# Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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### For which problems in P is it easier to verify the result than to find it?

For (search versions) of NP-complete problems, verifying a solution is clearly easier than finding it, since the verification can be done in polynomial time, while finding a witness takes (probably) ...
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### A reduction from the maximum $k$-closure problem to the clique problem

Fix a partially ordered set $(P, \le)$ with $N$ elements and real weights $w(p)$ for each $p \in P$. A subset $S \subset P$ is called closed if for any $x, y$ with $y \in S$ and $x \le y$ we also ...
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### Query about P/poly and Polynomial Hierarchy Collapse to $\Sigma _{2}$

I am not conversant in the complexity class $P/poly$. While reading about the class on wiki I encountered two conditional statements about it, namely: If $NP ⊆ P/poly$ then $PH$ (the polynomial ...
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### Computational Complexity of 3SAT variant with additional restrictions on variables/clauses

Given a 3SAT problem with the additional constraints that: No clause or set of clauses is the 3SAT instance is 'redundant'. Thus, this 3SAT cannot eliminate any clauses. For any/every clause, the ...
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### An easy computable injection with a hard inverse

I was reading Charles Bennett's Thermodynamics of Computer Science and a passage (p. 926) caught my eye The construction of a reversible machine from an irreversible machine implies that the open ...
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### Computational Complexity of MIN-EQ-3-CNF

Consider the decision problem: MIN-EQ-CNF= $\{\langle\phi, k\rangle |\exists \text{CNF formula } \psi \text{ of size }\leq k\text{ that is equivalent to the CNF formula }\phi\}$ CNF is a Boolean ...
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### Recovering the inputs to Boolean circuits after partial evaluation

This question discusses Boolean Circuits and Boolean functions from $n>1$ inputs to one Boolean output. Notation: $\textit{arity}(\mathcal{C})=n$ if $\mathcal{C}$ takes $n$ inputs, similarly for ...
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### What are the current best known upper and lower bounds on the (un)satisfiability threshold for random k-sat and/or 3-sat?

I would like to know the current state of the phase transition for random k-sat, given n variables and m clauses, what is the best known c=m/n for upper and lower bounds.
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### Fastest Known Algorithm for $k$-Dimensional Matching and $k$-Exact Cover

Given a $k$-uniform hypergraph $G$ (i.e., each edge of $G$ contains precisely $k$ vertices) on $n$ vertices, the $k$-Exact Cover problem is the task of deciding if there exists $n/k$ edges in $G$ ...
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### Derandomizing arbitrary width *read-many* and *ordered* branching programs?

Modifying following TedP We know that derandomizing width $5\leq k\in O(1)$ read many branching programs is equivalent to $BPNC^1=NC^1$ and derandomizing width $k\in\Omega(n)$ read once ordered ...
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### Fine-Grained Hardness for Undirected Hamiltonicity

The fastest known algorithm for detecting Hamiltonian cycles in directed graphs on $n$ nodes runs in essentially $2^n\text{poly}(n)$ time. However, for undirected graphs on $n$ nodes, there is an ...
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### Cook inspiration for NP completeness

An academic descendant of Cook just lectured on NP completeness. He said that the idea came from a well-known theorem in first-order logic that talks about completeness of satisfiability for ...
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### What would be the consequences of PH=PSPACE?

A recent question (see Consequences of NP=PSPACE) asked for the "nasty" consequences of $NP=PSPACE$. The answers list quite a few collapse consequences, including $NP=coNP$ and others, providing ...
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### Consequences of NP=PSPACE

What would be the nasty consequences of NP=PSPACE? I am surprised I did not found anything on this, given that these classes are among the most famous ones. In particular, would it have any ...
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### Are runtime bounds in P decidable? (answer: no)

The question asked is whether the following question is decidable: Problem  Given an integer $k$ and Turing machine $M$ promised to be in P, is the runtime of $M$ ${O}(n^k)$ with respect ...
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### Is polynomial-time the same in all classical computational models?

There are many models of computability, all giving the same notion of 'computable function'. To pick a few examples: Turing machines (with variants: one-ended, two-ended, multiple tapes...) RAM ...
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### Diophantine equations with bounds on variables

Solving Diophantine equations is famously known to be undecidable. What about Diophantine equations to be solved over a finite domain? In particular, if I put an upper bound $k$ over the value of the ...
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### complexity class of a function - linear combinations and reductions (Fermionant, immanant, $GL_n$ representations)

The fermionant is a matrix function from physics, which is indexed by a positive integer $k$: \begin{align} \operatorname{Ferm}_k(A) = \sum_{\lambda} d_{\lambda}^{(k)} \operatorname{Imm}_{\lambda^T}(A)...
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### Is finding the shortest consistent term to fill a missing line in a truth table still NP-hard?

