Maximal subsets of a point set which fit in a unit disk

Suppose that there are a set $ P$ of $ n$ points on the plane, and let $ P_1, \dots, P_k$ be not necessarily disjoint subsets of $ P$ such that every point in $ P_i|\ 1 \leq i \leq k$ fits inside a unit disk $ D_i$ .

Moreover, each $ P_i$ is maximal. This means that if the corresponding unit disk $ D_i$ moves to cover another point, then one point which was inside the disk will be uncovered.

Here is an example: enter image description here

In the above figure, there are three maximal subsets.

I don’t know whether this problem has a name or was studied before, but my question is:

  1. Can $ k$ be $ O(1)^n$ ?
  2. If not, then can we find those subsets in polynomial time w.r.t. $ n$ ?

Counting a property of every sub-sets

Consider a array $ a$ of length $ n$ and $ a_i<n$ and a array of length $ n$ ,$ b=[0,0….(n$ $ times)]$ .Consider every sub-set of array $ a$ and denote it by $ s$ i need to find a number $ k$ with the largest number of occurrences or the most frequent one, If there are several options, choose the smallest one and then increase $ b[k]$ by one .I need to final $ b$ after considering all sub sets.

My idea is that ,I make a array $ val$ of length $ n$ where $ val[i]$ is number of times $ i$ occurred in array $ a$ .Consider that i am trying to find value for $ b[k]$ then i will create a array $ temp$ of length where $ temp[i]=min(val[k],val[i])$ .Then consider the value of $ val[k]$ to be $ m$ then in a subset element $ k$ can occur from $ 0$ to $ m$ times and using this i will find answer for all of its occurrences from $ 0$ to $ m$ which is a simple number of sub sets problem and finally i will have my answer.

But my algorithms complexity is bad as it quadratic or worse.Could anyone help me.

Finding $l$ subsets such that their intersection has less or equal than $k$ elements NP-complete or in P?

I have a set $ M$ , subsets $ L_1,…,L_m$ and natural numbers $ k,l\leq m$ .

The problem is:

Are there l unique indices $ 1\leq i_1,…,i_l\leq m$ , such that

$ \hspace{5cm}\left|\bigcap_{j=1}^{l} L_{i_{j}}\right| \leq k$

Now my question is whether this problem is $ NP$ -complete or not. What irritates me is the two constraints $ l $ and $ k$ because the NP-complete problems that were conceptually close to it that I took a look on (set cover, vertex cover) only have one constraint respectively that also appears in this problem.

I then tried to write a polynomial time algorithm that looks at which of the sets $ L_1,…,L_m$ share more than $ k$ elements with other sets but even if all sets would share more than $ k$ elements with other this wouldn’t mean that their intersection has more than $ k$ elements…

This question kind of comes close but in it there is no restriction on the amount of subsets to use and the size of the intersection should be exactly $ k$ , but maybe this could be useful anyways.

Can somebody further enlighten me ?

Checking disjointness between subsets of a poset

If there is a poset $ (P, \le)$ and two sets $ X \subseteq P$ and $ Y \subseteq P$ , and we have a way $ f : P^2 \to 2$ to efficiently compute for any $ (x, y) \in P^2$ whether there exists a $ z \in P$ such that $ (x \le z) \wedge (y \le z)$ , we want to return $ \mathbf{T}$ if there exists a pair $ (x, y) \in X \times Y$ such that $ f(x, y) = 1$ and $ \mathbf{F}$ otherwise, using the fewest possible number of calls to $ f$ .

Literature request: Generating all vertex subsets of a graph

I made the same post here in Mathematics Exchange but maybe here is a better place.

I am working in an algorithm which finds a unique maximal independent set of vertices which generates all other vertex subsets. This is done using two measures of ‘importance’ for vertices. I assume this might have some applications outside mathematics, especially in computer science, but I am not sure where I can read about such applications. Does anyone have any good literature suggestions?

Optimal ordering – Dynamic programming on subsets

We have a set T of n elements and m subsets $ R_i \subset T i = 1,…,m$ . The $ S_i$ are not assumed to be different. We also define an ordering of T, a one-to-one mapping $ \pi$ of $ T$ onto the set of integers $ {1,…n}$ . That is, every $ \pi(v)$ is an integer from 1 to n and all $ \pi(v), v \in T$ are distinct. For a set $ R \subset T$ and an ordering of $ \pi$ of $ V$ , let $ l_\pi(S)$ and $ r_\pi(S)$ denote, respectively, the smallest and largest value $ \pi(v)$ among all $ v \in S$ . The problem is to find an ordering $ \pi$ that minimizes $ \sum_{i=1}^m (r_\pi(S_i) – l_\pi(S_i))$ .

This problem is NP-hard but is in FPT and can be solved with dynamic programming on subsets in $ O^*(2^n)$

The idea here is to optimize the ordering of the first $ k$ items and then go from $ k$ to $ k+1$ items and prove that the time complexity does not change when doing so.

But how do we optimize the ordering for the first $ k$ items in just $ O^*(2^n)$ time? It seems like the same problem as the initial one to me.

Functions cumputable in $\theta(n)$ and $\theta(|S| \cdot n)$ for calculating subsets of size |S|

I need to define a function $ f: \mathbb{N} \rightarrow \{0,1 \}$ that is computable (by a deterministic TM) in $ \theta(n)$ worst case time and such that the time complexity for calculating $ f(S)$ is $ \theta(|S| \cdot n)$ for all finite $ S \subset \mathbb{N}$ where $ S$ is of the form $ S = \{0,1,2, \dots ,k \}$ for some $ k \in \mathbb{N}$

A naive attempt was to define $ f$ by the positiveness of a polynomial of degree $ n$ but the problem is that it is unknown whether computing the roots of a polynomial of degree $ n$ can be done in polynomial time. If it can be done in polynomial time then $ f(S)$ can be done in $ \theta(n) \neq \theta(|S| \cdot n)$ worst case time by summing the size of ranges of positiveness.

Is there any better approach?