Optimizing egalitarian performance in the side-effects model of colocation for data center resource management

In data centers, up to dozens of tasks are colocated on a single physical machine. Machines are used more efficiently, but tasks' performance deteriorates, as colocated tasks compete for shared resources. As tasks are heterogeneous, the resulting performance dependencies are complex. In our previous work [18] we proposed a new combinatorial optimization model that uses two parameters of a task - its size and its type - to characterize how a task influences the performance of other tasks allocated to the same machine. In this paper, we study the egalitarian optimization goal: maximizing the worst-off performance. This problem generalizes the classic makespan minimization on multiple processors (P||Cmax). We prove that polynomially-solvable variants of multiprocessor scheduling are NP-hard and hard to approximate when the number of types is not constant. For a constant number of types, we propose a PTAS, a fast approximation algorithm, and a series of heuristics. We simulate the algorithms on instances derived from a trace of one of Google clusters. Algorithms aware of jobs' types lead to better performance compared with algorithms solving P||Cmax. The notion of type enables us to model degeneration of performance caused by using standard combinatorial optimization methods. Types add a layer of additional complexity. However, our results - approximation algorithms and good average-case performance - show that types can be handled efficiently.Comment: Author's version of a paper published in Euro-Par 2017 Proceedings, extends the published paper with addtional results and proof

A Technique for Obtaining True Approximations for kk-Center with Covering Constraints

There has been a recent surge of interest in incorporating fairness aspects into classical clustering problems. Two recently introduced variants of the $k$-Center problem in this spirit are Colorful $k$-Center, introduced by Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the Fair Robust $k$-Center problem introduced by Harris, Pensyl, Srinivasan, and Trinh. To address fairness aspects, these models, compared to traditional $k$-Center, include additional covering constraints. Prior approximation results for these models require to relax some of the normally hard constraints, like the number of centers to be opened or the involved covering constraints, and therefore, only obtain constant-factor pseudo-approximations. In this paper, we introduce a new approach to deal with such covering constraints that leads to (true) approximations, including a $4$-approximation for Colorful $k$-Center with constantly many colors---settling an open question raised by Bandyapadhyay, Inamdar, Pai, and Varadarajan---and a $4$-approximation for Fair Robust $k$-Center, for which the existence of a (true) constant-factor approximation was also open. We complement our results by showing that if one allows an unbounded number of colors, then Colorful $k$-Center admits no approximation algorithm with finite approximation guarantee, assuming that $\mathrm{P} \neq \mathrm{NP}$. Moreover, under the Exponential Time Hypothesis, the problem is inapproximable if the number of colors grows faster than logarithmic in the size of the ground set

Interval Selection in the Streaming Model

A set of intervals is independent when the intervals are pairwise disjoint. In the interval selection problem we are given a set $\mathbb{I}$ of intervals and we want to find an independent subset of intervals of largest cardinality. Let $\alpha(\mathbb{I})$ denote the cardinality of an optimal solution. We discuss the estimation of $\alpha(\mathbb{I})$ in the streaming model, where we only have one-time, sequential access to the input intervals, the endpoints of the intervals lie in $\{1,...,n \}$, and the amount of the memory is constrained. For intervals of different sizes, we provide an algorithm in the data stream model that computes an estimate $\hat\alpha$ of $\alpha(\mathbb{I})$ that, with probability at least $2/3$, satisfies $\tfrac 12(1-\varepsilon) \alpha(\mathbb{I}) \le \hat\alpha \le \alpha(\mathbb{I})$. For same-length intervals, we provide another algorithm in the data stream model that computes an estimate $\hat\alpha$ of $\alpha(\mathbb{I})$ that, with probability at least $2/3$, satisfies $\tfrac 23(1-\varepsilon) \alpha(\mathbb{I}) \le \hat\alpha \le \alpha(\mathbb{I})$. The space used by our algorithms is bounded by a polynomial in $\varepsilon^{-1}$ and $\log n$. We also show that no better estimations can be achieved using $o(n)$ bits of storage. We also develop new, approximate solutions to the interval selection problem, where we want to report a feasible solution, that use $O(\alpha(\mathbb{I}))$ space. Our algorithms for the interval selection problem match the optimal results by Emek, Halld{\'o}rsson and Ros{\'e}n [Space-Constrained Interval Selection, ICALP 2012], but are much simpler.Comment: Minor correction

Non-Preemptive Scheduling on Machines with Setup Times

Consider the problem in which n jobs that are classified into k types are to be scheduled on m identical machines without preemption. A machine requires a proper setup taking s time units before processing jobs of a given type. The objective is to minimize the makespan of the resulting schedule. We design and analyze an approximation algorithm that runs in time polynomial in n, m and k and computes a solution with an approximation factor that can be made arbitrarily close to 3/2.Comment: A conference version of this paper has been accepted for publication in the proceedings of the 14th Algorithms and Data Structures Symposium (WADS

