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Questions tagged [polynomial-time-hierarchy]

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Is there a notion similar to NP-intermediate, but for higher classes in PH?

It is well-known that (assuming P$\neq$NP), there are problems in NP that are not NP-hard neither in P. Such class of problems are called NP-intermediate (https://en.wikipedia.org/wiki/NP-intermediate), and one candidate problem that might be…
441Juggler
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Generalizations of integer-programming for the polynomial hierarchy?

Integer programming is known to be NP-complete. We also know that each class in the polynomial hierarchy contains elements not contained in the ones below, so Integer programming is not complete for classes higher up in the hierarchy. Is there a…
user56834
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complexity theory: polynomial hierarchy for function problems / TSP with output

I'm searching for equivalent problem classes from the polynomial hierarchy to function problems. I have this problem similar to traveling salesperson, which imo lies in the second order of polynomial hierarchy. However the existence of an answer can…
Duda
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Using time hierarchy theorem to show $Time(n^7)$ strictly contained in P

I'm relatively new to computational complexity and am trying to use the time hierarchy theorem to show that $Time(n^7)$ is strictly contained in P. I understand that the time hierarchy theorem says that if the limit as n tends to infinity of…
Lucas
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Toda's theorem without $\oplus P$

Toda's theorem says that the polynomial time hierarchy is contained in $P^{\#P}$, the class of problems solvable by polynomial time oracle Turing machines with access to an oracle in #P, which is the class of functions that return the number of…
user6767509
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Different ways definitions of $\Sigma_i \text{SAT}$

Some research paper define $\Sigma_i \text{SAT}$ in this way, $$\Sigma_i \text{QBF}=\left\{\enspace\substack{\exists x_1\forall x_2\exists x_3\cdots Q_ix_i\varphi(x_1\cdots x_i) \text{ where } \varphi(x_1\cdots x_i) \text{ is an unquantified boolean…
Monte_carlo
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EXACT-INDSET is $\Pi_2^P$-complete

Let $$\text{EXACT-INDSET} = \{\text{the largest independent set in G has size exactly k}\}.$$ We know that $\text{EXACT-INDSET}$ is in both $\Sigma_2^P$ and $\Pi_2^P$. My question is what would be the consequences if we prove that EXACT-INDSET is…
Adam Finch
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$\Sigma_i\text{SAT}$ is hard for $\Pi_{i−1}^P$

Let $$\Sigma_i\text{SAT} = \{\phi(u_1,\cdots, u_i): \exists{u_1}\forall{u_2}\exists\cdots Q_iu_i \phi(u_1,\cdots, u_i) = 1\}$$, where here $\phi$ is a Boolean formula (not necessarily in CNF form, although this does not make much difference), each…
Adam Finch
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How powerful is this 'non-deterministic' reduction?

Let us define NP.NP as a class of problems that non-deterministically reduces to 3SAT. Within the scope of this post, assume non-deterministic reduction is a non-deterministic Turing machine (or algorithm) that computes (in polytime) the function…
108_mk
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Evidence for or against the conjecture $QCMA\subseteq BQP^{NP}$

Is there some (complexity theoretic) argument for or against Quantum-classical Merlin Arthur $$QCMA\subseteq BQP^{NP}?$$ I am aware of one (weak!) supporting argument for posing it as a conjecture. This is due to quantum-classical PCP (QC-PCP)…
108_mk
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