Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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2answers
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Complexity of Topological Properties.

I am a computer scientist taking a course on Topology (a sprinkling of point-set topology heavily flavored with continuum theory). I have become interested in decision problems testing a description ...
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Deciding if a given $\mathsf{NC}^0$ circuit computes a permutation

What is the complexity of deciding whether an $\mathsf{NC}^0$ circuit with $n$ input bits and $n$ output bits computes a permutation of $\{0,1\}^n$? in the other words, whether every bit strings in ...
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Complexity of n-queens-completion?

The classical $n$-queens problems asks, given a positive integer $n$, whether there is an array $Q[1..n]$ of integers satisfying the following conditions: $1\le Q[i] \le n$ for all $i$ $Q[i] \ne Q[j]$...
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A Notion of Monotone Quantum Circuits

In computational complexity there is an important distinction between monotone and general computations and a famous theorem by Razborov asserts that 3-SAT and even MATCHING are not polynomial in the ...
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Reasons to believe $P \ne NP \cap coNP$ (or not)

It seems that many people believe that $P \ne NP \cap coNP$, in part because they believe that factoring is not polytime solvable. (Shiva Kintali has listed a few other candidate problems here). On ...
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Deciding whether an NC${}^0_3$ circuit computes a permutation or not

I would like to ask about a special case of the question “Deciding if a given NC0 circuit computes a permutation” by QiCheng that has been left unanswered. A Boolean circuit is called an NC0k circuit ...
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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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What would be the consequences of factoring being NP-complete?

Are there any references covering this?
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Consequences of #P = FP

Which would be the consequences of #P = FP? I'm interested in both practical and theoretical consequences. From a practical point of view, I'm particularly interested in consequences on Artificial ...
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876 views

What are the consequences of $L = \oplus L$?

Shiva Kintali has just announced a (cool!) result that graph isomorphism for bounded treewidth graphs of width $\geq 4$ is $\oplus L$-hard. Informally, my question is, "How hard is that?" We know ...
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What specific evidence is there for P = RP?

RP is the class of problems decidable by a nondeterministic Turing machine that terminates in polynomial time, but that is also allowed one-sided error. P is the usual class of problems decidable by ...
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Finding a prime greater than a given bound

Is a deterministic polynomial-time algorithm known for the following problem: Input: a natural number $n$ (in binary encoding) Output: a prime number $p > n$. (According to a list of open ...
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Is it NP-hard to play international draughts correctly?

Is the following problem NP-hard? Given a board configuration for $n\times n$ international draughts, find a single legal move. The corresponding problem for $n\times n$ American checkers (aka ...
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Computing any information about Max-3SAT

For a 3CNF formula $C$ let $M(C)$ be the maximal number of satisfied clauses in any assignment to $C$. It is known that Max-3SAT is hard to approximate (subject to P≠NP), i.e. there is no polytime ...
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Intermediate problems between L and NL

It is well-known that directed st-connectivity is $NL$-complete. Reingold's breakthrough result showed that undirected st-connectivity is in $L$. Planar directed st-connectivity is known to be in $UL \...
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4answers
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Best known deterministic time complexity lower bound for a natural problem in NP

This answer to Major unsolved problems in theoretical computer science? question states that it is open if a particular problem in NP requires $\Omega(n^2)$ time. Looking at the comments under answer ...
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2answers
988 views

Complexity of factoring in number fields

What is known about the computational complexity of factoring integers in general number fields? More specifically: Over the integers we represent integers via their binary expansions. What is the ...
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1answer
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Consequences of Complete problems for NP intersects coNP

What are the consequences of having complete problems in $NP\cap coNP$?
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Decide whether a matrix's kernel contains any non-zero vector all of whose entries are -1, 0, or 1

Given an $m$ by $n$ binary matrix $M$ (entries are $0$ or $1$), the problem is to determine if there exists two binary vectors $v_1 \ne v_2$ such that $Mv_1 = Mv_2$ (all operations performed over $\...
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1answer
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Complexity of matrix powering

Let $M$ be a square integer matrix, and let $n$ be a positive integer. I am interested in the complexity of the following decision problem: Is the top-right entry of $M^n$ positive? Note that the ...
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What is the complexity of distinguishing a true Fourier spectra from a fake one?

A $PH$ machine is given oracle access to a random Boolean function $f:\{0,1\}^n \to \{ -1,1 \}$ , and two Fourier spectra $g$ and $h$. The Fourier spectra of a function $f$ is defined as $F:\{0,1\}^...
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Is Hankelability NP-hard?

I asked this question on SO on April 7 and added a bounty which has now expired but no poly time solution has been found yet. I am trying to write code to detect if a matrix is a permutation of a ...
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11answers
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Example where equivalence is easy but finding class representative is hard

Suppose we have a class of objects (say graphs, strings), and an equivalence relation on these objects. For graphs this could be graph isomorphism. For strings, we could declare two strings equivalent ...
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Subexponentially solvable hard graph problems

In light of the recent result of Arora, Barak, and Steurer, Subexponential Algorithms for Unique Games and Related Problems, I'm interested in graph problems that have subexponential time algorithms ...
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5answers
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Verifying unique solutions of SAT

Consider the following problem: given a CNF formula and an assignment that satisfies this formula, is there another satisfying assignment for this formula ? What is the complexity of this problem ? (...
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Why do equalities between complexity classes translate upwards and not downwards?

Hey Guys, I understand that the padding trick allows us to translate complexity classes upwards - for example $P=NP \rightarrow EXP=NEXP$. Padding works by "inflating" the input, running the ...
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4answers
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What's the “smallest” complexity class for which a superlinear circuit bound is known?

