A Constant Factor Approximation Algorithm for Unsplittable Flow on Paths

In the unsplittable flow problem on a path, we are given a capacitated path $P$ and $n$ tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks, such that for each edge $e$ of $P$, the total demand of selected tasks that use $e$ does not exceed the capacity of $e$. This is a well-studied problem that has been studied under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack and interval packing. We present a polynomial time constant-factor approximation algorithm for this problem. This improves on the previous best known approximation ratio of $O(\log n)$. The approximation ratio of our algorithm is $7+\epsilon$ for any $\epsilon>0$. We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves a special case of the maximum weight independent set of rectangles problem to optimality. In the setting of resource augmentation, wherein the capacities can be slightly violated, we give a $(2+\epsilon)$-approximation algorithm. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either~1,~2, or~3.Comment: 37 pages, 5 figures Version 2 contains the same results as version 1, but the presentation has been greatly revised and improved. References have been adde

Assessing the ability of the 14C projection-age method to constrain the circulation of the past in a 3-D ocean model

Radiocarbon differences between benthic and planktonic foraminifera (B-P ages) and radiocarbon projection ages are both used to determine changes of the past ocean circulation rate. A global 3-D ocean circulation model with a constant modern ocean circulation is used to study which method is less influenced by atmospheric Δ14C variations. Three factors cause uncertainties: first, the long equilibration time of the ocean after atmospheric Δ14C changes; second, different mixing processes in the ocean, which cause an ocean response of smaller amplitude than the atmospheric forcing; and third, the unknown source region and corresponding initial surface 14C reservoir age of subsurface waters. The model suggests that B-P ages and projection ages have lower uncertainties the closer they are to deepwater formation zones. In the North Atlantic the B-P age method is less influenced by atmospheric Δ14C variations than the projection-age method. Projections ages vary less in the Pacific as long as atmospheric Δ14C decreases linearly. A more irregular atmospheric Δ14C evolution leads to age variations of similar magnitude with both methods. On the basis of the model experiment, we suggest a potential improvement of the projection-age method

Nhetoric: Rhetorical power in cyberspace

The Alpha group is a faction of the white supremacy movement that has established a virtual presence in cyberspace. Rhetorical strategies of agitation are practiced by the Alpha group on the World Wide Web in an effort to encourage visiting avatars to join the white supremacy movement. This study explores the rhetorical strategies of power and promulgation which Alpha uses in cyberspace. Analysis of Alpha\u27s digital discourse provide an opportunity to understand and evaluate the unique potentials that information technologies such as the World Wide Web bring to the rhetorical environment

Decay of the (2\u3ci\u3es\u3c/i\u3e\u3csup\u3e2\u3c/sup\u3e2\u3ci\u3ep\u3c/i\u3e)\u3csup\u3e2\u3ci\u3ep\u3c/i\u3e\u3c/sup\u3e Compound State in Helium by Two-Electron Emission

Compound states of atoms are known to decay by the emission of a single electron into many states lying at lower energies. Compound states associated with doubly excited states lie at energies above the ionization potential, and therefore the decay into an ion by the emission of two electrons is energetically possible. This paper presents evidence for such a two-electron decay in the case of the (2s22p)2p compound state of helium which derives from the (2s2)1S atomic state