A PTAS for Minimizing Average Weighted Completion Time With Release Dates on Uniformly Related Machines

A classical scheduling problem is to find schedules that minimize average weighted completion time of jobs with release dates. When multiple machines are available, the machine environments may range from identical machines (the processing time required by a job is invariant across the machines) at one end, to unrelated machines (the processing time required by a job on any machine is an arbitrary function of the specific machine) at the other end of the spectrum. While the problem is strongly NP-hard even in the case of a single machine, constant factor approximation algorithms have been known for even the most general machine environment of unrelated machines. Recently, a polynomial-time approximation scheme (PTAS) was discovered for the case of identical parallel machines [1]. In contrast, it is known that this problem is MAX SNP-hard for unrelated machines [10]. An important open problem is to determine the approximability of the intermediate case of uniformly related machines where each machine i has a speed si and it takes p/si time to executing a job of processing size pIn this paper, we resolve this problem by obtaining a PTAS for the problem. This improves the earlier known ratio of (2 + ∈) for the problem

A PTAS for the Multiple Knapsack Problem

The Multiple Knapsack problem (MKP) is a natural and well known generalization of the single knapsack problem and is defined as follows. We are given a set of n items and m bins (knapsacks) such that each item i has a profit p(i) and a size s(i), and each bin j has a capacity c(j). The goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins. MKP is a special case of the Generalized Assignment problem (GAP) where the profit and the size of an item can vary based on the specific bin that it is assigned to. GAP is APX-hard and a 2-approximation for it is implicit in the work of Shmoys and Tardos [26], and thus far, this was also the best known approximation for MKP. The main result of this paper is a polynomial time approximation scheme for MKP. Apart from its inherent theoretical interest as a common generalization of the well-studied knapsack and bin packing problems, it appears to be the strongest special case of GAP that is not APX-hard. We substantiate this by showing that slight generalizations of MKP are APX-hard. Thus our results help demarcate the boundary at which instances of GAP become APX-hard. An interesting aspect of our approach is a ptas-preserving reduction from an arbitrary instance of MKP to an instance with O(log n) distinct sizes and profits

On Multi-dimensional Packing Problems

We study the approximability of multi-dimensional generalizations of three classical packing problems: multiprocessor scheduling, bin packing, and the knapsack problem. Specifically, we study the vector scheduling problem, its dual problem, namely, the vector bin packing problem, and a class of packing integer programs. The vector scheduling problem is to schedule n d-dimensional tasks on m machines such that the maximum load over all dimensions and all machines is minimized. The vector bin packing problem, on the other hand, seeks to minimize the number of bins needed to schedule all n tasks such that the maximum load on any dimension accross all bins is bounded by a fixed quantity, say 1. Such problems naturally arise when scheduling tasks that have multiple resource requirements. Finally, packing integer programs capture a core problem that directly relates to both vector scheduling and vector bin packing, namely, the problem of packing a miximum number of vectors in a single bin of unit height. We obtain a variety of new algorithmic as well as inapproximability results for these three problems

Water Droplet Impact And Spreading On A Narrow Gap

Paper presented at 2018 Canadian Society of Mechanical Engineers International Congress, 27-30 May 2018.The impact and spreading of a water droplet on a gap between two parallel plates has been studied experimentally. A deionized water droplet (2.03 mm diameter) impacted the plates at velocities of 0.06, 0.5, 1.0, and 1.5 m/s; the tested gap spacings of the parallel glass plates were 50, 100, and 150 μm. Using a high-speed camera, we simultaneously photographed the drop spreading both above and within the gap. We show that water begins to penetrate the gap immediately after impact. On the largest spacing tested, up to 10% of the initial drop volume can penetrate the gap before the maximum spreading diameter is reached

Algorithms for Minimizing Weighted Flow Time

We study the problem of minimizing weighted flow time on a single machine in the preemptive setting. We present an O(log2 P)-competitive semi-online algorithm where P is the ratio of the maximum and minimum processing times of jobs in the system. In the offline setting we show that a (2 + ε)-approximation is achievable in quasi-polynomial time. These are the first non-trivial results for the weighted versions of minimizing flow time. For multiple machines we show that no competitive randomized online algorithm exists for weighted flow time. We also present an improved online algorithm for minimizing total stretch (a special case of weighted flow time) on multiple machines

Potential of Laceyella sacchari strain B42 crude xylanase in biobleaching of kraft pulp

Xylanase producing thermophilic actinomycetes strain B42 was isolated from bagasse. This strain was enriched on oat spelt xylan agar medium and screened onto xylan-congo red agar plate by the xylanolysis method. The phylogenetic analysis using 16S rDNA sequence data confirmed that strain B42 showed highest homology (99.0%) with Laceyella sacchari and was identified as Laceyella sacchari strain B42. Maximal xylanase production was achieved at the incubation period of 48 h with the xylanase and cellulase activities as 24.5 and 0.08 U/ml, respectively. The optimum pH and optimum temperature of L. sacchari strain B42 xylanase was found to be 9.0 and 60°C, respectively. Xylanase was thermostable at 60°C for 1 h and retained 90% of its activity up to 6 h at this temperature, and subsequently enzyme retained 75 and 60% activity at 70 to 80°C, respectively after 6 h. At biobleaching of kraft pulp, enzyme released sufficient amount of phenolic and hydrophobic compounds. The ultraviolet (UV absorption spectrum of the compounds released by enzyme treatment indicated the presence of lignin in the released coloring matter. The enzymatic biobleaching of kraft pulp caused ~12% reduction of kappa number, 6.67 fold releases of reducing sugars and 10% decrease of lignin content at xylanase optimum dose (60 U/g) and time (4 h).Keywords: Biobleaching, kappa number, pulp and paper industry, thermostable, xylanase, 16S rDNAAfrican Journal of Biotechnology Vol. 12(6), pp. 570-57

