On the optimality of exact and approximation algorithms for scheduling problems

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Tight running times for minimum ℓq-norm load balancing: beyond exponential dependencies on 1/∊Lin Chen, Liangde Tao, and José VerschaeLin Chen, Liangde Tao, and José Verschaepp.275 - 315Chapter DOI:https://doi.org/10.1137/1.9781611977073.14PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We consider a classical scheduling problem on m identical machines. For an arbitrary constant q > 1, the aim is to assign jobs to machines such that is minimized, where Ci is the total processing time of jobs assigned to machine i. It is well known that this problem is strongly NP-hard. Under mild assumptions, the running time of an (1 + ∊)-approximation algorithm for a strongly NP-hard problem cannot be polynomial on 1/∊, unless P = NP. For most problems in the literature, this translates into algorithms with running time at least as large as 2Ω(1/∊) + nO(1). For the natural scheduling problem above, we establish the existence of an algorithm which violates this threshold. More precisely, we design a PTAS that runs in time. This result is in sharp contrast to the closely related minimum makespan variant, where an exponential lower bound is known under the exponential time hypothesis (ETH). We complement our result with an essentially matching lower bound on the running time, showing that our algorithm is best-possible under ETH. The lower bound proof exploits new number-theoretical constructions for variants of progression-free sets, which might be of independent interest. Furthermore, we provide a fine-grained characterization on the running time of a PTAS for this problem depending on the relation between ∊ and the number of machines m. More precisely, our lower bound only holds when . Better algorithms, that go beyond the lower bound, exist for other values of m. In particular, there even exists an algorithm with running time polynomial in 1/∊ if we restrict ourselves to instances with m = Ω(1/∊ log2 1/∊). Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771

We consider a classical scheduling problem on m identical machines. For an arbitrary constant [Formula: see text], the aim is to assign jobs to machines such that [Formula: see text] is minimized, where [Formula: see text] is the total processing time of jobs assigned to machine i. It is well known that this problem is strongly NP-hard. Under mild assumptions, the running time of a [Formula: see text]-approximation algorithm for a strongly NP-hard problem cannot be polynomial on [Formula: see text], unless [Formula: see text]. For most problems in the literature, this translates into algorithms with running time at least as large as [Formula: see text]. For the natural scheduling problem above, we establish the existence of an algorithm that violates this threshold. More precisely, we design a PTAS that runs in [Formula: see text] time. This result is in sharp contrast to the closely related minimum makespan variant, where an exponential lower bound is known under the exponential time hypothesis (ETH). We complement our result with an essentially matching lower bound on the running time, showing that our algorithm is best possible under ETH. The lower bound proof exploits new number-theoretical constructions for variants of progression-free sets, which might be of independent interest. Furthermore, we provide a fine-grained characterization on the running time of a PTAS for this problem depending on the relation between [Formula: see text] and the number of machines m. More precisely, our lower bound only holds when [Formula: see text]. Better algorithms that go beyond the lower bound exist for other values of m. In particular, there even exists an algorithm with running time polynomial in [Formula: see text] if we restrict ourselves to instances with [Formula: see text]. Funding: This research was supported by the Center for Mathematical Modeling BASAL [Fund FB210005], ANID-Chile, and ANID/Fondecyt [Regular 1221460]. National Science Foundation China [12271477], National Science Foundation [1756014].

