An Efficient PTAS for Stochastic Load Balancing with Poisson Jobs

We give the first polynomial-time approximation scheme (PTAS) for the stochastic load balancing problem when the job sizes follow Poisson distributions. This improves upon the 2-approximation algorithm due to Goel and Indyk (FOCS\u2799). Moreover, our approximation scheme is an efficient PTAS that has a running time double exponential in 1/? but nearly-linear in n, where n is the number of jobs and ? is the target error. Previously, a PTAS (not efficient) was only known for jobs that obey exponential distributions (Goel and Indyk, FOCS\u2799). Our algorithm relies on several probabilistic ingredients including some (seemingly) new results on scaling and the so-called "focusing effect" of maximum of Poisson random variables which might be of independent interest

Minimum-Norm Load Balancing Is (Almost) as Easy as Minimizing Makespan

We consider the minimum-norm load-balancing (MinNormLB) problem, wherein there are n jobs, each of which needs to be assigned to one of m machines, and we are given the processing times {p_{ij}} of the jobs on the machines. We also have a monotone, symmetric norm f:?^m ? ?_{? 0}. We seek an assignment ? of jobs to machines that minimizes the f-norm of the induced load vector load->_? ? ?_{? 0}^m, where load_?(i) = ?_{j:?(j) = i}p_{ij}. This problem was introduced by [Deeparnab Chakrabarty and Chaitanya Swamy, 2019], and the current-best result for MinNormLB is a (4+?)-approximation [Deeparnab Chakrabarty and Chaitanya Swamy, 2019]. In the stochastic version of MinNormLB, the job processing times are given by nonnegative random variables X_{ij}, and jobs are independent; the goal is to find an assignment ? that minimizes the expected f-norm of the induced random load vector. We obtain results that (essentially) match the best-known guarantees for deterministic makespan minimization (MinNormLB with ?_? norm). For MinNormLB, we obtain a (2+?)-approximation for unrelated machines, and a PTAS for identical machines. For stochastic MinNormLB, we consider the setting where the X_{ij}s are Poisson random variables, denoted PoisNormLB. Our main result here is a novel and powerful reduction showing that, for any machine environment (e.g., unrelated/identical machines), any ?-approximation algorithm for MinNormLB in that machine environment yields a randomized ?(1+?)-approximation for PoisNormLB in that machine environment. Combining this with our results for MinNormLB, we immediately obtain a (2+?)-approximation for PoisNormLB on unrelated machines, and a PTAS for PoisNormLB on identical machines. The latter result substantially generalizes a PTAS for makespan minimization with Poisson jobs obtained recently by [Anindya De et al., 2020]

Restricted Adaptivity in Stochastic Scheduling

We consider the stochastic scheduling problem of minimizing the expected makespan on m parallel identical machines. While the (adaptive) list scheduling policy achieves an approximation ratio of 2, any (non-adaptive) fixed assignment policy has performance guarantee ?((log m)/(log log m)). Although the performance of the latter class of policies are worse, there are applications in which non-adaptive policies are desired. In this work, we introduce the two classes of ?-delay and ?-shift policies whose degree of adaptivity can be controlled by a parameter. We present a policy - belonging to both classes - which is an ?(log log m)-approximation for reasonably bounded parameters. In other words, an exponential improvement on the performance of any fixed assignment policy can be achieved when allowing a small degree of adaptivity. Moreover, we provide a matching lower bound for any ?-delay and ?-shift policy when both parameters, respectively, are in the order of the expected makespan of an optimal non-anticipatory policy

