Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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9
votes
3answers
492 views

Why exactly are complexity theorists interested in closed timelike curves?

Context: There are several papers that study the implications of closed timelike curves (CTCs) to quantum complexity. In 2008, Aaronson and Watrous published their famous paper on this topic which ...
31
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6answers
2k views

Is there a natural problem on the naturals that is NP-complete?

Any natural number can be regarded as a bit sequence, so inputting a natural number is the same as inputting a 0-1 sequence, so NP-complete problems with natural inputs obviously exist. But are there ...
20
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1answer
454 views

Is prime-counting function #P-complete?

Recall $\pi(n)$ the number of primes $\le n$ is the prime-counting function. By "PRIMES in P", computing $\pi(n)$ is in #P. Is the problem #P-complete? Or, perhaps, there is a complexity reason to ...
4
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1answer
210 views

Why is counting the number of hamiltonian subgraphs $\sharp P $ hard?

I'm confused about how to prove either of the following closely related statements. They are both from this paper: https://epubs.siam.org/doi/10.1137/0208032 1) "A further problem that can be shown ...
3
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0answers
376 views

Convention for RAM machine models

When algorithm asymptotic runtimes are given without explicitly noting the computational model, what is the convention for the exact model used? My understanding is that most problems use unit-cost ...
3
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0answers
72 views

$XP_{\text{uniform}}=FPT$ and update to $EPTAS$ section in complexity zoo?

Complexity zoo in https://complexityzoo.uwaterloo.ca/Complexity_Zoo:E#eptas has the following: $FPT = XPuniform\implies EPTAS = PTAS$. Fundamentals of Parametrized complexity on page $534$ has ...
2
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0answers
142 views

Is the following problem in $coNP$?

Given an $n\times n$ matrix $M$ with $\mathbb Z$ entries is 'does an $\frac n2\times\frac n2$ minor of $M$ vanish?' in $\bf{coNP}$? At least one $\frac n2\times\frac n2$ minor non-vanish implies rank ...
2
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1answer
132 views

Lower bound on alternations needed in $BQP$ versus $PH$ result?

What is the fastest $f(n)$ the relatively new result of oracle separation of $\mathsf{BQP}$ from $\mathsf{PH}$ provides such that ${\#\mathsf{SAT}}\not\subseteq\mathsf{FP}^{\mathsf{PH}[O(f(n))]}$ ...
10
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1answer
412 views

An obstruction like ETH

We know under $ETH$ we cannot solve $K$-SUM in $f(K)poly(nK)$ time under any function $f(K)$ (usually $2^{O(K)}$). Is there any conjecture that prevents a $(\log n)^{O(K)}$ complexity (this is ...
3
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1answer
82 views

Hardness of LWE on not-uniform vector samples

The "usual decisional LWE": The challenger and the adversary get a common random matrix $A \in F_{q}^{m \times n}$. The challenger chooses a secret $s \in F_{q}^{n}$ and generates random (small) ...
-1
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1answer
95 views

Why is $BPP^{NP}$ in polynomial hierarchy? [closed]

Why is $BPP^{NP}$ in the polynomial hierarchy? I know that $BPP$ is contained in $NP^{NP}$, so $BPP$ is inside $PH$. However, how does that imply $BPP^{NP}$ is inside the polynomial hierarchy?
2
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1answer
156 views

How to use a 𝑝-coin so a TM can decide an undecidable language in polynomial time? [closed]

In "Computational complexity- A modern approach" book (page 117) for the lemma 7.12 (following) the author mentioned that if the ρ is efficiently computable ρ-coin cannot give probabilistic algorithm ...
10
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1answer
333 views

How hard is deciding the existence of Red-Blue perfect matching?