Apologies for asking a question that must surely be in a lot of standard references. I'm curious about exactly the question in the title, in particular I am thinking of Boolean circuits, no depth ...
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6answers
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Advanced techniques for determining complexity lower bounds

Some of you may have been following this question, which was closed due to not being research level. So, I'm extracting the part of the question which is at a research level. Beyond the "simpler" ...
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3answers
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Constructivity in Natural Proof and Geometric Complexity

Recently, Ryan Willams proved that Constructivity in Natural Proof is unavoidable to derive a separation of complexity classes : $\mathsf{NEXP}$ and $\mathsf{TC}^{0}$. Constructivity in Natural ...
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4answers
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Proofs, Barriers and P vs NP

It is well known that any proof resolving the P vs NP question must overcome relativization, natural proofs and algebrization barriers. The following diagram partitions the "proof space" into ...
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Has there been any research on $k$-SAT above the satisfiability threshold?

A well known characteristic of $k$-SAT instances is the ratio of the number of clauses $m$ over the number of variables $n$, i.e., the quotient $\rho = m/n$. For every $k$, there is a threshold value ...
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3answers
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Complexity of “is a graph a product”

This question arises out of pure curiosity (it came up while thinking about unshuffling a string, but I'm not sure if it's actually related) so I hope it's appropriate. There are various graph ...
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4answers
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Separating Logspace from Polynomial time

It is clear that any problem that is decidable in deterministic logspace ($L$) runs in at most polynomial time ($P$). There is a wealth of complexity classes between $L$ and $P$. Examples include $NL$,...
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Complexity zoo for unary languages

Of course, some complexity results may collapse for unary languages but I wonder if there is somewhere a survey summarizing the known results in this case: a kind of complexity zoo for unary ...
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2answers
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Is there an NP-complete language that contains precisely half of the n-bit instances?

Is there a (preferably natural) NP-complete language $L\subseteq \{0,1\}^*$, such that for every $n\geq 1$ $$|L\cap \{0,1\}^n|=2^{n-1}$$ holds? In other words, $L$ contains precisely half of all $n$-...
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1answer
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What is $\mathsf{NP}$ restricted to linear size witnesses?

This is related to the question Is the Witness Size of Membership for Every NP Language Already Known? Some natural $\mathsf{NP}$(-complete) problems have linear length witnesses: a satisfying ...
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2answers
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Formula size lower bounds for AC0 functions

Question: What is the best known formula size lower bound for an explicit function in AC0? Is there an explicit function with an $\Omega(n^2)$ lower bound? Background: Like most lower bounds, ...
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Mathematical implications of complexity theory conjectures outside TCS

Do you know interesting consequences of (standard) conjectures in complexity theory in other fields of mathematics (i.e. outside of theoretical computer science)? I would prefer answers where: the ...
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4answers
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Arguments for existence of one-way functions

I have read in several papers that the existence of one-way functions is widely believed. Can someone shed light on why this is the case? What arguments do we have for supporting the existence of one-...
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1answer
756 views

Recognizing sequences with all permutations of $\{1, \ldots, n\}$ as subsequences

For any $n > 0$, I say that a sequence $s$ of integers in $\{1, \ldots, n\}$ is $n$-complete if, for every permutation $\mathbf{p}$ of $\{1, \ldots, n\}$, written as a sequence of pairwise distinct ...
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Computational complexity of counting induced subgraphs which admit perfect matchings

Given an undirected and unweighted graph $G=(V,E)$ and an even integer $k$, what is the computational complexity of counting sets of vertices $S\subseteq V$ such that $|S|=k$ and the subgraph of $G$ ...
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2answers
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Natural CLIQUE to k-Color reduction

There is clearly a reduction from CLIQUE to k-Color because they're both NP-Complete. In fact, I can construct one by composing a reduction from CLIQUE to 3-SAT with a reduction from 3-SAT to k-Color. ...
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1answer
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Is cubic complexity still the state of the art for LP?

According to D. den Hertog, Interior Point Approach to Linear, Quadratic and Convex Programming, 1994, a linear program with $n$ variables, $n$ constraints and precision $L$ is solvable in $O(n^3L)$ ...
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Are there NP-complete problems with polynomial expected time solutions?

Are there any NP-complete problems for which an algorithm is known that the expected running time is polynomial (for some sensible distribution over the instances)? If not, are there problems for ...
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5answers
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Implications of unprovability of $P\neq NP$

I was reading "Is P Versus NP Formally Independent?" but I got puzzled. It is widely believed in complexity theory that $\mathsf{P} \neq \mathsf{NP}$. My question is about what if this is ...
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3answers
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Hardness of approximation - additive error

There is a rich literature and at least one very good book setting out the known hardness of approximation results for NP-hard problems in the context of multiplicative error (e.g. 2-approximation for ...
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2answers
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Sum-of-squares proof system

Recently I have seen several articles on arxiv that refer to a proof system called sum-of-squares. Can someone explain what is a sum-of-squares proof and why such proofs are important/interesting? ...
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3answers
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What are the relationships between those hypotheses in Fine-Grained Complexity Theory?

Complexity theory, through such concepts as NP-completeness, distinguishes between computational problems that have relatively efficient solutions and those that are intractable. "Fine-grained" ...
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1answer
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Is $AC^0$ with bounded fanout weaker than $AC^0$?

In the survey "Small Depth Quantum Circuits" by D. Bera, F. Green and S. Homer (p. 36 of ACM SIGACT News, June 2007 vol. 38, no. 2), I read the following sentence: The classical version of $QAC^0$ (...
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Formally Verified Complexity Theory

Is there any ongoing project to formally verify the theorems and proofs of complexity theory using a proof assistant like Coq? Are there any boundaries to doing this?

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