Makespan scheduling on identical machines is one of the most basic and fundamental packing problem studied in the discrete optimization literature. It asks for an assignment of n jobs to a set of m identical machines that minimizes the makespan. The problem is strongly NPhard, and thus we do not expect a (1 + epsilon)-approximation algorithm with a running time that depends polynomially on 1/epsilon. Furthermore, Chen et al. [Chen/JansenZhang, SODA'13] recently showed that a running time of 2^{1/epsilon}^{1-delta} + poly(n) for any delta > 0 would imply that the Exponential Time Hypothesis (ETH) fails. A long sequence of algorithms have been developed that try to obtain low dependencies on 1/epsilon, the better of which achieves a running time of 2^{~O(1/epsilon^{2})} + O(n*log(n)) [Jansen, SIAM J. Disc. Math. 2010]. In this paper we obtain an algorithm with a running time of 2^{~O(1/epsilon)} + O(n*log(n)), which is tight under ETH up to logarithmic factors on the exponent. Our main technical contribution is a new structural result on the configuration-IP. More precisely, we show the existence of a highly symmetric and sparse optimal solution, in which all but a constant number of machines are assigned a configuration with small support. This structure can then be exploited by integer programming techniques and enumeration. We believe that our structural result is of independent interest and should find applications to other settings. In particular, we show how the structure can be applied to the minimum makespan problem on related machines and to a larger class of objective functions on parallel machines. For all these cases we obtain an efficient PTAS with running time 2^{~O(1/epsilon)} + poly(n).

Community search on attributed networks has recently attracted great deal of research interest. However, most of existing works require query users to specify some community structure parameters. This may not be always practical as sometimes a user does not have the knowledge and experience to decide the suitable parameters. In this paper, we propose a novel parameter-free contextual community model for attributed community search. The proposed model only requires a query context, i.e., a set of keywords describing the desired matching community context, while the community returned is both structure and attribute cohesive w.r.t. the provided query context. We theoretically show that both our exact and approximate contextual community search algorithms can be executed in worst case polynomial time. The exact algorithm is based on an elegant parametric maximum flow technique and the approximation algorithm that significantly improves the search efficiency is analyzed to have an approximation factor of 1/3. In the experiment, we use six real networks with ground-truth communities to evaluate the effectiveness of our contextual community model. Experimental results demonstrate that the proposed model can find near ground-truth communities. We also test both our exact and approximate algorithms using eight large real networks to demonstrate the high efficiency of the proposed algorithms.

The Random Shortest Path Problem with the second moment criterion is discussed in this paper. After the formulation of the problem, exact algorithms, based on general concepts for solving the Multi-objective Shortest Path Problem, are described. Next, several approximate algorithms are proposed. It is shown that the complexity of the exact algorithms is exponential, while the complexity of the approximate algorithms is only polynomial. Computational results for the exact and approximate algorithms, which were performed on large graphs, are shown.

In this paper we gather several improvements in the field of exact and approximate exponential-time algorithms for the Bandwidth problem. For graphs with treewidth t we present a O(n O(t) 2n) exact algorithm. Moreover for the same class of graphs we introduce a subexponential constant-approximation scheme – for any α> 0 there exists a (1 + α)-approximation algorithm running in \(O(\exp(c(t + \sqrt{n/\alpha})\log n))\) time where c is a universal constant. These results seem interesting since Unger has proved that Bandwidth does not belong to APX even when the input graph is a tree (assuming P ≠ NP). So somewhat surprisingly, despite Unger’s result it turns out that not only a subexponential constant approximation is possible but also a subexponential approximation scheme exists. Furthermore, for any positive integer r, we present a (4r − 1)-approximation algorithm that solves Bandwidth for an arbitrary input graph in \(O^*(2^{n\over r})\) time and polynomial space. Finally we improve the currently best known exact algorithm for arbitrary graphs with a O(4.473n) time and space algorithm.In the algorithms for the small treewidth we develop a technique based on the Fast Fourier Transform, parallel to the Fast Subset Convolution techniques introduced by Björklund et al. This technique can be also used as a simple method of finding a chromatic number of all subgraphs of a given graph in O *(2n) time and space, what matches the best known results.KeywordsFast Fourier TransformApproximation AlgorithmChromatic NumberExact AlgorithmInput GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We design new exact exponential time algorithms for the well known NP-hard allocation problems- Makespan minimization, the Santa Claus problem (with and without capacity constraints) and the Bin packing problem. These problems are very well-studied in the paradigm of approximation algorithms. However the best known exact, exponential-time algorithms for all of the above problems has complexity of \(O^*(3^m)\) [6], where m is the number of jobs except for Bin Packing which has a \(O^*(2^m)\) inclusion exclusion based algorithm (where m is the number of items) [8]. We introduce a new dynamic programming formulation which helps solve Makespan minimization and Santa Claus problem more efficiently in \(O^*(2^m)\) time and gives a completely different approach with the same time complexity in case of Bin Packing. In addition, Jansen et al. [6] showed that unless the ETH (exponential time hypothesis) is false, there is no exact algorithm that runs in time \(2^{o(m)}\).

