-approximation result mentioned above. We proceed to give a pseudo-polynomial time dynamic program, establishing the ordinary NP-hardness of the problem in general. We then present a fully-polynomial time approximation scheme as well. Finally, we report the findings from a limited computational study on the heuristic solution of the general problem. Our results specialize easily to the unweighted case; they also lead to an approximation of the set of schedules that are efficient with respect to both the weighted mean absolute deviation and the weighted mean completion time. © 2011 Elsevier Inc. All rights reserved.
Open Access
Minimum maximum-degree publish-subscribe overlay network design
(IEEE, 2011) Onus, Melih; Richa, A.W.
Designing an overlay network for publish/subscribe communication in a system where nodes may subscribe to many different topics of interest is of fundamental importance. For scalability and efficiency, it is important to keep the degree of the nodes in the publish/subscribe system low. It is only natural then to formalize the following problem: Given a collection of nodes and their topic subscriptions, connect the nodes into a graph that has least possible maximum degree in such a way that for each topic t, the graph induced by the nodes interested in t is connected. We present the first polynomial-time logarithmic approximation algorithm for this problem and prove an almost tight lower bound on the approximation ratio. Our experimental results show that our algorithm drastically improves the maximum degree of publish/subscribe overlay systems. We also propose a variation of the problem by enforcing that each topic-connected overlay network be of constant diameter while keeping the average degree low. We present three heuristics for this problem that guarantee that each topic-connected overlay network will be of diameter 2 and that aim at keeping the overall average node degree low. Our experimental results validate our algorithms, showing that our algorithms are able to achieve very low diameter without increasing the average degree by much. © 2011 IEEE.
Open Access
Network-aware virtual machine placement in cloud data centers with multiple traffic-intensive components
(Elsevier BV, 2015) Ilkhechi, A. R.; Korpeoglu, I.; Ulusoy, Özgür
Following a shift from computing as a purchasable product to computing as a deliverable service to consumers over the Internet, cloud computing has emerged as a novel paradigm with an unprecedented success in turning utility computing into a reality. Like any emerging technology, with its advent, it also brought new challenges to be addressed. This work studies network and traffic aware virtual machine (VM) placement in a special cloud computing scenario from a provider's perspective, where certain infrastructure components have a predisposition to be the endpoints of a large number of intensive flows whose other endpoints are VMs located in physical machines (PMs). In the scenarios of interest, the performance of any VM is strictly dependent on the infrastructure's ability to meet their intensive traffic demands. We first introduce and attempt to maximize the total value of a metric named "satisfaction" that reflects the performance of a VM when placed on a particular PM. The problem of finding a perfect assignment for a set of given VMs is NP-hard and there is no polynomial time algorithm that can yield optimal solutions for large problems. Therefore, we introduce several off-line heuristic-based algorithms that yield nearly optimal solutions given the communication pattern and flow demand profiles of subject VMs. With extensive simulation experiments we evaluate and compare the effectiveness of our proposed algorithms against each other and also against naïve approaches.
Open Access
Noise-enhanced M-ary hypothesis-testing in the minimax framework
(IEEE, 2009-09) Bayram, Suat; Gezici, Sinan
In this study, the effects of adding independent noise to observations of a suboptimal detector are studied for M-ary hypothesis-testing problems according to the minimax criterion. It is shown that the optimal additional noise can be represented by a randomization of at most M signal values under certain conditions. In addition, a convex relaxation approach is proposed to obtain an accurate approximation to the noise probability distribution in polynomial time. Furthermore, sufficient conditions are presented to determine when additional noise can or cannot improve the performance of a given detector. Finally, a numerical example is presented. © 2009 IEEE.
Open Access
One-dimensional partitioning for heterogeneous systems: theory and practice
(Academic Press, 2008-11) Pınar, A.; Tabak, E. K.; Aykanat, Cevdet
We study the problem of one-dimensional partitioning of nonuniform workload arrays, with optimal load balancing for heterogeneous systems. We look at two cases: chain-on-chain partitioning, where the order of the processors is specified, and chain partitioning, where processor permutation is allowed. We present polynomial time algorithms to solve the chain-on-chain partitioning problem optimally, while we prove that the chain partitioning problem is NP-complete. Our empirical studies show that our proposed exact algorithms produce substantially better results than heuristics, while solution times remain comparable. © 2008 Elsevier Inc. All rights reserved.