Optimal Composition Ordering Problems for Piecewise Linear Functions

In this paper, we introduce maximum composition ordering problems. The input is $n$ real functions $f_1,\dots,f_n:\mathbb{R}\to\mathbb{R}$ and a constant $c\in\mathbb{R}$. We consider two settings: total and partial compositions. The maximum total composition ordering problem is to compute a permutation $\sigma:[n]\to[n]$ which maximizes $f_{\sigma(n)}\circ f_{\sigma(n-1)}\circ\dots\circ f_{\sigma(1)}(c)$, where $[n]=\{1,\dots,n\}$. The maximum partial composition ordering problem is to compute a permutation $\sigma:[n]\to[n]$ and a nonnegative integer $k~(0\le k\le n)$ which maximize $f_{\sigma(k)}\circ f_{\sigma(k-1)}\circ\dots\circ f_{\sigma(1)}(c)$. We propose $O(n\log n)$ time algorithms for the maximum total and partial composition ordering problems for monotone linear functions $f_i$, which generalize linear deterioration and shortening models for the time-dependent scheduling problem. We also show that the maximum partial composition ordering problem can be solved in polynomial time if $f_i$ is of form $\max\{a_ix+b_i,c_i\}$ for some constants $a_i\,(\ge 0)$, $b_i$ and $c_i$. We finally prove that there exists no constant-factor approximation algorithm for the problems, even if $f_i$'s are monotone, piecewise linear functions with at most two pieces, unless P=NP.Comment: 19 pages, 4 figure

Scheduling of a parcel delivery system consisting of an aerial drone interacting with public transportation vehicles

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This paper proposes a novel parcel delivery system which consists of a drone and public transportation vehicles such as trains, trams, etc. This system involves two delivery schemes: drone-direct scheme referring to delivering to a customer by a drone directly and drone–vehicle collaborating scheme referring to delivering a customer based on the collaboration of a drone and public transportation vehicles. The fundamental characteristics including the delivery time, energy consumption and battery recharging are modelled, based on which a time-dependent scheduling problem for a single drone is formulated. It is shown to be NP-complete and a dynamic programming-based exact algorithm is presented. Since its computational complexity is exponential with respect to the number of customers, a sub-optimal algorithm is further developed. This algorithm accounts the time for delivery and recharging, and it first schedules the customer which leads to the earliest return. Its computational complexity is also discussed. Moreover, extensive computer simulations are conducted to demonstrate the scheduling performance of the proposed algorithms and the impacts of several key system parameters are investigated

Scheduling Algorithms for Procrastinators

This paper presents scheduling algorithms for procrastinators, where the speed that a procrastinator executes a job increases as the due date approaches. We give optimal off-line scheduling policies for linearly increasing speed functions. We then explain the computational/numerical issues involved in implementing this policy. We next explore the online setting, showing that there exist adversaries that force any online scheduling policy to miss due dates. This impossibility result motivates the problem of minimizing the maximum interval stretch of any job; the interval stretch of a job is the job's flow time divided by the job's due date minus release time. We show that several common scheduling strategies, including the "hit-the-highest-nail" strategy beloved by procrastinators, have arbitrarily large maximum interval stretch. Then we give the "thrashing" scheduling policy and show that it is a \Theta(1) approximation algorithm for the maximum interval stretch.Comment: 12 pages, 3 figure

The safety case and the lessons learned for the reliability and maintainability case

This paper examine the safety case and the lessons learned for the reliability and maintainability case

ERA: A Framework for Economic Resource Allocation for the Cloud

Cloud computing has reached significant maturity from a systems perspective, but currently deployed solutions rely on rather basic economics mechanisms that yield suboptimal allocation of the costly hardware resources. In this paper we present Economic Resource Allocation (ERA), a complete framework for scheduling and pricing cloud resources, aimed at increasing the efficiency of cloud resources usage by allocating resources according to economic principles. The ERA architecture carefully abstracts the underlying cloud infrastructure, enabling the development of scheduling and pricing algorithms independently of the concrete lower-level cloud infrastructure and independently of its concerns. Specifically, ERA is designed as a flexible layer that can sit on top of any cloud system and interfaces with both the cloud resource manager and with the users who reserve resources to run their jobs. The jobs are scheduled based on prices that are dynamically calculated according to the predicted demand. Additionally, ERA provides a key internal API to pluggable algorithmic modules that include scheduling, pricing and demand prediction. We provide a proof-of-concept software and demonstrate the effectiveness of the architecture by testing ERA over both public and private cloud systems -- Azure Batch of Microsoft and Hadoop/YARN. A broader intent of our work is to foster collaborations between economics and system communities. To that end, we have developed a simulation platform via which economics and system experts can test their algorithmic implementations

Composition Orderings for Linear Functions and Matrix Multiplication Orderings

We consider composition orderings for linear functions of one variable. Given $n$ linear functions $f_1,\dots,f_n$ and a constant $c$, the objective is to find a permutation $\sigma$ that minimizes/maximizes $f_{\sigma(n)}\circ\dots\circ f_{\sigma(1)}(c)$. It was first studied in the area of time-dependent scheduling, and known to be solvable in $O(n\log n)$ time if all functions are nondecreasing. In this paper, we present a complete characterization of optimal composition orderings for this case, by regarding linear functions as two-dimensional vectors. We also show several interesting properties on optimal composition orderings such as the equivalence between local and global optimality. Furthermore, by using the characterization above, we provide a fixed-parameter tractable (FPT) algorithm for the composition ordering problem for general linear functions, with respect to the number of decreasing linear functions. We next deal with matrix multiplication orderings as a generalization of composition of linear functions. Given $n$ matrices $M_1,\dots,M_n\in\mathbb{R}^{m\times m}$ and two vectors $w,y\in\mathbb{R}^m$, where $m$ denotes a positive integer, the objective is to find a permutation $\sigma$ that minimizes/maximizes $w^\top M_{\sigma(n)}\dots M_{\sigma(1)} y$. The problem is also viewed as a generalization of flow shop scheduling through a limit. By this extension, we show that the multiplication ordering problem for $2\times 2$ matrices is solvable in $O(n\log n)$ time if all the matrices are simultaneously triangularizable and have nonnegative determinants, and FPT with respect to the number of matrices with negative determinants, if all the matrices are simultaneously triangularizable. As the negative side, we finally prove that three possible natural generalizations are NP-hard: 1) when $m=2$, 2) when $m\geq 3$, and 3) the target version of the problem.Comment: 38 page