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I presume that $L$ is the length of $D_m$ for all machines. Hence your input is of size $O(|M|\cdot (L+|A|))$ because it contains $|A|$ activities for all machines and the $D_m$ array of length $L$ fo …

You may think that your algo is pseudo-polynomial because the outer loop repeats $T$ times and the possible size of $T$ grows exponentially with a bit string describing $T$. But notice that $R_k(t)$ i …

In case we desire that all indices are distinct one can use balanced BSTs such as an AVL-Tree to check if indices overlap. For each of the $O(n^{k/2})$ sums you also store the set of indices (using a …

Well an algorithm with $O(0)$ fulfills the criterion. It basically does nothing. As soon as your algorithm does at least one operation on execution it has a runtime cost $t(n) > 0$. Since $$t(n)\in O …

If $P$ and $Q$ are languages closed under prefix, then $P\cup Q$, $P\cap Q$, $P \cdot Q$, $P/Q$ are also closed under prefix. Also you can think of such a language $L$ as a directed tree. Define the g …

Let $AND:\{0,1\}^n\mapsto\{0,1\}$ be a polynomial in $GF(q), q\geq 2$. Notice that the polynomial can't have a constant term (the constant term is zero), because $AND(0,...,0) = 0$. Meaning we can wri …

$$ IP = PSPACE $$ According to wikipedia. Thus $NP \subseteq IP$ and all NP-complete problems are in $IP$.

The answer is unknown. If $X\in NP$ and $\bar X\in \mathcal{NP}$ than by definition it's a so called "co-NP" problem ($X\in \mathcal{coNP}$). It's still an open problem if $\mathcal{NP} = \mathcal{coN …

This is more of a comment, but I want to append an image. The following graph is maximal planar and every node has degree 4, thus the proposed algorithm does not work. The maximal independent set of t …

Here is how I would do it: Algo: Verifier for DNF-SAT problem given as array of closures and a possible mapping of literals to booleans Input: Array of closures: [$C_1$,$C_2$,...,$C_m$] where $C_j \ …

You want to prove a tight $\Theta(n)$ bound as @vonbrand commented you could to show: $$am+b \leq T(m)\leq cm+d\quad (1)$$ Now start by assuming $(1)$ holds for all $m < n$ and conclude: $$T(n) = 2T( …

Big hint based on Yuvals Hint: Consider the language $\{\langle S, f \rangle | S \text{ is a SAT formula and } f\text{ a satisfying assingment}\} \in P$