Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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21 views

How to deal with witness size when collapsing PH with $\Sigma_k \neq \Pi_k$

I'm reading Goldreich's note on PH and it notes that if $\Sigma_k \neq \Pi_k$ for some $k$, then PH collapses to the corresponding level (so $\mathcal{PH} = \Sigma_k$). At least one formulation of the ...
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An idea for finding simple cases for $\mathsf{SAT}$ problem

As I know there are relatively small number of cases when we know a polynomial-time for $\mathsf{SAT}$. One such example is $2$-$\mathsf{SAT}$. (Is there a reference for a good survey of such ...
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On definition and study of software delivery algorithms and related computational complexity

Recent developments around software delivery automation (often related to as continuous integration/devops) have emphasized on standardization of phases required to deliver software, differing though ...
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112 views

Is unary $\Pi_2$-SUBSETSUM coNP-complete?

Consider the following problem: for given integers $a_1, \ldots, a_{2n}$ and $A$ that are given in unary representation define is it true that for every $S \subseteq \{1, ..., 2n \}$ such that $|...
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Conditional separations of $\exists\mathbb{R}$ from $\mathbf{PSPACE}$

As pointed out explicitly by Emil Jeřábek here: Even with Turing reductions, $\mathbf{PSPACE}=\mathbf{P}^{\exists\mathbb{R}}$ would still be a breakthrough (and completely unexpected) result. So ...
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113 views

Evidence integer multiplication is in linear time?

After millenia of quest we have identified two $n$ bit integers can be multiplied in $O(n\log n)$ time. Please refer details in https://www.quantamagazine.org/mathematicians-discover-the-perfect-way-...
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Volume computation of special polytopes

I'm interested in computing the volume of a special class of $\mathcal{H}$-polytopes and the complexity of doing so. I know that in general it is #P-hard to compute the volume of $\mathcal{H}$ -...
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1answer
96 views

Satisfiability problems with restricted (not bounded) number of occurrences per variable

Intro It is known that SAT is hard even when restricted to, e.g., formulas with exactly 3 literals per clause and at most 4 occurrences per variable. On the other hand it is easy if there are exactly ...
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Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time

It is commonly believed that for all $\epsilon > 0$, it is possible to multiply two $n \times n$ matrices in $O(n^{2 + \epsilon})$ time. Some discussion is here. I have asked some people who are ...
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Evidence for $\mathsf{P} \neq \mathsf{PP}$ if the polynomial hierarchy collapses?

We think that $\mathsf{PH}$ does not collapse, and that $\mathsf{PP}$ is not in $\mathsf{P}$. Suppose on the contrary that $\mathsf{PH}$ does collapse, say even $\mathsf{P}= \mathsf{NP}$. $\mathsf{...
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409 views

PPAD and Quantum

Today in New York and all over the world Christos Papadimitriou's birthday is celebrated. This is a good opportunity to ask about the relations between Christos' complexity class PPAD (and his other ...
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Is $\Sigma_2^{NP}=NP^{\Sigma_2}$? [migrated]

Disclaimer: If not interested in my background, skip directly to the question below! I am a complete newbie when it comes to complexity theory. I come from a physics background and I am currently ...
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Is the variant of Exact-Cover By 3 Sets (X3C) with |C| <=2q, NP-complete? [duplicate]

In general case |X|=3q, C={C_1,C_2,...,C_m},|C|=m. Question: Is there exists an exact cover C* of X with |C*|=q? And this problem is known to be NP-complete. But, the restricted version of X3C ...
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178 views

Lower bound on pebbling numbers

Out of curiosity, I tried finding the original paper showing that there are graphs that require $n/\log n$ pebbles in the sense of Hopcroft, Paul, and Valiant’s seminal paper “On Time Versus Space”. (...
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Sparsity related to linear programming in $NC$?

Linear Programming is $P$-complete. Results like here and here implies certain non-trivial cases have been parallelized. How does complexity theory characterize $NC$ cases of linear programming? In ...
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95 views

How to prove a general convex set is nonempty or empty in polynomial time?

