Estimation of vertex cover in a constant boundsry

I would really appreciate your assistance with this:

for the following function: $ f\left(G,v\right)\:=\:size\:of\:minimal\:vertex\:cover\:v\:belongs\:to$ .

The function gets an undirected graph G and a vertex v and returns a natural number, which is the size of the smallest vertex cover in G that v belongs to.

problem: proving that if it is possible to estimate f in a constant boundary of 5 in polynomial time, then P=NP. meaning, if it is possible to compute in polynomial time a function $ g(G,v)$ and it is guaranteed that $ f(G,v)-5 \leq g(G,v) \leq f(G,v) + 5$ then P=NP.

I don’t understand why it happens and why if known that $ f(G,v)$ can be computed in polynomial time and $ f(G,v)-5 \leq g(G,v) \leq f(G,v) + 5$ then P=NP

What is an $O(n \log(n))$ binary sorting algorithm with a guaranteed low scaling constant on the run-time?

Let $ O_c(f(n))$ denote that $ c$ is the scaling constant for the run-time (e.g. $ \text{run time} \leq c\cdot f(n) + B$ if $ n$ is large enough)

The absolute lower limit on the run-time for a binary sorting algorithm is $ \log_2(n!) \in O_{1}(n \log_2(n))$ .

My question is, is there an actual sorting algorithm guaranteed to achieve that optimal scaling of $ c=1$ on average? Or what’s the lowest possible scaling constant? And what sorting algorithm achieves the lowest (or a very low) scaling constant?

As a baseline, Quicksort achieves a scaling constant of $ c = 2 \ln(2) \approx 1.39$ , as explained on Wikipedia (see recurrences subsection).

what happens to max flow if we decrease the capacity of every edge by some constant?

Given a graph $ G = (V,A)$ , with source $ s$ , sink $ t$ , edge capacity larger than 1 (but not all equal), I know that if we decrease the capacity of one edge by 1, the $ s,t$ -maximum flow decreases by at most 1. But I would like to know what happens to max flow if we decrease (or increase) the capacity of all edges by 1. I’d appreciate any comments/insights on this. Thanks!

Conditionally returning a constant if function returns Undefined

Often times functions might return Undefined in Mathematica, e.g. when the Volume@RegionIntersection is called for 2 objects which have no overlapping volume. A simple If statement checking if the return has been indeed Undefined does not seem to work. For example, a=Undefined; If[a == Undefined, Print["yes"]] ends up printing the command itself.

  • How can we check if a variable (or a function return) such as a has been assigned as Undefined and assign a different value to it in that case?

Move Unity NavMeshAgent at a constant speed

My NavMeshAgent randomly changes speed while moving along it’s path. It seems to be slower when a segment of a path is shorter (between two waypoints/corners) or when there are many close continuous corners ahead. I have tried changing the acceleration and angular speed but that didn’t work. I have also tried changing the agent.velocity and even though I set it to vectors of the same magnitude each frame, it still doesn’t move at a constant speed. I have also checked if everything is on the same Y coordinates since my game is top down orthographic.

Why can we ignore the constant factor in Weis’s proof of the Master Theorem

In the 4th edition of his Data Structures textbook, Weis gives a proof of part of the Master Theorem. This proof says “Let us … ignore the constant factor in $ \theta(N^k)$ … I don’t understand why it ignoring that constant is valid. (I know that the real Master Theorem is broader than the version presented in the text. I’m just having trouble understanding the argument presented in the text.)

Specifically, the relevant part of Theorem 10.6 on page 469 is

The solution to the equation $ T(N) = aT(N/b) + \Theta(N^k)$ is $ T(N) = O(N^{log_b} a)$ if $ a > b^k$

The proof includes the phrase “ignore the constant factor in $ \theta(N^k)$ …” and then goes on to use a telescoping sum to get

$ T(N) = T(b^m) = a^m \sum_{i=0}^m (\frac{b^k}{a})^i$

At this point, he argues that because $ a > b^k$ , then the sum is a geometric series with a ratio less than 1. However, if we don’t ignore the constant in the $ \Theta$ expression, isn’t there a constant inside the summation that would affect whether the geometric series converges?

Constant console errors in SharePoint Online

I just finished building my first SharePoint Online page. Just a simple page with a few standard web parts (image, news, document library, list, events, file embed). Everything seems to work okay (except for what appears to be a margin-right CSS issue on the news web part) but I’m getting a nonstop trickle of console errors coming in the longer I stay on the page.

For example, I’m in Chrome and did a fresh reload of the page when starting to type up this question, and in 2-3 minutes I’m up to 85 errors and 12 warnings. The errors are all over the place from

Warning! Use of this tool exposes you to potential security threats which can result in others gaining access to your personal Office 365 data (documents, emails, conversations and more). Make sure you trust the person or organization that asked you to access this tool before proceeding.

warning security threat

to mixed content errors related to a browser.pipe.aria.microsoft.com:

Mixed Content: The page at ” was loaded over HTTPS, but requested an insecure XMLHttpRequest endpoint ”. This request has been blocked; the content must be served over HTTPS.

to errors that seem to be related to the PowerPoint embed I have:

Access to XMLHttpRequest at ‘https://browser.pipe.aria.microsoft.com/ping’ from origin ‘https://powerpoint.officeapps.live.com’ has been blocked by CORS policy: No ‘Access-Control-Allow-Origin’ header is present on the requested resource.

Given my level of experience with SharePoint (literally nonexistent), I wouldn’t be totally surprised if I’m doing something wrong but given that I haven’t used any custom web parts or done any other theming, I’m a little perplexed as to what it could be.

Is it normal to get all these console errors?