Packing While Traveling: Mixed Integer Programming for a Class of Nonlinear Knapsack Problems

Packing and vehicle routing problems play an important role in the area of supply chain management. In this paper, we introduce a non-linear knapsack problem that occurs when packing items along a fixed route and taking into account travel time. We investigate constrained and unconstrained versions of the problem and show that both are NP-hard. In order to solve the problems, we provide a pre-processing scheme as well as exact and approximate mixed integer programming (MIP) solutions. Our experimental results show the effectiveness of the MIP solutions and in particular point out that the approximate MIP approach often leads to near optimal results within far less computation time than the exact approach

A 0.821-ratio purely combinatorial algorithm for maximum k-vertex cover in bipartite graphs

We study the polynomial time approximation of the max k-vertex cover problem in bipartite graphs and propose a purely combinatorial algorithm that beats the only such known algorithm, namely the greedy approach. We present a computer-assisted analysis of our algorithm, establishing that the worst case approximation guarantee is bounded below by 0.821. © Springer-Verlag Berlin Heidelberg 2016

An EPTAS for Scheduling on Unrelated Machines of Few Different Types

In the classical problem of scheduling on unrelated parallel machines, a set of jobs has to be assigned to a set of machines. The jobs have a processing time depending on the machine and the goal is to minimize the makespan, that is the maximum machine load. It is well known that this problem is NP-hard and does not allow polynomial time approximation algorithms with approximation guarantees smaller than $1.5$ unless P$=$NP. We consider the case that there are only a constant number $K$ of machine types. Two machines have the same type if all jobs have the same processing time for them. This variant of the problem is strongly NP-hard already for $K=1$. We present an efficient polynomial time approximation scheme (EPTAS) for the problem, that is, for any $\varepsilon > 0$ an assignment with makespan of length at most $(1+\varepsilon)$ times the optimum can be found in polynomial time in the input length and the exponent is independent of $1/\varepsilon$. In particular we achieve a running time of $2^{\mathcal{O}(K\log(K) \frac{1}{\varepsilon}\log^4 \frac{1}{\varepsilon})}+\mathrm{poly}(|I|)$, where $|I|$ denotes the input length. Furthermore, we study three other problem variants and present an EPTAS for each of them: The Santa Claus problem, where the minimum machine load has to be maximized; the case of scheduling on unrelated parallel machines with a constant number of uniform types, where machines of the same type behave like uniformly related machines; and the multidimensional vector scheduling variant of the problem where both the dimension and the number of machine types are constant. For the Santa Claus problem we achieve the same running time. The results are achieved, using mixed integer linear programming and rounding techniques

Capacitated Center Problems with Two-Sided Bounds and Outliers

In recent years, the capacitated center problems have attracted a lot of research interest. Given a set of vertices $V$, we want to find a subset of vertices $S$, called centers, such that the maximum cluster radius is minimized. Moreover, each center in $S$ should satisfy some capacity constraint, which could be an upper or lower bound on the number of vertices it can serve. Capacitated $k$-center problems with one-sided bounds (upper or lower) have been well studied in previous work, and a constant factor approximation was obtained. We are the first to study the capacitated center problem with both capacity lower and upper bounds (with or without outliers). We assume each vertex has a uniform lower bound and a non-uniform upper bound. For the case of opening exactly $k$ centers, we note that a generalization of a recent LP approach can achieve constant factor approximation algorithms for our problems. Our main contribution is a simple combinatorial algorithm for the case where there is no cardinality constraint on the number of open centers. Our combinatorial algorithm is simpler and achieves better constant approximation factor compared to the LP approach

Interference-Aware Scheduling Using Geometric Constraints

The large scale parallel and distributed platforms produce a continuously increasing amount of data which have to be stored, exchanged and used by various jobs allocated on different nodes of the platform. The management of this huge communication demand is crucial for the performance of the system. Meanwhile, we have to deal with more interferences as the trend is to use a single all-purpose interconnection network. In this paper, we consider two different types of communications: the flows induced by data exchanges during computations and the flows related to Input/Output operations. We propose a general model for interference-aware scheduling, where explicit communications are replaced by external topological constraints. Specifically, we limit the interferences of both communication types by adding geometric constraints on the allocation of jobs into machines. The proposed constraints reduce implicitly the data movements by restricting the set of possible allocations for each job. We present this methodology on the case study of simple network topologies, namely the line and the ring. We propose theoretical lower and upper bounds under different assumptions with respect to the platform and jobs characteristics. The obtained results illustrate well the difficulty of the problem even on simple topologies