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3.205 complexity-theory questions.

Let $G = (V, E)$ be a connected undirected graph with $v > 6$ nodes $V = \{v_1, v_2, \dots, v_n\}$ and $n$ edges. Let $\{e_1, e_2, \dots , e_m\}$ be all the edges of $G$ listed in some specific ...

I have two lists say $A$ and $B$, each consisting of n positive integers. I make a list $C$ such that each element of $C$ i.e., $C_i=A_iB_i$ for each element $A_i \in A$ and $B_i \in B$. Now I have ...

I have the following language : $L=\{\langle C_1,C_2\rangle \text{ | } C_1 \text{ and } C_2 \text{ are two circuits that calculate different function}\}$. We can call this language SAT-2C. Prove that ...

Show that NP is closed under concatenation. This is a homework problem and I would appreciate some guidance. I began by saying the following: Let $A$ and $B$ exist in NP. Let $V_1$ and $V_2$ be ...

So let's say I've implemented an algorithm running in $O(n^2)$ on my 3-tape TM. What kind of time complexity would I expect for a single-tape TM? I just don't know where to get started...

Is the class $\sf NP$ closed under complement or is it unknown? I have looked online, but I couldn't find anything.

For which $c, d$ is $Gap2SAT[c, d]$ in $P$ (such that $0<c<d<1$)? (I know if d=1 then for each c it will be in P, however with which c,d such that $0<c<d<1$ can I simply return ...

I know that CNF SAT is in NP (and also NP-complete), because SAT is in NP and NP-complete. But what I don't understand is why? Is there anyone that can explain this?

I was doing a practice question. As you can see below there is an Implication graph. To check whether the problem is satisfiable, I checked whether there were any 'bad loops'. To do so, for each ...

In practical applications, search algorithms are often strengthened using heuristics. e.g., Deep Blue beat gary kasparov by searching through possible chess moves by "guiding" its search with human-...

I'm looking for a kind of way to create a minimal perfect hash function given a set known integers. More specifically, I have M numbers within the range 0-N with N > M. Does it exist a way to create ...

Give the recurrence relation: $T(n)=n^2+T(\frac{n}{2})+T(\frac{n}{4})+T(\frac{n}{8})+...+T(\frac{n}{2^k})$ ($k$ is some constant and assume $n$ is $2^t$ for some $t\in \mathbb{Z}$) I'm trying to ...

Lemma 4 in How easy is local search by Johnson, Papadimitriou, and Yannakakis, states: If a PLS problem is NP-hard then NP = P So assuming L is a PLS problem (polynomial local search problem) that ...

Is it $NP$-complete given $c(x),g(x)\in\mathbb{F}_2[x]$ where $g$ generates a cyclic code of length $n$ (so $g\mid x^n-1$), and $\deg c<n$ to find the nearest codeword to $c$? This is related to ...

A friend of mine and I are trying to teach a bot play a card game (bela) We are using monte carlo tree search (MCTS) to estimate the probability of winning hand in regards to multiple possible (!...

Prove/disprove that the class of decidable (resp. partially decidable) languages is closed under symmetric difference. A symmetric difference of sets A and B is the set (A \ B) ∪ (B \ A). I know that ...

The Problem: I am currently analyzing a simple program that takes a file of length $n$, splits it into its individual words (seperated by white space) and adds those words to a set: ...

I have some objects $x\in X$ and a metric $s:X\times X\to\mathbb{R_{+}}$. For each $x$, there is a $y\in Y$. Note that $x$ and $y$ are highly structured and we cannot consider neural networks for ...

I am having troubles understanding Levin's universal search method. In Scholarpedia, http://www.scholarpedia.org/article/Universal_search, it is claimed that “If there exists a program $p$, of length $...

I think that these two classes should be the same, but I can't find any literature about this and have a limited background on the topic. This is my reasoning, and I would like to know if (1) this is ...

What are the differences between CYK and Earley algorithms? Please provide a thorough rundown.

In Aurora and Barak, they give the following alternative definition of $BPP$: What is the meaning of the subscript to $Pr$? Is that $Pr_{r \in_R \{0,1\}^{p(|x|)}}$? My guess is this is supposed to ...

Are there any known problems in $UP \cap co-UP$ other than integer factorization and parity games (or a problem that can be reduced in polynomial time to either problem), that aren't known to be in $P$...

