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I seems, your problem is Turing-complete for the class ${\mathsf{P}}^{\mathsf{NP}[O(\log n])]}$. As mentioned in the question, you already know that it falls in this class. To show Turing-completeness …

In many cases, approximating NP-optimization problems give rise to sharp complexity jumps. For example, SET COVER can be approximated within a factor of $\ln n$ in polynomial time (by the Greedy Algor …

Here is an answer to a special case, when we restrict ourselves to the case when the Karp reduction can also be made length-increasing, along with making it injective. (Length-increasing means that $ …

Due to Babai's recent result (see the paper) $GI$ is in quasi-polynomial time ($QP$). If $GI$ is $NP$-complete, then it implies $NP\subseteq QP=DTIME(n^{polylog\, n})$. This, in turn, implies $EXP=NE …

While I don't know a specific (conjectured) example in $QP-\beta P$, there is still rather compelling evidence that $\beta P$ is a proper subset of $QP$. Namely, these classes behave very differently …

Let $L$ be an NP-complete language. Let $W(x)$ denote the set of (polynomially length bounded) witnesses that certify $x\in L$. That is, $x\in L$ if and only if there exists a $w$, such that $w\in W( …

An 'easy' SAT case (although not expressed by the clause to variable ratio) is this: A $k$-SAT formula is satisfiable if every clause overlaps with at most $2^{k-2}$ other clauses in it. Overlap means …

Consider an undirected graph, with non-negative weights on the edges, and two distinguished nodes $s\neq t$. If $P$ is a simple $s-t$ path in this graph, let $W_1(P)$ denote the sum of the weights o …

We can write ${\mathsf {NP}}-{\mathsf P}= {\mathsf {NPC}}\cup {\mathsf {NPI}}$ where ${\mathsf {NPC}}$ is the set of ${\mathsf {NP}}$-complete languages (not in ${\mathsf {P}}$ by this partition), an …

Instance: An undirected graph $G$ with two distinguished vertices $s\neq t$, and an integer $k\geq 0$. Question: Does there exist an $s-t$ path in $G$, such that the path intersects at most $k$ trian …

I am looking for conjectures about algorithms and complexity that were viewed credible by many at some point in time, but later they were either disproved, or at least disbelieved, due to mounting cou …

I wonder, what is (currently) the largest number $k$, such that a natural problem is known with the following properties: An $O(n^k)$ algorithm has been already found for the problem. For any fixed …

Diagonalization is a frequently used technique in complexity theory. However, the problems (sets) that are created by diagonalization rarely correspond to anything natural. It would be interesting to …

It is well known that many NP-complete problems exhibit phase transition. I am interested here in phase transition with respect to containment in the language, rather than the hardness of the input, r …

It is true that if the function $f$ is given by an oracle, then a randomized algorithm is exponentially faster than any deterministic algorithm. With an oracle function, however, this is not a $BPP$ p …