Two-colorable perfect matching problem is to decide whether a graph has coloring with two colors such that each node has exactly one neighbor the same color as itself. The problem was proven to be NP-...
2
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0answers
124 views

A complexity-class of problems that cannot be solved in finite time

Consider the following game: Alice chooses a real function number $x\in [0,1]$; Bob has to guess the number by asking Alice any number of queries of the form "is $x > a$?" [where Bob can choose ...
11
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0answers
168 views

$\exists \mathbb R$ and IP

We know NP$\subseteq$ $\exists \mathbb R$$\subseteq$ PSPACE=IP, but is there some more direct proof for $\exists \mathbb R\subseteq$ IP? What about the other direction, are there some Arthur-Merlin ...
1
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0answers
86 views

Minimum cut with nonlinear objective function

Let $G$ be an undirected graph. The classic minimum (cardinality) cut problem asks for a cut $C\subseteq E(G)$, such that $|C|$ is minimum. Let us generalize it the following way: let $f$ be a ...
4
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0answers
94 views

Counting quotient graphs, but not exactly

All graphs considered will be directed graphs $G=(V,E)$, with $E \subseteq V \times V$ (so possibly with self-loops). For $k \in \mathbb{N}_{\geq 1}$, I will write $[k]$ the set $\{1,\ldots,k\}$. A $k$...
29
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1answer
2k views

Fourier coefficients Boolean Functions described by Bounded Depth Circuits with AND OR and XOR gates

Let $f$ be a Boolean function and let's think about f as a function from $\{-1,1\}^n$ to $\{ -1,1 \}$. In this language the Fourier expansion of f is simply the expansion of f in terms of square free ...
2
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1answer
160 views

Is balanced Hamiltonian cycle NP complete on maximal plane graphs?

I know that the Hamiltonian cycle is NP complete on the class of maximal plane graphs. If we instead ask about balanced Hamiltonian cycles (i.e. same number of faces on both sides) on maximal plane ...
1
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1answer
174 views

Does Hadwiger conjecture imply that NP = coNP?

(Disclaimer: I suspect the answer is no, but I fail to see why) Here is a nice picture by David Epstein (taken from Wikipedia) illustrating Hadwiger's conjecture: The point is that if in a given ...
1
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0answers
123 views

Canonical complete problem for $\mathrm{FP}^{\Sigma^p_2}$

Given a $\Sigma^p_2$-complete oracle (i.e., $\Sigma_2 \mathrm{SAT}$), I have a problem that requires to call this oracle polynomially many times and returns an integer. Essentially, this is a function ...
2
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0answers
59 views

Complexity of comparing extended integer power towers

Inspired by this stackexchange question, is it an open problem to compare two power towers of positive integers if we additionally allow numbers lower in the tower to themselves be represented by ...
4
votes
1answer
164 views

How small can extension complexity be?

In this article on extension complexity of regular polygons https://arxiv.org/pdf/1505.08031.pdf it is mentioned that extension complexity of $n$ regular polygons should be $\theta(\log n)$. This is ...
1
vote
1answer
74 views

Complexity status of the Edge Deletion problem to bounded degree graphs

I'm interested in the complexity status of the following problem. Input: a graph $G=(V,E)$ and two natural numbers $k$ and $d$. Output: Yes, if there exists a subset $E' \subseteq E$ of cardinality ...
2
votes
2answers
779 views

Strongly NP-complete variants of subset sum or partition problem

Some problems have variants that appear to be harder. For instance, Graph Automorphism (GA) problem has quasi-polynomial time algorithm ( by Babai's GI result). However, the fixed-point free GA ...
1
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0answers
85 views

Entropy bounds on solutions to problems in BPP and other complexity classes based on entropy demands

Has anyone studied the asymptotics of problems in complexity classes like $BPP$? The thought came to me that if a problem in $BPP$ only requires $O(log(n))$ bits of entropy to solve then, intuitively, ...
15
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1answer
283 views

Does requiring uniqueness of valid answers for Merlin limit the power of Arthur-Merlin protocols?