<span lang="EN-US">This research focuses on the k-center problem and its applications. Different methods for solving this problem are analyzed. The implementations of an exact algorithm and of an approximate algorithm are presented. The source code and the computation complexity of these algorithms are presented and analyzed. The multitasking mode of the operating system is taken into account considering the execution time of the algorithms. The results show that the approximate algorithm finds solutions that are not worse than two times optimal. In some case these solutions are very close to the optimal solutions, but this is true only for graphs with a smaller number of nodes. As the number of nodes in the graph increases (respectively the number of edges increases), the approximate solutions deviate from the optimal ones, but remain acceptable. These results give reason to conclude that for graphs with a small number of nodes the approximate algorithm finds comparable solutions with those founds by the exact algorithm.</span>

The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+epsilon)-approximations in f(k,epsilon)n^O(1) time where k is some parameter of the input. The goal is to overcome lower bounds from either of the areas. We obtain the following results on parameterized approximability: - In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA'05]. The best-known polynomial-time approximation factor is O(log log n) [Chalermsook and Chuzhoy, SODA'09] and it admits a QPTAS [Adamaszek and Wiese, FOCS'13; Chuzhoy and Ene, FOCS'16]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant epsilon>0 and integer k>0, in time f(k,epsilon)n^g(epsilon), either outputs a solution of size at least k/(1+epsilon), or declares that the optimum solution has size less than k. - In the (2-dimensional) geometric knapsack problem (2DK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of 2DK with rotations (2DKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factor is 2+epsilon [Jansen and Zhang, SODA'04]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA'15]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for 2DKR. For all considered problems, getting time f(k,epsilon)n^O(1), rather than f(k,epsilon)n^g(epsilon), would give FPT time f'(k)n^O(1) exact algorithms by setting epsilon=1/(k+1), contradicting W[1]-hardness. Instead, for each fixed epsilon>0, our PASs give (1+epsilon)-approximate solutions in FPT time. For both MISR and 2DKR our techniques also give rise to preprocessing algorithms that take n^g(epsilon) time and return a subset of at most k^g(epsilon) rectangles/items that contains a solution of size at least k/(1+epsilon) if a solution of size k exists. This is a special case of the recently introduced notion of a polynomial-size approximate kernelization scheme [Lokshtanov et al., STOC'17].

Article Free Access Share on Exact sampling and approximate counting techniques Author: Mark Huber School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NYView Profile Authors Info & Claims STOC '98: Proceedings of the thirtieth annual ACM symposium on Theory of computingMay 1998 Pages 31–40https://doi.org/10.1145/276698.276709Published:23 May 1998Publication History 25citation666DownloadsMetricsTotal Citations25Total Downloads666Last 12 Months66Last 6 weeks15 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

In this thesis, we study the theoretical approach on energy-efficient scheduling problems arising in demand response management in the modern electrical smart grid. Consumers send in power requests with flexible feasible timeslots during which their requests can be served. The grid controller, upon receiving power requests, schedules each request within the specified interval. The electricity cost is measured by a convex function of the load in each timeslot. The objective is to schedule all requests with the minimum total electricity cost. We study the smart grid scheduling problem in different models. For the offline model, we prove the problem is NP-hard for the general case. We propose a polynomial time algorithm for special input where jobs have unit power request and unit time duration. By adapting the polynomial time algorithm for unit-size jobs, we propose an approximation algorithm for more general input. On the other hand, we also present an exact algorithm to find the optimal schedule for the problem with general input. For the online model, we propose an online algorithm for jobs with jobs with arbitrary power request, arbitrary time duration, and arbitrary contiguous feasible intervals. We also show a lower bound of the competitive ratio for the smart grid scheduling problem with unit height and arbitrary width. For special cases, we design different online algorithms with better competitive ratios. Finally, we look at other optimization problems and show how to solve them by adapting our techniques. We prove that our online algorithm can solve the machine minimization problem with an asymptotically optimal competitive ratio. We also show that our exact algorithm can be adapted to solve other demand response management problems.