Stochastic Minimum Norm Combinatorial Optimization

Motivated by growing interest in optimization under uncertainty, we undertake a systematic study of designing approximation algorithms for a wide class of 1-stage stochastic-optimization problems with norm-based objective functions. We introduce the model of stochastic minimum norm combinatorial optimization, denoted StochNormOpt. We have a combinatorial-optimization problem where the costs involved are random variables with given distributions, and we are given a monotone, symmetric norm f. Each feasible solution induces a random multidimensional cost vector whose entries are independent random variables, and the goal is to find an oblivious solution (i.e., one that does not depend on the realizations of the costs) that minimizes the expected f-norm of the induced cost vector. We consider two concrete problem settings. In stochastic load balancing, jobs with random processing times need to be assigned to machines, and the induced cost vector is the machine-load vector, where the load on a machine is given by the sum of job random variables that are assigned to it. In stochastic spanning tree, we have a graph whose edges have stochastic weights, and the induced cost vector consists of edge-weight variables of edges that belong to the spanning tree. The class of monotone, symmetric norms is broad: it includes frequently-used objectives such as max-cost (infinity-norm) and sum-of-costs (1-norm), and more generally all p-norms and Top-k-norms (sum of k largest coordinates in absolute value). Closure properties under taking nonnegative linear combinations and pointwise maximums offer versatility to this class of norms. In particular, the latter closure-property can be used to incorporate multiple norm budget constraints through a single norm-minimization objective. Our chief contribution is a framework for designing approximation algorithms for stochastic minimum norm optimization, a significant generalization of the framework of Chakrabarty and Swamy [5] for the deterministic version of StochNormOpt. Our framework has two key components: (i) A reduction from the minimization of expected f-norm to the simultaneous minimization of a (small) collection of expected Top-k-norms; and (ii) Showing how to tackle the minimization of a single expected Top-k-norm by leveraging techniques used to deal with minimizing the expected maximum, circumventing the difficulties posed by the non-separable nature of Top-k norms. We apply our framework to obtain approximation algorithms for stochastic min-norm versions of load balancing (StochNormLB) and spanning tree (StochNormTree) problems. We highlight the following approximation guarantees. 1) An O(1)-approximation for StochNormLB on unrelated machines with: (i) arbitrary monotone symmetric norms and job sizes that are weighted Bernoulli random variables; and (ii) Top-k norms and arbitrary job-size distributions. 2) An O(log log m/log log log m)-approximation for general StochNormLB, where m is the number of machines. 3) For identical machines, the above approximation guarantees are in fact simultaneous approximations that hold with respect to every monotone, symmetric norm. 4) An O(1)-approximation for StochNormTree with an arbitrary monotone, symmetric norm and arbitrary edge-weight distributions; this guarantee extends to stochastic minimum-norm matroid basis. We also consider the special setting of StochNormOpt when the underlying random variables follow Poisson distributions. Our main result here is a novel and powerful reduction showing that, in essence, the stochastic minimum-norm problem can be reduced to a deterministic min-norm version of the same problem. Applying this reduction to (Poisson versions of) spanning tree and load balancing problems yields: (i) an optimal algorithm for StochNormTree; (ii) an almost 2-approximation for StochNormLB when the machines are unrelated, and (iii) a PTAS for StochNormLB when the machines are identical. Results (ii) and (iii) utilize approximation algorithms for (deterministic) min-norm load balancing from the work of Ibrahimpur and Swamy [19] in a black-box fashion

Online Algorithms for Dynamic Resource Allocation Problems

Dynamic resource allocation problems are everywhere. Airlines reserve flight seats for those who purchase flight tickets. Healthcare facilities reserve appointment slots for patients who request them. Freight carriers such as motor carriers, railroad companies, and shipping companies pack containers with loads from specific origins to destinations. We focus on optimizing such allocation problems where resources need to be assigned to customers in real time. These problems are particularly difficult to solve because they depend on random external information that unfolds gradually over time, and the number of potential solutions is overwhelming to search through by conventional methods. In this dissertation, we propose viable allocation algorithms for industrial use, by fully leveraging data and technology to produce gains in efficiency, productivity, and usability of new systems. The first chapter presents a summary of major methodologies used in modeling and algorithm design, and how the methodologies are driven by the size of accessible data. Chapters 2 to 5 present genuine research results of resource allocation problems that are based on Wang and Truong (2017); Wang et al. (2015); Stein et al. (2017); Wang et al. (2016). The algorithms and models cover problems in multiple industries, from a small clinic that aims to better utilize its expensive medical devices, to a technology giant that needs a cost-effective, distributed resource-allocation algorithm in order to maintain the relevance of its advertisements to hundreds of millions of consumers

Efficiency and fairness in ambulance planning

Mei, R.D. van der [Promotor]Bhulai, S. [Promotor