An O(log⁡n)O(\log n)-approximation for the Set Cover Problem with Set Ownership

In highly distributed Internet measurement systems distributed agents periodically measure the Internet using a tool called {\tt traceroute}, which discovers a path in the network graph. Each agent performs many traceroute measurement to a set of destinations in the network, and thus reveals a portion of the Internet graph as it is seen from the agent locations. In every period we need to check whether previously discovered edges still exist in this period, a process termed {\em validation}. For this end we maintain a database of all the different measurements performed by each agent. Our aim is to be able to {\em validate} the existence of all previously discovered edges in the minimum possible time. In this work we formulate the validation problem as a generalization of the well know set cover problem. We reduce the set cover problem to the validation problem, thus proving that the validation problem is ${\cal NP}$-hard. We present a $O(\log n)$-approximation algorithm to the validation problem, where $n$ in the number of edges that need to be validated. We also show that unless ${\cal P = NP}$ the approximation ratio of the validation problem is $\Omega(\log n)$

MicroED structure of the NaK ion channel reveals a Na+ partition process into the selectivity filter.

Sodium (Na+) is a ubiquitous and important inorganic salt mediating many critical biological processes such as neuronal excitation, signaling, and facilitation of various transporters. The hydration states of Na+ are proposed to play critical roles in determining the conductance and the selectivity of Na+ channels, yet they are rarely captured by conventional structural biology means. Here we use the emerging cryo-electron microscopy (cryoEM) method micro-electron diffraction (MicroED) to study the structure of a prototypical tetrameric Na+-conducting channel, NaK, to 2.5 Å resolution from nano-crystals. Two new conformations at the external site of NaK are identified, allowing us to visualize a partially hydrated Na+ ion at the entrance of the channel pore. A process of dilation coupled with Na+ movement is identified leading to valuable insights into the mechanism of ion conduction and gating. This study lays the ground work for future studies using MicroED in membrane protein biophysics