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Questions related to the (computational) complexity of solving problems

Complexity classes are (usually) expressed using Landau notation, which makes precise what you state. See also our reference questions. Yes, by definition (usual) complexity classes don't say anythin …
answered Nov 2 '18 by Raphael
Let $S \subseteq \mathbb{N}^3$ with $|S| < \infty$ and $P_1(S)$ where $\quad \displaystyle P_1(S)\ \ :\Longleftrightarrow\ \ \forall (i,j,k)\in S. i,j,k \text{ mutually coprime }.$ $P_1$ is clearly …
answered Mar 22 '12 by Raphael
Luke illustrates one point of confusion; let me point out another one. Algorithm $A$ for problem $P \in \mathsf{NP}$ uses algorithm $B$ for problem $Q \in \mathsf{PSPACE}\text{-complete}$. Is it poss …
answered Jul 17 '12 by Raphael
No, it doesn't mean that. The set of NP-hard problems is upward-closed in the sense that there is no upper bound on the hardness of problems in it. You are correct to say that the class of NP-hard pr …
answered Mar 2 '18 by Raphael
Be very mindful of who takes "efficient parallel algorithms" to mean what, exactly. The older answers explain the perspective of complexity theory. There, "efficient" usually means something vague li …
answered Jun 5 '15 by Raphael
You are missing one thing, namely that the optimal algorithm and worst instance are not simply functions of the instance size. Also, I am not sure that minimising/maximising algorithm and instance ind …
answered Oct 5 '12 by Raphael
The first parameter is the number of random bits, and the second the number of queries to the oracle string $\Pi$ the verifier may use. The length of $\Pi$ is not bounded; therefore, having only acce …
answered Apr 24 '15 by Raphael
Assume the new problem was polynomially solvable. Take any instance of the old problem and add release dates $r(t)=0$ for all $t \in T$. Then the new problem has a "yes" answer for the modified instan …
answered Jul 9 '12 by Raphael
Given input $x = \langle M \rangle$, decide whether $M$ halts after $|x|!$ steps on input $x$.
answered Aug 3 '14 by Raphael
As Artem notes in his comment, the question is rather unmeaning as you can define whatever you like. Let me illustrate where things start to be "kind of silly". Some notation: For two problems $P,Q$, …
answered Jun 5 '12 by Raphael
When using as the set of coins all logarithms of the prime numbers or numbers in general The change-making typically assumes a finite set of coins. You can, of course, extend it to infinite set …
answered Jul 6 '16 by Raphael
According to the definition on Wikipedia, $T = \{(i,x,t) \mid \varphi_i \text{ tosses } t \text{ coins on } x\}$ has to be recursive if the measure you propose is to be a Blum complexity measure. As t …
answered Mar 27 '12 by Raphael
If there were an algorithm that factored in polynomial time by means of examining each possible factor of a complex number efficiently Starting from a self-contradicting statement like this, you …
answered May 14 '12 by Raphael
What you have in the case of DHC is really just a reduction: assuming you could solve DHC in polynomial time, you can solve HC in polynomial time, too, because it is a special case (use the same algor …
answered Jan 14 '13 by Raphael
We have $M$ columns $C_1, \dots, C_M$ in the original table, each with a finite set of distinct values $V_1, \dots, V_M$. Given the cross-table frequencies $c_{i,j}(a,b)$ (frequency of $a$ in $V_i$ a …
answered Apr 7 '13 by Raphael

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