Edge-Disjoint Paths in Planar Graphs

We study the maximum edge-disjoint paths problem (MEDP). We are given a graph G = (V,E) and a set Τ = {s1t1, s2t2, . . . , sktk} of pairs of vertices: the objective is to find the maximum number of pairs in Τ that can be connected via edge-disjoint paths. Our main result is a poly-logarithmic approximation for MEDP on undirected planar graphs if a congestion of 2 is allowed, that is, we allow up to 2 paths to share an edge. Prior to our work, for any constant congestion, only a polynomial-factor approximation was known for planar graphs although much stronger results are known for some special cases such as grids and grid-like graphs. We note that the natural multicommodity flow relaxation of the problem has an integrality gap of Ω(√|V|) even on planar graphs when no congestion is allowed. Our starting point is the same relaxation and our result implies that the integrality gap shrinks to a poly-logarithmic factor once 2 paths are allowed per edge. Our result also extends to the unsplittable flow problem and the maximum integer multicommodity flow problem. A set X ⊆ V is well-linked if for each S ⊂ V , |δ(S)| ≥ min{|S ∩ X|, |(V - S) ∩ X|}. The heart of our approach is to show that in any undirected planar graph, given any matching M on a well-linked set X, we can route Ω(|M|) pairs in M with a congestion of 2. Moreover, all pairs in M can be routed with constant congestion for a sufficiently large constant. This results also yields a different proof of a theorem of Klein, Plotkin, and Rao that shows an O(1) maxflow-mincut gap for uniform multicommodity flow instances in planar graphs. The framework developed in this paper applies to general graphs as well. If a certain graph theoretic conjecture is true, it will yield poly-logarithmic integrality gap for MEDP with constant congestion

Transient Performance of a Liquid Desiccant Solar Regenerator

A solar liquid desiccant cooling system uses renewable energy and natural refrigerant (water) which makes it attractive. Its main components are air dehumidifier, solar liquid desiccant regenerator and direct/indirect evaporative cooler. In this cooling system, strong liquid desiccant absorbs water from air in the dehumidifier that must be rejected in the solar regenerator. This makes regeneration of a liquid desiccant one of the key processes in the cooling system that requires thermal energy. The energy needed for regeneration can be obtained from sun with the help of an open type solar collector cum regenerator. It also enables storage of energy in the form of regenerated desiccant for use during non-sunshine hours. Detailed analysis of the performance of this device has prime importance in integrating it with a liquid desiccant air cooling system. This entails the development and experimental performance testing of a liquid desiccant solar regenerator under actual weather conditions over the sunshine hours. A solar collector cum regenerator of effective solar area of 4 m2 was fabricated using corrugated sheet metals, layers of thermocol insulation, insulation wood box, glass supporting frames and glasses. The solar collector cum regenerator was mounted on a metal supporting structure inclined at 14o. The corrugated absorber was coated with iron oxide, before black paint, to minimize corrosion. This paper presents transient regeneration performance of the solar collector cum regenerator in terms of increase in concentration, mass of water evaporated and solar collector cum regenerator efficiency. Desiccant concentration increase, total mass of water evaporated and mean daily solar collector cum regenerator efficiency during regeneration of LiCl and CaCl2 solutions were found to be 0.33-0.46 & 0.31-0.47; 13 & 17 kg and 36 & 43%, respectively. These typical results were obtained from experiments carried out on separate days between 9 am and 4 pm in the month of May. The mass of water evaporated was estimated using an equation derived by applying conservation of solution mass. The experimental procedure and the performance analysis technique used are useful in designing solar components of open cycle liquid desiccant cooling systems

Multi-processor Scheduling to Minimize Flow Time with epsilon Resource Augmentation

We investigate the problem of online scheduling of jobs to minimize flow time and stretch on m identical machines. We consider the case where the algorithm is given either (1 + ε)m machines or m machines of speed (1 + ε), for arbitrarily small ε \u3e 0. We show that simple randomized and deterministic load balancing algorithms, coupled with simple single machine scheduling strategies such as SRPT (shortest remaining processing time) and SJF (shortest job first), are O(poly(1/ε))-competitive for both flow time and stretch. These are the first results which prove constant factor competitive ratios for flow time or stretch with arbitrarily small resource augmentation. Both the randomized and the deterministic load balancing algorithms are non- migratory and do immediate dispatch of jobs. The randomized algorithm just allocates each incoming job to a random machine. Hence this algorithm is non- clairvoyant, and coupled with SETF (shortest elapsed time first), yields the first non-clairvoyant algorithm which is con- stant competitive for minimizing flow time with arbitrarily small resource augmentation. The deterministic algorithm that we analyze is due to Avrahami and Azar. For this algorithm, we show O(1/ε)-competitiveness for total flow time and stretch, and also for their Lp norms, for any fixed p ≥ 1