Total completion time minimization in a 2-stage differentiation flowshop with fixed sequences per job type

This paper addresses the total completion time minimization in a two-stage differentiation flowshop where the sequences of jobs per type are predetermined. The two-stage differentiation flowshop consists of a stage-1 common machine and m stage-2 parallel dedicated machines. The goal is to determine an optimal interleaved processing sequence of all jobs at the first stage. We propose an dynamic programming algorithm, where nk is the number of type-k jobs. The running time is polynomial when m is constant.     

A simpler competitive analysis for scheduling equal-length jobs on one machine with restarts

We consider the online problem of scheduling jobs with equal processing times on a single machine. Each job has a release time and a deadline, and the goal is to maximize the number of jobs completed by their deadlines. Chrobak et al. (2007, SICOMP 36:6) introduce a preempt-restart model in which progress toward completing a preempted job is lost, yet that job can be restarted from scratch. They provide a -competitive deterministic algorithm and show that this is the optimal competitiveness. Their analysis is based on a complex charging scheme among individual jobs and the use of several partial functions and mappings for assigning the charges. In this paper, we provide an alternative proof of the result using a more global potential argument to compare the relative progress of the algorithm versus the optimal schedule over time. This new proof is significantly simpler and more intuitive that the original, and our technique is applicable to related problems.     

Explicit convergence rates for MRAC-type systems

It is well known since many years ago that for the linear time-varying system , with A Hurwitz, and B(t) bounded and globally Lipschitz, it is necessary and sufficient for global exponential stability, that B(t) satisfy the so-called persistency of excitation condition. In this note, we revisit this question and provide explicit bounds for the convergence rate and on the overshoot of the transient behaviour of the solutions e(t), θ(t) as functions of the richness of B(·). We believe that the result that we present is useful since knowing convergence rates aids in the construction of converse Lyapunov functions. Moreover, the type of systems that we study here appear for instance in model reference adaptive control (MRAC).     

Resource constrained scheduling on multiple machines

This paper is devoted to Resource Constrained Scheduling. An instance for this problem is given by a set of n jobs with lengths and weights and a set of m machines with capacities. At every time each machine can run arbitrary many jobs in parallel but the total weight of these jobs must not exceed the capacity of the machine. Furthermore the m machines work in parallel and we wish to find a schedule that minimizes the makespan or the sum of completion times. Thus load balancing on a cluster of servers is a typical application for scheduling under resource constraints. For both measures the problem is -complete even for m=1. We show that the problem is approximable within factors of 2 (makespan) and 14.85 (sum of completion times) for arbitrary m.     

A best online algorithm for unbounded parallel-batch scheduling with restarts to minimize makespan

We consider online scheduling with restarts in an unbounded parallel-batch processing system to minimize the makespan. By online we mean that jobs arrive over time and all the information on a job is unknown before its arrival time (release date) and restart means that a running batch may be interrupted, losing all the work done on it, and the jobs in the interrupted batch are released and become independently unscheduled jobs. It is known in the literature that the considered problem has no online algorithm with a competitive ratio less than \((5-\sqrt{5})/2\). We give an online algorithm for the considered problem with a competitive ratio \((5-\sqrt{5})/2\approx 1.382\).     

Online batch scheduling on parallel machines with delivery times

We study the online batch scheduling problem on parallel machines with delivery times. Online algorithms are designed on m parallel batch machines to minimize the time by which all jobs have been delivered. When all jobs have identical processing times, we provide the optimal online algorithms for both bounded and unbounded versions of this problem. For the general case of processing time on unbounded batch machines, an online algorithm with a competitive ratio of 2 is given when the number of machines m=2 or m=3, respectively. When m≥4, we present an online algorithm with a competitive ratio of 1.5+o(1).     

