251 research outputs found
Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines
In the paper we consider the problem of scheduling identical jobs on 4
uniform machines with speeds respectively.
Our aim is to find a schedule with a minimum possible length. We assume that
jobs are subject to some kind of mutual exclusion constraints modeled by a
bipartite incompatibility graph of degree , where two incompatible jobs
cannot be processed on the same machine. We show that the problem is NP-hard
even if . If, however, and ,
, then the problem can be solved to optimality in time
. The same algorithm returns a solution of value at most 2 times
optimal provided that . Finally, we study the case and give an -time -approximation algorithm in
all such situations
The Damage Is Done: Ordering a New Trial Based Only on Damages
Federal Rule of Civil Procedure 59(a) allows trial courts to grant new trials to any or all of the parties, on any or all of the issues, including damages. However, the federal circuits are split on how to handle new trials based solely on damages. One croup of circuits grants partial new trials on damages alone only if the erroneous damage amount did not in any way affect the determination of any other issue. Under this standard, a new trial on damages is allowed when the second jury can evaluate the first damage award without also re-examining other issues, such as liability. Another group of courts, however, permits a new trial on damages even where the issue of damages is interwoven with other issues, and allows both parties to present evidence and testimony from the first trial if the re-presented evidence or testimony is related to damages in some way. This article argues that courts should follow the latter approach because it prevents injustice to one or both parties by making it easier to obtain a new trial based solely on damages, and it allows both parties to present evidence necessary to their arguments
Quantitative Assessment of Seven Transmembrane Receptors (7TMRs) Oligomerization by Bioluminescence Resonance Energy Transfer (BRET) Technology
On bipartization of cubic graphs by removal of an independent set
We study a new problem for cubic graphs: bipartization of a cubic graph Q by deleting sufficiently large independent set I. It can be expressed as follows: Given a connected n-vertex tripartite cubic graph Q = (V, E) with independence number Ξ±(Q), does Q contain an independent set I of size k such that Q β I is bipartite? We are interested for which value of k the answer to this question is affirmative. We prove constructively that if Ξ±(Q) β₯ 4n/10, then the answer is positive for each k fulfilling β(n β Ξ±(Q))/2β β€ k β€ Ξ±(Q). It remains an open question if a similar construction is possible for cubic graphs with Ξ±(Q) \u3c 4n/10. Next, we show that this problem with Ξ±(Q) β₯ 4n/10 and k fulfilling inequalities βn/3β β€ k β€ Ξ±(Q) can be related to semi-equitable graph 3-coloring, where one color class is of size k, and the subgraph induced by the remaining vertices is equitably 2-colored. This means that Q has a coloring of type (k, β(n β k)/2β, β(n β k)/2β)
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