I understand the logic minimization problem is NP-hard when given the onset, since the last step is equivalent to set cover optimization. If instead we are given a partial truth table, and we just ...
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### EXPSPACE-complete problems involving numbers

This is a subset of this question. I'm looking for EXPSPACE-complete problems, for using in a reduction, which involve numbers in some ways, since my target problem involves numbers and linear ...
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### Is the Kolmogorov complexity of the truth tables of the halting problem known asymptotically?

Let $HALT_n$ denote the string of length $2^n$ corresponding to the truth table of the halting problem for inputs of length $n$. If the sequence of Kolmogorov complexities $K(HALT_n)$ were $O(1)$, ...
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### Asymptotic complexity lower bounds of proof checking

This paper on universal search mentions (on pp. 6-7) that proof checking can be done in $O(n^2)$ where $n$ is the length of the proof. Is this optimal? I don't want to specify the problem too ...
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### Problems Between P and NPC

Factoring and graph isomorphism are problems in NP that are not known to be in P nor to be NP-Complete. What are some other (sufficiently different) natural problems that share this property? ...
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### Does Horn SAT (Horn formula in CNF) have an integral polytope?

In some ways, my question is related to this: Is the matching polytope integral? Matching and Horn-SAT are both polynomial time solvable.. So I wonder if there is a Horn polytope, similar to the ...
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### minimizing size of regular expression for finite sets

It is known that minimizing the size of a regular expression is PSPACE-complete even if we have a DFA as the language's specification. What are the results if the language is finite? One can ...
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### Sorting with an average of $\mathrm{lg}(n!)+o(n)$ comparisons

Is there a comparison-based sorting algorithm that uses an average of $\mathrm{lg}(n!)+o(n)$ comparisons? Existence of a worst-case $\mathrm{lg}(n!)+o(n)$ comparison algorithm is an open problem (...
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### Useful primitives CPUs could provide, from TC0 (or NC1)

I just listened to XYZ talking about "How Universal Is the Idea of Numbers?", and bashing the concept as an accidental historical artifact. He suggested that totally different computational ...
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The simplest Reduction for 3-SAT to 1-in-3-SAT reduction is as follows: For each 3SAT clause: $x+y+z=1$ Introduce 4 new variables $\{a, b, c, d\}$ and replace original clause with below 3 clauses: $R(... 1answer 569 views ### Is it known if$\mathrm{CFL} \subseteq\mathrm{ NSPACE}(o(log^2(n)))$?$\mathrm{CFL}$is the class of context-free languages. Question Is$\mathrm{CFL}$known to be solvable in$o(log^{2}(n))$non-deterministic space? What about$\mathrm{DCFL}$? 1answer 135 views ### Is QMA known to contain Co-NP? Is QMA known to contain Co-NP? If not, would Co-NP being contained in QMA have any implications for other complexity classes. (e.g. Causing the polynomial heirachy to collapse.) 4answers 2k views ### Communication Complexity ...Classes? Discussion: I've been spending some personal time lately learning various things in communication complexity. For instance, I've re-familiarized myself with the relevant chapter in Arora/Barak, ... 2answers 247 views ### Complexity of "can we get a cycle by stacking directed bipartite graphs?" Preliminaries We consider directed bipartite graphs of the form$G = (V,V',E)$, in which the nodes are partitioned into$V = \{1,\ldots,n\}$and$V'=\{1',\ldots,n'\}$, with$|V|=|V'|=n$, and$E\...
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I recently started reading about Descriptive Complexity, the branch of Complexity Theory studying the logic languages needed to express complexity classes. The main milestone in the area seems to be ...
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### Minimum cut with size bounds $k\leq |S| \leq |V|-k$

It is known by the max flow min cut theorem that the minimum cut problem is in $P$. I am interested in knowing what is known on the complexity of the minimum cut with size $k\leq |S| \leq , |V|- k$. ...
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### Questions about the equivalence between PH and depth-d circuits with respect to an oracle

Consider an oracle $A$ and the language \begin{equation} P(A) = \{x\in \{0, 1\}^{n}: \text{the number of strings in }~A~\text{of length}~|x|~\text{is odd}\}. \end{equation} I am trying to make sense ...
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### Graph classes where giving a q-clique edge cover makes testing for q-colouring easy

A $q$-clique of a graph is a complete subgraph on $q$ vertices. A $q$-clique edge cover of $G$ is a set of subgraphs of $G$ such that each subgraph is a $q$-clique and each edge of G is contained in ...
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### The complexity of checking whether two DAG have the same number of topological sorts

This problem is highly related to the CNF one. Here is the problem: given two DAG (directed acyclic graphs), if they have the same counting of topological sorts, answer "Yes", otherwise, answer "No". ...
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### An NP-hard Hidden Subgroup Problem

I've encountered a model which can be thought of as a version of the Hidden Subgroup Problem (https://en.wikipedia.org/wiki/Hidden_subgroup_problem), but that doesn't quite meet the standard problem's ...