The general convex set should be represented by a set of (generalized) inequalities $f_{i}(x)\leq 0 $ with $ f_{i}(x) $ being convex in $ x $. I know ellipsoid method and interior method, but I do ...
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1answer
219 views

3-coloring planar graphs in $O\left(3^{n^.5}\right)$?

I was wondering if the task of searching for planar 3-colorings is known to be of complexity $O\left(c^{\sqrt{n}}\right)$ or lower? This feels like it would be an intuitive consequence based from ...
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178 views

Nondeterminstic Linear Time vs Other Complexity Classes

Is it known whether or not nondeterministic linear time contains $P$ and/or smaller classes such as Uniform-$NC^1$?
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521 views

A game on several graphs

Consider the following game on a directed weighted graph $G$ with a chip at some node. All nodes of $G$ are marked by A or B. There are two players Alice and Bob. The goal of Alice (Bob) is to ...
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1answer
82 views

Complexity of existence of simple polygonalization with prescribed area?

This is a followup on my previous question. Fekete proved the NP-completeness of deciding the existence of simple polygonalization with minimum (or maximum) enclosed area (simple polygonalization is ...
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1answer
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On the sensitivity conjecture?

The recent establishment of the relation $bs(f)=O(s(f)^4)$ goes through Gotsman,Linial . Can the same approach get to $O(s(f)^2)$ or is there an essential limitation to the approach?
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Counting solutions to extended MSO formulas, and sampling — do these appear in the literature?

I am trying to determine if the literature contains various extensions of Courcelle's theorem. Since I haven't been able to find these in the literature, I guess that these are folklore results, or ...
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108 views

Sensitivity and Low-Degree Approximation under Non-Uniform Distribution

I am searching for generalizations of analysis of Boolean functions when the input strings are distributed according to a general non-uniform distribution, possibly with arbitrary dependencies between ...
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2answers
143 views

reference request for construction of expanders

I'm looking for a good exposition of the explicit constructive proof of the existence of expander graph families due to Reingold Vadhan and Wigderson. Arora/Barak has a chapter on it, but i find it ...
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Is there any research on the complexity of producing given a problem description?

It is straightforward enough to analyze the complexity of a particular algorithm as a function of input size or other variables in terms of runtime or space used or whatever else. I am wondering if ...
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95 views

Separation of AM and SZK

Are any results on the separation of AM from SZK known (e.g. relativized separation, or a separation assuming one-way functions exist, etc.)?
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311 views

Hidden Constants in Complexity of Algorithms

For many problems, the algorithm with the best asymptotic complexity has a very large constant factor that is hidden by big O notation. This occurs in matrix multiplication, integer multiplication (...
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1answer
147 views

Possibility of hierarchy with $UP$ class?

I am not sure if this is a cheap query. However I am unable to find this myself. So I am posting here. The standard complexity class is built with $NP$ and $coNP$ and leads up to $PSPACE$. The ...
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1answer
254 views

What is the reason to use NP instead of EXP as 'the' class of intractable problems?

What is the rationale for not using EXP has the main class for intractable problems but NP? Why is it important that solutions are verifiable in polynomial time?
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322 views

Bootstrapping results that really bootstrap

There is a type of results in TCS usually called bootstrapping results. In general, it is of the form If proposition $A$ holds, then proposition $A'$ holds. where $A$ and $A'$ are propositions ...
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Bipartite formula complexity lower bound

I'm trying to understand the paper The Bipartite Formula Complexity of Inner Product is Quadratic, by Avishay Tal. The argument is recapped here. I am having trouble understanding the proof Theorem 3 ...
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Is equivalence of uniform AC0 decidable?

Is there a representation of the functions from $\mathsf{DLogTime}$-uniform $\mathsf{AC}^0$ for which equivalence is decidable? Since the languages defined by nondeterministic pushdown automata ...
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137 views

Is sorting pairwise distances as hard as sorting arbitrary points?