Approximation and inapproximability results for maximum clique of disc graphs in high dimensions

We prove algorithmic and hardness results for the problem of finding the largest set of a fixed diameter in the Euclidean space. In particular, we prove that if A^∗ is the largest subset of diameter r of n points in the Euclidean space, then for every ε>0 there exists a polynomial time algorithm that outputs a set B of size at least |A^∗| and of diameter at most . On the hardness side, roughly speaking, we show that unless P=NP for every ε>0 it is not possible to guarantee the diameter for B even if the algorithm is allowed to output a set of size .     

Online chasing problems for regular polygons

We consider a server location problem with only one server to move. In this paper we assume that a request is given as a region and that the service can be done anywhere inside the region. Namely, for each request an online algorithm chooses an arbitrary point in the region and moves the server there. Note that if every request is a single point and the server must exactly go there in the given order as conventional server problems, there is no choice for the online player and the problem is trivial. Our main result shows that if the region is a regular n-gon, the competitive ratio of the greedy algorithm is for odd n, and for even n. In particular for a square region, the greedy algorithm turns out to be optimal.     

Three results on frequency assignment in linear cellular networks

In the frequency assignment problem we are given a graph representing a wireless network and a sequence of requests, where each request is associated with a vertex. Each request has two more attributes: its arrival and departure times, and it is considered active from the time of arrival to the time of departure. We want to assign frequencies to all requests so that at each time step any two active requests associated with the same or adjacent vertices use different frequencies. The objective is to minimize the number of frequencies used.We focus exclusively on the special case of the problem when the underlying graph is a linear network (path). For this case, we consider both the offline and online versions of the problem, and we present three results. First, in the incremental online case, where the requests arrive over time, but never depart, we give an algorithm with an optimal (asymptotic) competitive ratio . Second, in the general online case, where the requests arrive and depart over time, we improve the current lower bound on the (asymptotic) competitive ratio to . Third, we prove that the offline version of this problem is NP-complete.     

Minimum distance between a canal surface and a simple surface

The computation of the minimum distance between two objects is an important problem in the applications such as haptic rendering, CAD/CAM, NC verification, robotics and computer graphics. This paper presents a method to compute the minimum distance between a canal surface and a simple surface (i.e. a plane, a natural quadric, or a torus) by finding roots of a function of a single parameter. We utilize the fact that the normals at the closest points between two surfaces are collinear. Given the spine curve C(t), t_min≤t≤t_max, and the radius function r(t) for a canal surface, a point on the spine curve uniquely determines a characteristic circle on the surface. Normals to the canal surface at points on form a cone with a vertex and an axis which is parallel to Then we construct a function of t which expresses the condition that the perpendicular from C(t) to a given simple surface is embedded in the cone of normals to the canal surface at points on K(t). By solving this equation, we find characteristic circles which contain the points of locally minimum distance from the simple surface. Based on these circles, we can compute the minimum distance between given surfaces.     

A running time analysis of an Ant Colony Optimization algorithm for shortest paths in directed acyclic graphs

In this paper, we prove polynomial running time bounds for an Ant Colony Optimization (ACO) algorithm for the single-destination shortest path problem on directed acyclic graphs. More specifically, we show that the expected number of iterations required for an ACO-based algorithm with n ants is for graphs with n nodes and m edges, where ρ is an evaporation rate. This result can be modified to show that an ACO-based algorithm for One-Max with multiple ants converges in expected iterations, where n is the number of variables. This result stands in sharp contrast with that of Neumann and Witt, where a single-ant algorithm is shown to require an exponential running time if ρ=O(n−1−ε) for any ε>0.     

One conjecture of bubble-sort graphs

The bubble-sort graph is an important interconnection network designed from Cayley graph model. One conjecture is proposed in Shi and Lu (2008) [10] as follows: for any integer n?2, if n is odd then bubble-sort graph Bn is a union of edge-disjoint hamiltonian cycles; if n is even then bubble-sort graph Bn is a union of edge-disjoint hamiltonian cycles and its perfect matching that has no edges in common with the hamiltonian cycles. In this paper, we prove that conjecture is true for n=5,6.