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.

NP-hard problems but only for n≥3

  • 2-SAT is in P; 3-SAT is NP-complete.
  • Exact cover by 2-sets is in P; exact cover by 3-sets is NP-complete
  • Two-dimensional matching is in P; three-dimensional matching is NP-complete
  • Graph 2-coloring is trivially in P; graph 3-coloring is NP-complete

Are there other well-known or important examples that are similar, in that there is a family of problems with a natural parameter $ n$ , where the problem is known to be in P when $ n<3$ and known to be NP-hard when $ n≥3$ ?

Approximate algorithms for class P problems

As a part of my Algorithm course we studied Approximate Algorithms for NP-complete or NP-hard problems, e.g. “set cover”, “vertex cover”, “load balancing”, etc. My professor asked us as an extra activity to learn an approximate algorithm for a P problem. I searched a lot on google but all I found were NP problems.

I was wondering if anybody can help and tell me a approximate algorithm for a problem with already a exact polynomial algorithm? It will be very much better if the algorithm be short and easy.

Are SAT problems with at most two false clauses NP-complete?

Is the problem of deciding whether a SAT instance, where at most two clauses are false (that is, any given variable assignment will either lead to all clauses being true, all but one, or all but two), is satisfiable solvable in polynomial time?

This is a follow-up from my previous question. The problem presented in that question (at most one false clause) was solvable in polynomial time, because we could take advantage of the fact that the set of truth assignments that make a clause false is disjoint to that of any other clause. With two or more clauses, the sets that make a clause false can overlap with each other. Does this mean that problems with at most two (or more) false clauses (when I say “or more,” I do not mean that the number can change with the problem size. A problem with five clauses where at most five are false is a trivial version of regular SAT, but what about a problem of a few million clauses where are most five can be false?) are NP-complete, or are all versions of SAT where you limit the number of false clauses solvable in polynomial time?

What potential problems are there with dumping dex in bard build? [on hold]

I’ve recently gotten into a group for DnD 5e, and I’m looking for some advice on character build. I’m not new to Tabletop RPGs in general, but I don’t know the tricks of the trade for 5e.

What I’m trying to do:

I’m working on a roleplay build. The character I’m playing is an old gnome, around 400 years old, who is the sort of unofficial mentor to the younger adventurers. An important thing about this character is that I want him to be knowledgable about history, arcana religion and insight, so Intelligence is very important to me unlike most Bards.

I favor a playstyle of getting past obstacles in the most creative way possible, and I prefer the ‘in between’ parts of tabletop roleplay over the fighting.

Ability Scores

STR: 8 DEX: 9 CON: 10 INT: 16 WIS: 14 CHA*: 16

Cantrips:

  • Dancing Lights
  • Vicious Mockery

1st level spells:

  • Healing word
  • Dissonant Whispers
  • Heroism
  • Tasha’s Hideous Laughter

My question:

I’ve read in bard guides like this one (https://www.tribality.com/2016/02/01/dd-5e-character-optimization-bard/) that DEX is the second most important stat for a bard, and INT is a dump stat.

What potential problems are there with dumping DEX in a bard build?

What are some ways I can work around it?

Are there ways I can use INT to my advantage as a bard?

*Note: the DM allows us to buy a 16 for 11 points, which is how I got my Charisma up to +3. The high INT is thanks to my race bonus.

How can I avoid problems that arise from rolling ability scores?

Rolling ability scores is a time-honored tradition across many editions of D&D. However, it can sometimes cause problems for players and/or the DM. For example, one player character may end up much weaker or much stronger than the rest of the party, which can result in a poor experience for some of the players. In other cases, a player may have their characters repeatedly commit suicide-by-monster so they can try to reroll for higher stats, which can be quite frustrating for the GM and other players.

What approaches are available to mitigate these problems?

Note: Answers should ideally be able to prevent both the “Joe rolled all 7s, and his character is useless” problem and the “Karen rolled all 18s and her character makes everyone else’s character useless” problem. That is, an answer that only avoids very low average/total scores is not as good as one that avoids both very low and very high average/total scores.

How can I avoid problems that arise from rolling ability scores?

Rolling ability scores is a time-honored tradition across many editions of D&D. However, it can sometimes cause problems for players and/or the DM. For example, one player character may end up much weaker or much stronger than the rest of the party, which can result in a poor experience for some of the players. In other cases, a player may have their characters repeatedly commit suicide-by-monster so they can try to reroll for higher stats, which can be quite frustrating for the GM and other players.

What approaches are available to mitigate these problems?

Note: Answers should ideally be able to prevent both the “Joe rolled all 7s, and his character is useless” problem and the “Karen rolled all 18s and her character makes everyone else’s character useless” problem. That is, an answer that only avoids very low average/total scores is not as good as one that avoids both very low and very high average/total scores.

How can I avoid problems that arise from rolling ability scores?

Rolling ability scores is a time-honored tradition across many editions of D&D. However, it can sometimes cause problems for players and/or the DM. For example, one player character may end up much weaker or much stronger than the rest of the party, which can result in a poor experience for some of the players. In other cases, a player may have their characters repeatedly commit suicide-by-monster so they can try to reroll for higher stats, which can be quite frustrating for the GM and other players.

What approaches are available to mitigate these problems?

Note: Answers should ideally be able to prevent both the “Joe rolled all 7s, and his character is useless” problem and the “Karen rolled all 18s and her character makes everyone else’s character useless” problem. That is, an answer that only avoids very low average/total scores is not as good as one that avoids both very low and very high average/total scores.

How can I avoid problems that arise from rolling ability scores?

Rolling ability scores is a time-honored tradition across many editions of D&D. However, it can sometimes cause problems for players and/or the DM. For example, one player character may end up much weaker or much stronger than the rest of the party, which can result in a poor experience for some of the players. In other cases, a player may have their characters repeatedly commit suicide-by-monster so they can try to reroll for higher stats, which can be quite frustrating for the GM and other players.

What approaches are available to mitigate these problems?

Note: Answers should ideally be able to prevent both the “Joe rolled all 7s, and his character is useless” problem and the “Karen rolled all 18s and her character makes everyone else’s character useless” problem. That is, an answer that only avoids very low average/total scores is not as good as one that avoids both very low and very high average/total scores.