There are lots of attempts at proving either $\mathsf{P} = \mathsf{NP} $ or $\mathsf{P} \neq \mathsf{NP}$, and naturally many people think about the question, having ideas for proving either direction....

In Parameterized Complexity the Independent Set Problem for a Parameter $k$ ist $W[1]$-complete, and Dominating set is $W[2]$-complete. Now the prototypical $W[1]$ problem is deciding by a single-tape ...

Suppose you have a function which calls to itself twice. So both go down recusively until they reach some condition. But one of them will go less times than the other. For example one would call ...

If P is a program that can be run in exponential time on a deterministic RAM. Can P always be run in polynomial time on a non-deterministic RAM?

I have a homework question about the properties (decidability, Turing-recognizability, etc.) of the language $$ L = \{ \langle M \rangle | \text{$M$ is a TM and $M$ accepts some string $w$ which has ...

Given a graph G, how can I check that its diameter doesn't exceed log(n) (n is the number of vertices)- by using only O(log(n)) space? (adjacency matrix doesn't seems to help...)

The question asks to provide an algorithm to compute $(i)$ The product of $n$-bit numbers using reciprocation operation and addition operation but not using multiplication and squaring. $(ii)$ The ...

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.\ \...

I'm interested in the following scheduling problem: Time is divided in $T$ slots as represented by the left circle below (where $T = 8$). A minimum number of employees $b_t \geq 0$ need to be present ...

Let $T=(V,E)$ be tree and each edge has a positive scalar weight. I need to print all paths in the tree and then sort the weight of edges in each paths. it needs $O(n^3\log(n))$ time. To solve this ...

The $\text{NP-Complete}$ class of problems is defined w.r.t Karp Reductions, which are polytime many-one reductions. However, they need not necessarily preserve the number of solutions. A more ...

I'm trying to understand this claim. I see that if there are $S$ vertices, then we can identify each vertex using $\log S$ bits. Now each vertex can be connected to, let's say, $S$ other ones (is ...

Let A = {w|w $\in$ {0,1}, such that w=0 iff P=NP w=1 iff P!=NP Would the language itself be decidable?

Suppose you have a box full of stamps. There are $a$ stamps in the box. You want to get the $b$ oldest stamps from the box. Whereby $a$ is much bigger than $b$. What would be the best algorithm (worst-...

Consider the following image: The problem is: can we cover the bigger rectangle with small rectangles such that no two rectangles overlap and no gap opens up? Prove that this problem is $NP-Hard$. I ...

Let $G$ be an arbitrary graph with $n$ vertices and we want to find a simple cycle with $\log n$ length. Is there exists a known polynomial algorithm for this problem?

I've got problem on integer programming, specifically with the following knapsack problem. I'd be happy to get some suggestions on how to solve the problem in a time efficient way. There are 120 ...

Let $f$ be a continuous function and $[a,b]$ be an interval where $f(c)=0$ for some unique number $c \in [a,b]$ and where $f(a) f(b) \leq 0$. Suppose there exists a sub-interval $[a_0,b_0]\subset [a,b]...

In this link https://techdifferences.com/difference-between-bubble-sort-and-selection-sort.html it says that the best case of bubble sort is order of n due to the fact that there would be only ...

I want to test if this clause: {x,y,z},{¬x,y},{¬y},{¬z} is satisfiable. However, I noticed that the clauses contain more than two literals. How can I check whether the formula is satisfiable. So ...

Many important (non-deterministic) complexity classes like NP are believed not to be closed under complement. But have any of them been proven not to be? I'm sure one could construct some contrived ...

I have the following boolean formula: {x,y},{-x,-y},{x,-y}, and below is the corresponding Implication graph: I know that the next step is to check for 'bad loops'/Strongly Connected Components. ...

I am trying to convert the following 2-sat clauses to implications and then draw the implication graph. The clauses are: ...

I'm trying to understand the runtime of this code:def f(n): if (n <= 1): return 1 else return f(n-1)*f(n-1) + f(n-1) At first, my logic said ...

We have seen various languages proven to be outside of REG and CFL by corresponding pumping lemmas. Has something similar been done for SAT?

Normally to show that an optimization problem is hard, we show the corresponding decision version of the problem is hard. However, is this sufficient to support the conclusion? Does there exist any ...

The problem is as follows: given a list of cities, a list of distances between them, and upper bounds for date/times that the cities must have been visited by, compute the shortest (optimal) going ...

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