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Questions related to the (computational) complexity of solving problems

3
votes
Schaefer's dichotomy theorem gives you the answer. Apply it and see what you get, and let us know. Looking at the modern formulation, in your case $\Gamma$ contains the three relations $\lnot x \lor …
answered Apr 19 '15 by Yuval Filmus
0
votes
Complexity theory is based on analyzing the asymptotic running time of algorithms. In order to talk about asymptotics, the input length should be unbounded. However, real CPUs can only handle inputs o …
answered Sep 16 '16 by Yuval Filmus
2
votes
Let's assume for simplicity that each negated literal appears exactly once. Let $C_i$ be the clause containing $\lnot x_i$. Consider the graph whose vertices are the variables, and $x_i$ is connected …
answered Jun 7 '16 by Yuval Filmus
4
votes
When people say polynomial time, they mean that the time has polynomial growth, that is if we denote the time by $T(n)$, then $T(n) = O(n^c)$ for some real $c$. We get the same definition if we only a …
answered Mar 2 '14 by Yuval Filmus
2
votes
Let's take SAT, the quintessential NP-complete problem, as an example. An instance of SAT is a formula, and the answer is whether the formula is satisfiable. The corresponding #P-hard problem #SAT is …
answered Feb 14 '16 by Yuval Filmus
2
votes
Many problems involve integers, for example in the form of integer weights. There are two common ways to encode integers: Binary encoding, or more generally base B encoding for some constant B. Unar …
answered Nov 19 '17 by Yuval Filmus
1
vote
How about $(x_1 \lor x_2 \lor x_3) \land (\bar{x}_1 \lor \bar{x}_2 \lor \bar{x}_3)$?
answered Feb 8 '18 by Yuval Filmus
1
vote
The difficulty here is how to check using non-deterministic logarithmic space that there is no path from $s$ to $t$ of some length $\ell$. We cannot just go over all possible paths of length $\ell$ si …
answered Aug 28 '17 by Yuval Filmus
3
votes
You can check out these lecture notes, as well as countless more. The proof of Fagin's theorem is divided into two parts: showing that every language expressible in existential second-order language i …
answered Jun 8 '14 by Yuval Filmus
6
votes
Use the Sipser-Lautemann theorem: $\mathrm{BPP} \subseteq \Sigma_2^p \cap \Pi_2^p$.
answered Oct 20 '14 by Yuval Filmus
3
votes
You have described an optimization problem, namely, find the arrangement that minimizes the average deviation from the mean. The decision version of this problem is, given an input to your problem and …
answered Jun 30 '18 by Yuval Filmus
0
votes
You're asking several questions. Here are answers for some of them. The OR function has degree $n$, but it can be approximated by the function $(x_1+\cdots+x_n)/n$. This shows that $d_{1,\text{OR},1 …
answered Jan 6 '15 by Yuval Filmus
1
vote
As you get further from Germany, you will get points which are farther and farther away. So unless you are taking into account the spherical nature of Earth, we need to change the problem, and look fo …
answered Feb 21 by Yuval Filmus
2
votes
If the running time of the algorithm is bounded, in the sense that for every input $x$ there is a number $T(x)$ such that whatever the coin tosses are, the algorithm always terminates within $T(x)$ st …
answered Jun 20 by Yuval Filmus
0
votes
The definition of $g$ should be: $g(x_1,\ldots,x_k) = 1$ if there exists $i \in [k-1]$ such that $x_i=x_{i+1}=1$ and $x_j = 0$ for $j \neq i,i+1$. As for your question, whether $g=1$ iff the conditio …
answered Jul 8 by Yuval Filmus

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