Unanswered Questions
487 questions with no upvoted or accepted answers
32
votes
1
answer
886
views
Largest set of cocircular points
Given $n$ points with integer coordinates in the plane, determine the maximum number of points that lie on the same circle (on its circumference, not its interior).
This can be done in $O(n^3)$ easily ...
CommunityBot
- 1
26
votes
1
answer
784
views
Complexity of deciding whether there is a winning strategy in the following game
The sum divider game for $n$ starts with the set $M_0 = \{1,\dots,n\}$. Player A chooses a number $m_1$ from $M_0 \setminus \{1\}$ and B has to choose a divider $m_2$ of $m_1$ from $M_1 = M_0 \...
10
votes
0
answers
1k
views
Complexity of Sorting Integers on a Multitape Turing Machine
How expensive is sorting integers on a Multitape Turing Machine? Well known sorting algorithms, like quicksort, tend to rely on jumping / indirect-access being cheap. But MTMs have no indirect access.....
- 5,962
9
votes
0
answers
209
views
Complexity of frog game on graphs is exponential, or can we do better?
Frog game initializes by placing one frog on every vertex of a simple connected graph $G$ with $n$ vertices. A move consists of moving all $x\gt 0$ frogs from one vertex to another non-empty vertex to ...
- 117
9
votes
0
answers
372
views
P vs NP and the Time Hierarchy
Assuming $P\neq NP$, is it possible that there exists a $k$ such that $P\subseteq\textsf{NTIME}(t^k)$?
There reason I ask this is that I assume the following:
$$P=NP \implies \forall k\ \exists j.\ \...
CommunityBot
- 1
8
votes
0
answers
1k
views
Non-deterministic time hierarchy theorem: universal TM overhead
I am currently reading the book of Arora and Barak on computational complexity.
In the third chapter (p69-70), two classic theorems regarding time complexity hierarchies are introduced:
$\left[f(n)\...
- 81
7
votes
0
answers
172
views
Overlap Maximization problem
Here's the problem:
I have a collection of collections, $C$, where each $c\in C$ is a collection of sets $X\subset U$. Denote $c_i$ as the i-th $X$ in $c$. Informally, I want to map all the sets in ...
- 378
6
votes
0
answers
115
views
Polynomial time algorithms for graphs and cycles
For a given undirected graph $G$ , let $ c(G) $ denote the length of the longest cycle in $ G $ (by cycle, we mean a closed path without repetitions). Prove that if there exists a polynomial-time ...
- 7,603
6
votes
0
answers
1k
views
Time complexity of obtaining the support set of an unsorted sequence?
Consider a sequence $s$ of $n$ integers (let's ignore the specifics of their representation and just suppose we can read, write and compare them in O(1) time with arbitrary positions). What's known ...
- 1,025
6
votes
0
answers
327
views
Prove/disprove the existance of a data structure that has O(log N) inserts/deletes and get k-th largest element in O(1)
Consider a sorted array. We can get the $k$-th largest element in $O(1)$, but insertions and deletions cost $O(n)$.
Consider an order statistic tree. Insertions and deletions cost $O(\log{N})$, but ...
CommunityBot
- 1
6
votes
0
answers
130
views
Problems with Θ(n³) complexity on TMs with lower bounds by communication complexity arguments
One of the most used simple examples of application of Communication Complexity is the $\Omega(n^2)$ lower bound for recognizing palindromes of length $2n$ on a single tape Turing machine.
Is there a ...
CommunityBot
- 1
5
votes
0
answers
71
views
Complexity of recognizing a covering system
Suppose we a given a set $S$ of residues $a_i\pmod{m_i}$, $i=1,2,\dots,k$. I wonder how hard it is to recognize whether they form a covering system, that is, whether for every integer $n$ there exists ...
- 211
5
votes
0
answers
73
views
Data structure that supports adding to evenly spaced indices
I need an array-like data structure that stores integers and supports fast addition to multiple evenly spaced elements on given interval. Formally, if $n$ is length of the array, it has to support ...
D.W.♦
- 167k
5
votes
0
answers
103
views
Given $n=pq=a^2+b^2$, can we factor $n$?
Just to be clear, $a$ and $b$ are known, while $p$ and $q$ are unknown prime numbers, both congruent to $1$ modulo $4$. Can we design an efficient algorithm to retrieve $p$ and $q$?
It is a known ...
D.W.♦
- 167k
5
votes
0
answers
115
views
Specific quadratic 0-1 knapsack problem solvable in linear time?
I am interested in a simple variant of the quadratic knapsack problem.
Let $\{w_1, \ldots, w_n\} \in \{0,1\}$ be $n$ weights and $\{v_1, \ldots, v_n\} \in \mathbb{R}$ be $n$ values. Furthermore, ...
- 39.2k