A belief network comprises a graphical representation of dependencies between variables of a domain and a set of conditional probabilities associated with each dependency. Unless rho =NP, an efficient, exact algorithm does not exist to compute probabilistic inference in belief networks. Stochastic simulation methods, which often improve run times, provide an alternative to exact inference algorithms. Such a stochastic simulation algorithm, D-BNRAS, which is a randomized approximation scheme is presented. To analyze the run time, belief networks are parameterized, by the dependence value D/sub xi /, which is a measure of the cumulative strengths of the belief network dependencies given background evidence xi . This parameterization defines the class of f-dependence networks. The run time of D-BNRAS is polynomial when f is a polynomial function. Thus, the results prove the existence of a class of belief networks for which inference approximation is polynomial and, hence, provably faster than any exact algorithm. >

A heuristic algorithm is developed for the classical scheduling problem in which independent jobs are nonpreemptively scheduled on identical parallel machines with the objective of minimizing the makespan with load balancing. The algorithm uses iteratively the LPT procedure to be successful computational tools in many hard Combinatorial Optimization problems. We present an approximation algorithm based on balance the Hopfild with binary decision variables which are a kind of assignment variables for the classical deterministic scheduling problem of minimizing the makespan on identical machines. These problems are characterized by their complexity, known as NP-hard in most cases. Computational results indicate that the proposed algorithm is very competitive with respect to well-known constructive algorithms for a large number of benchmark.

In this paper we study the densest subgraph problem, which plays a key role in many graph mining applications. The goal of the problem is to find a subset of nodes that induces a graph with maximum average degree. The problem has been extensively studied in the past few decades under a variety of different settings. Several exact and approximation algorithms were proposed. However, as normal graph can only model objects with pairwise relationships, the densest subgraph problem fails in identifying communities under relationships that involve more than 2 objects, e.g., in a network connecting authors by publications. We consider in this work the densest subgraph problem in hypergraphs, which generalizes the problem to a wider class of networks in which edges might have different cardinalities and contain more than 2 nodes. We present two exact algorithms and a near-linear time r-approximation algorithm for the problem, where r is the maximum cardinality of an edge in the hypergraph. We also consider the dynamic version of the problem, in which an adversary can insert or delete an edge from the hypergraph in each round and the goal is to maintain efficiently an approximation of the densest subgraph. We present two dynamic approximation algorithms in this paper with amortized polog update time, for any e > 0. For the case when there are only insertions, the approximation ratio we maintain is r(1+e), while for the fully dynamic case, the ratio is r2(1+e). Extensive experiments are performed on large real datasets to validate the effectiveness and efficiency of our algorithms.

We consider the problem of scheduling with renewable speed-up resources. Given m identical machines, n jobs and c different discrete resources, the task is to schedule each job non-preemptively onto one of the machines so as to minimize the makespan. In our problem, a job has its original processing time, which could be reduced by utilizing one of the resources. As resources are different, the amount of the time reduced for each job is different depending on the resource it uses. Once a resource is being used by one job, it can not be used simultaneously by any other job until this job is finished, hence the scheduler should take into account the job-to-machine assignment together with the resource-to-job assignment. We observe that, the classical unrelated machine scheduling problem is actually a special case of our problem when m=c, i.e., the number of resources equals the number of machines. Extending the techniques for the unrelated machine scheduling, we give a 2-approximation algorithm when both m and c are part of the input. We then consider two special cases for the problem, with m or c being a constant, and derive PTASes (Polynomial Time Approximation Schemes) respectively. We also establish the relationship between the two parameters m and c, through which we are able to transform the PTAS for the case when m is constant to the case when c is a constant. The relationship between the two parameters reveals the structure within the problem, and may be of independent interest.