If we have $n$ points in $\mathbb{R^d}$, what is the complexity of sorting the $O(n^2)$ pairwise distances? Clearly the complexity is $\Omega(n^2)$ but is there a reduction to show it is as hard as ...
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67 views

reduction from SAT to approximate set cover

I read this neat result proved in the early 90s: For any $c>1$, There's a poly time map from boolean formulas $\varphi$ to pairs $K, \mathcal S$ where $K$ is a positive integer and $\mathcal S$ ...
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1answer
153 views

complexity of deciding whether there's a small polynomial with a given root

Let $f\in (\mathbb{Z}/p\mathbb{Z})^\ast$ be a nonzero element of a prime finite field. For $d, r\in \mathbb{N}$ consider the problem of deciding whether there is a nonzero polynomial $$P(x) = a_0 +...
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65 views

Unknown gaps in computation models

I'm looking for computatuon models where it is known that there are problems that we can solve in time T1 and T2. where T1 is smaller then T2 and it is unknown if there are problems where their ...
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1answer
89 views

How to build comparison operator (comparator) in an arithmetic circuit

I am trying to convert a basic program into an arithmetic circuit. I am stuck on the step of converting the greater than operator into an arithmetic circuit. To be specific, I do not know how to ...
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1answer
86 views

Upper bounds on the circut depth

Suppose $f:\{0,1\}^n \to \{0,1\}$ is a function such that it can be computed by a circuit of size $n^c$ for some constant $c>0$. Q. Is there any nontrivial upper bound on the depth of a circuit ...
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10answers
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Most important new papers in computational complexity

We often hear about classic research and publications in the field of computational complexity (Turing, Cook, Karp, Hartmanis, Razborov etc). I was wondering if there are recently published papers ...
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“Planar graph coloring is not self-reducible” is this about all $p$-relations encoding that problem?

I have a question about the paper "Planar graph coloring is not self-reducible" by Samir Khuller and Vijay Vazirani. The final theorem in that paper states that "Planar Graph k-coloring is not self ...
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1answer
159 views

Language in $PSPACE$ and not necessarily in $P$ if $P=PP$?

If $P=PP$ then the counting hierarchy collapses to $CH=P$. Because so many complexity classes are contained in $CH$, this causes most classes to now be contained in $P$. My question is whether this is ...
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Example of a hardness-of-approximation proof which improves the approximation factor?

Are there any examples of hardness-of-approximation reductions where we get a better hardness bound for the problem we've reduced to than the problem we've reduced from? In the examples I've seen so ...
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84 views

Is it possible to sort by only knowing the sign of pairwise sums?

I am currently thinking of how much structure one actually needs in order to be able to sort things at all. All comparison-based algorithms need a direct comparability, but are we able to remove this ...
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81 views

Is $MSB$ of permanent and certifying half number of witnesses easy?

Can there be a $P$ algorithm to decide if number of perfect matchings is at least $(n!/2)+1$ for a bipartite graph on $n+n$ vertices? Can there be a $P$ algorithm to decide if number of witnesses ...
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46 views

Problem of determining if a $4$ connected graph has $k$ Hamiltonian cycles

Definition: Define the $k$-HamiltonianCycles problem as the decision problem that asks if a given graph has at least $k$ distinct Hamiltonian cycles. Question: Is there some constant $k$ so that the $...
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1answer
118 views

Count satisfying assignments of CNF formulas over all possible negation assignments

Consider the set of all CNF instances that can be generated by adding negations to a single monotone CNF instance. How hard is it to compute the sum of the counts of satisfying assignments for the set?...
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1answer
301 views

Planar Exact Cover by even-size sets

Major edit on June 6, 2019: Replaced the target problem with a simpler (but equivalent) one. Is the following problem NP-complete? Planar Exact Cover by even-size sets Input: A set $U$, a ...
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228 views

Razborov-Smolensky polynomial argument on $\textrm{ACC}[q]$ where $q$ is a prime power

It seems to be a folklore that we can "handle" $\textrm{ACC}[q]$ circuits not only for prime $q$ but prime power $q$. For example, authors of this paper say that ... any constant depth circuit ...
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56 views

Graph automorphism with prescribed values

Consider a graph $G$ with vertices labeled $1,...,n$ and edge weights $w_{ij}$. Recall an automorphism of G is a permutation $\sigma$ of the vertex labels such that $w_{\sigma(i),\sigma(j)}=w_{ij}$ ...
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2answers
807 views

Attempted proofs of P vs NP

What are the most recent (say in the last 3 years) attempts at disproving $P = NP$, and where can I find the papers?