Harper-type lower bounds and the bandwidths of the compositions of graphs

AbstractWe give a general Harper-type lower bound for the bandwidth of a graph which is a common generalization of several known results. As applications we get a lower bound for the bandwidth of the composition of two graphs. By using this we determine the bandwidths of some composition graphs such as (Pr × Ps)[H], (Pr × Cs)[H] (2r ≠ s), (Cr × Cs)[H] (6 ⩽ 2r ⩽ s), etc., for any graph H. Interestingly, the bandwidths of these graphs have nothing to do with the structure of H in general

On-line scheduling with delivery time on a single batch machine

AbstractWe consider a single batch machine on-line scheduling problem with jobs arriving over time. A batch processing machine can handle up to B jobs simultaneously as a batch, and the processing time for a batch is equal to the longest processing time among the jobs in it. Each job becomes available at its arrival time, which is not known in advance, and its characteristics, such as processing time and delivery time, become known at its arrival. Once the processing of a job is completed we deliver it to the destination. The objective is to minimize the time by which all jobs have been delivered. In this paper, we deal with two variants: the unbound model where B is sufficiently large and the bounded model where B is finite. We provide on-line algorithms with competitive ratio 2 for the unbounded model and with competitive ratio 3 for the bounded model. For when each job has the same processing time, we provide on-line algorithms with competitive ratios (5+1)/2, and these results are the best possible

Spectral radius and spanning trees of graphs

For integer $k\geq2,$ a spanning $k$-ended-tree is a spanning tree with at most $k$ leaves. Motivated by the closure theorem of Broersma and Tuinstra [Independence trees and Hamilton cycles, J. Graph Theory 29 (1998) 227--237], we provide tight spectral conditions to guarantee the existence of a spanning $k$-ended-tree in a connected graph of order $n$ with extremal graphs being characterized. Moreover, by adopting Kaneko's theorem [Spanning trees with constraints on the leaf degree, Discrete Appl. Math. 115 (2001) 73--76], we also present tight spectral conditions for the existence of a spanning tree with leaf degree at most $k$ in a connected graph of order $n$ with extremal graphs being determined, where $k\geq1$ is an integer

Single machine scheduling with release dates and job delivery to minimize the makespan

AbstractIn single machine scheduling with release dates and job delivery, jobs are processed on a single machine and then delivered by a capacitated vehicle to a single customer. Only one vehicle is employed to deliver these jobs. The vehicle can deliver at most c jobs at a shipment. The delivery completion time of a job is defined as the time at which the delivery batch containing the job is delivered to the customer and the vehicle returns to the machine. The objective is to minimize the makespan, i.e., the maximum delivery completion time of the jobs. When preemption is allowed to all jobs, we give a polynomial-time algorithm for this problem. When preemption is not allowed, we show that this problem is strongly NP-hard for each fixed c≥1. We also provide a 53-approximation algorithm for this problem, and the bound is tight

Even cycles and perfect matchings in claw-free plane graphs

Lov{\'a}sz showed that a matching covered graph $G$ has an ear decomposition starting with an arbitrary edge of $G$. Let $G$ be a graph which has a perfect matching. We call $G$ cycle-nice if for each even cycle $C$ of $G$, $G-V(C)$ has a perfect matching. If $G$ is a cycle-nice matching covered graph, then $G$ has ear decompositions starting with an arbitrary even cycle of $G$. In this paper, we characterize cycle-nice claw-free plane graphs. We show that the only cycle-nice simple 3-connected claw-free plane graphs are $K_4$, $W_5$ and $\overline C_6$. Furthermore, every cycle-nice 2-connected claw-free plane graph can be obtained from a graph in the family ${\cal F}$ by a sequence of three types of operations, where ${\cal F}$ consists of even cycles, a diamond, $K_4$, and $\overline C_6$.Comment: 12 page

Single machine parallel-batch scheduling with deteriorating jobs

AbstractWe consider several single machine parallel-batch scheduling problems in which the processing time of a job is a linear function of its starting time. We give a polynomial-time algorithm for minimizing the maximum cost, an O(n5) time algorithm for minimizing the number of tardy jobs, and an O(n2) time algorithm for minimizing the total weighted completion time. Furthermore, we prove that the problem for minimizing the weighted number of tardy jobs is binary NP-hard

Interfacial Stress in the Development of Biologics: Fundamental Understanding, Current Practice, and Future Perspective

Biologic products encounter various types of interfacial stress during development, manufacturing, and clinical administration. When proteins come in contact with vapor–liquid, solid–liquid, and liquid–liquid surfaces, these interfaces can significantly impact the protein drug product quality attributes, including formation of visible particles, subvisible particles, or soluble aggregates, or changes in target protein concentration due to adsorption of the molecule to various interfaces. Protein aggregation at interfaces is often accompanied by changes in conformation, as proteins modify their higher order structure in response to interfacial stresses such as hydrophobicity, charge, and mechanical stress. Formation of aggregates may elicit immunogenicity concerns; therefore, it is important to minimize opportunities for aggregation by performing a systematic evaluation of interfacial stress throughout the product development cycle and to develop appropriate mitigation strategies. The purpose of this white paper is to provide an understanding of protein interfacial stability, explore methods to understand interfacial behavior of proteins, then describe current industry approaches to address interfacial stability concerns. Specifically, we will discuss interfacial stresses to which proteins are exposed from drug substance manufacture through clinical administration, as well as the analytical techniques used to evaluate the resulting impact on the stability of the protein. A high-level mechanistic understanding of the relationship between interfacial stress and aggregation will be introduced, as well as some novel techniques for measuring and better understanding the interfacial behavior of proteins. Finally, some best practices in the evaluation and minimization of interfacial stress will be recommended

Transportation and Batching Scheduling for Minimizing Total Weighted Completion Time

We consider the coordination of transportation and batching scheduling with one single vehicle for minimizing total weighted completion time. The computational complexity of the problem with batch capacity of at least 2 was posed as open in the literature. For this problem, we show the unary NP-hardness for every batch capacity at least 3 and present a polynomial-time 3-approximation algorithm when the batch capacity is at least 2