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Decision tree: how to decide the next node?

data set

I have to decide for which value of “Klasse”

How do i do it?

I know that I have to decide on maximum information gain

So first off I’ve calculated the entropy of “Klasse”

That is E(Klasse)= -(3/11*log(3/11)+3/11*log(3/11)+5/11*log(5/11)) = 1.067

So how do I proceed from that?

I now need to find the first decision node, yes?

And how do I proceed if I’ve found a decision node ?

Thanks

How to resolve fundamental differences in perspective between players and DM about the roles each has in decision making?

To give context: I am a DM running a D&D 5e campaign set in a home-brew setting for the better part of a year. A few months ago I suggested having an alternate “gaiden” style campaign set in the same setting in which the same set of characters go on short adventures with rotating DMs that could be used during sessions where some players were not available to play. During this winter holiday, one of my players was unable to make sessions and I implemented this system with myself taking the first turn as DM. All players were given invitations to the campaign and given character sheets on our role play app, including the non-available player. The adventure was expanded as the players came to enjoy the narrative and I had new ideas, and is now reaching towards its conclusion. My absent player recently became available again and expressed interest in joining in this campaign. I said yes, some members of the group said no.

While the exact argument was long and circuitous with some hurt feelings, the basic salient points from each side were:

Against Joining

  • Inclusion of new members of the campaign are group decisions.
  • Introducing new characters towards the end of the campaign disrupts the narrative.
  • By making an executive decision under DM purview from the get go, I am undermining their perspectives and feelings by “laying down the law” and not listening to their side. (This might be true.)

For Joining

  • The player is part of the pre-existing group and is not a new member as this adventure is an extension of main campaign.
  • While players are encouraged to add to the narrative, what is and is not narrative appropriate falls under the domain of the DM as stated on pages 5 & 6 of the 5e Players Handbook.
  • Additionally, I have interesting narrative plans involved with his character, that several players and DM’s outside of this group approve of.
  • As the DM, I serve as referee. As I view this player as a pre-existing group member, it comes under my purview to make judgement calls how certain player on player issues are resolved. This has been the case before when other players had issues with class abilities overlapping over each others roles and could not come to a compromise, and certain players having issue with role-play issues overshadowing other players. In both cases I had to make official DM statements to resolve the issue.
  • As a DM I have a prerogative to enjoy myself during these sessions else I lose interest in the game and it dies. Excluding a player who I see as part of the group who has done nothing wrong do to what I feel is pressure from a large minority of players does not sit right with me and would impede in my enjoyment.

In the end, I said that I will not exclude anyone member of the group at the behest of the others for what I see as a non-issue, and several player were upset with this decision, seeing it as inappropriate, with at least one player leaving the group outright.

So how do I this resolve fundamental differences in perspective between players and DM about the roles each has in decision making? Am I in the right for making the decision I did? If so, how do I resolve this group dynamic issue? If not, how do I repair these ingresses?

Restriction: polynomial time decision of instance is why needed to “decision Problem”?

I am reading book “combinatorial optimization 3rd edition(Bernhard Korte、 Jens Vygen)”.

(latest version is sixth.)

There are some discriptions in this book that I don’t understand

Not all binary strings are instances of Hamiltonian Circuit but only those representing an undirected graph. For most interesting decision problems the in- stances are a proper subset of the 0-1-strings. We require that we can decide in polynomial time whether an arbitrary string is an instance or not:

  • quote from p350

decision problem is pair P = (X,Y), where X is a language decidable in polynomial time.

  • quote p351

Why decision problem required that decide in polynomial time whether an arbitrary string is an instance or not?

I can’t found any reasons of this restriction in the book.

Using decision oracle to solve optimization problem of maximum polyomino tiling

So, this problem is a kind of variant of polyomino packing which has been discussed frequently elsewhere, but I haven’t been able to find anything on my particular problem. Suppose we have a list of polyominos $ p_1, p_2, …, p_n$ (not necessarily distinct), and we want to find a tiling of a rectangle of dimension $ a \times b$ with $ a, b \leq n$ that maximizes the number of squares covered, where we can use each $ p_i$ at most once, and polyominos must be fully contained within the rectangle. Now, we have the decision problem which tells us, for a given $ t$ , if there is some tiling covering at least $ t$ squares, and the optimization problem which is finding a tiling that covers the maximum number of squares. There are two parts: first, if you can solve the optimization problem in polynomial time, can you solve the decision problem in polynomial time? And secondly, if you can solve the decision problem in polynomial problem, can you solve the optimization problem in polynomial time?

If we have an oracle that solves the optimization in polynomial time, solving the decision problem in polynomial time is easy. However, given an oracle for the decision problem, I was unable to find a way to solve the optimization problem in polynomial time. The main issue I’m facing is that the decision oracle only works for rectangular boards, which means we can’t just place pieces and then use the oracle to see if the placement works, since we won’t have a rectangular board if we want to exclude the piece we just placed. It isn’t hard to determine the actual maximum number of tiles you can cover, and you can even find the actual pieces you need to use, but I haven’t been able to figure out a way to find an arrangement of the pieces in polynomial time using the oracle. I assume there is some trick here, but I don’t see it.

Excerpt length decision

What should I take into account when limiting the text visible on a listing? I’m using a list layout with images on the left side, a title, a meta line and the excerpt, And I’ve wondering if there are any best practices like:

  • limiting the excerpt to full sentences only without breaking the sentences
  • limiting the text to fill n lines
  • limiting to words and add three dots and read more link afterwards
  • limiting the words and using justified text alignment …

What do you think?

Reductions from non decision problems

I want to show a minimization problem $ Y$ has no approximation factor of 1.36. To be more specific the problem $ Y$ is the exemplar distance problem between two genomes. Could I reduce from the min vertex cover problem instead of the decision version of the vertex cover problem. The problem I am having with reducing from the decision version is that a vertex cover of size k maps to the $ Y$ of size $ \leq ck$ , where $ c$ is a constant. A decision version for problem $ Y$ for me makes no sense, as there will always be a brekpoint distance between two genomes. I tried to research on the internet but I always only find reductions from decision problems. Could we reduce from non-decision problems.

Also when doing reductions from the vertex cover problem. I can’t assume the given instance $ G,k$ is such that k is the size of the optimal vertex cover right? $ k$ is just any size of a Vertex cover.