Preamble. The complexity class AM are those problems which can be solved by a two-round interactive proof system between a prover "Merlin" and a verifier "Arthur". A problem — which tests some ...
2
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0answers
78 views

how is time complexity defined in computational learning theory

In general, when we say an algorithm $A$ PAC learns $C$ in time $t$, we say $A$ takes time $t$ before outputting a hypothesis $h$, and the hypothesis can be evaluated (on every $x$) in time $t$. Now ...
8
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0answers
118 views

Can we define a meaningful concept of exptime reductions (as opposed to polytime reductions) for classes like NEXP or NEEXP?

Typically we are only interested in polytime reductions as we are usually interested in showing a reduction from one NP-problem to another. However, if we consider larger complexity classes such as ...
1
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0answers
99 views

Complexity of planted root of a system of quadratic homogeneous polynomials?

Given homogeneous degree $2$ randomly chosen polynomials $f_1,\dots,f_{m}$ in $\mathbb Z[x_1,\dots,x_n,y_1,\dots,y_n]$ each with only monomials $x_iy_j$ with condition that the system $f_1=\dots=f_{m}=...
1
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0answers
84 views

Problems rephrased as quadratic unconstrained binary optimization

I was impressed when i came across Quadratic unconstrained binary optimization (QUBO) recently, and saw how one can rephrase many combinatorial problems into questions about optima of binary functions....
8
votes
4answers
402 views

In-approximability results in severely restricted graph classes

Longest path problem is not polynomial-time approximable to any constant factor in cubic Hamiltonian graphs (Longest path $\notin APX$ unless $P=NP$). I don't know if it remains in-approximable in ...
4
votes
6answers
433 views

Is there a linear space lower bound for streaming set equality?

Consider two streams. In each stream one string arrives at a time. A query asks: Is the set of strings that has arrived so far the same in both streams? Is there a linear space randomized lower ...
3
votes
0answers
172 views

Take a NEXP-complete problem and then have the input in unary. Why is this not NP-complete?

It is known that if any unary language is NP-complete, then P=NP. Suppose we take a NEXP-complete language with input $x$ in binary and witness $y\in\{0,1\}^{2^{poly(|x|)}}$ such that the verifying ...
11
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0answers
206 views

Computational Complexity of the Frobenius Problem

The Frobenius problem takes as input $n$ positive integers $a_1,\ldots,a_n$ with $\gcd(a_1,\ldots,a_n)=1$ and asks for the largest integer $F$ that cannot be written in the form $F=a_1x_1+a_2x_2+\...
28
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1answer
1k views

Functions that are Not Efficiently Computable but Learnable

We know that (see, e.g., Theorems 1 and 3 of [1]), roughly speaking, under suitable conditions, functions that can be efficiently computed by Turing machine in polynomial time ("efficiently computable"...
1
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2answers
110 views

When is a problem specified on a TM contained in non-uniform classes such as P/poly? [closed]

In this paper by Gottesman and Irani: https://arxiv.org/abs/0905.2419 , they prove NEXP-hardness of tiling an $N\times N$ grid. They do so by encoding a TM in the tiles making up the grid. However, ...
4
votes
0answers
73 views

Refinement of the hierarchy theorem of a complexity class

Say $\mathrm{CTIME}$ is some complexity measure, syntactic or semantic (e.g. $\mathrm{DTIME}$ or $\mathrm{BPTIME}$). If we already know that for some $f(n) = \omega(n)$, $\mathrm{CTIME}(n) \subsetneq \...
13
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2answers
1k views

Is ALogTime != PH hard to prove (and unknown)?

Lance Fortnow recently claimed that proving L != NP should be easier than proving P != NP: Separate NP from Logarithmic space. I gave four approaches in a pre-blog 2001 survey on diagonalization ...
0
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1answer
79 views

Question on deduction that a certain problem requires exponential space

My question concern's a statement from the classic paper The equivalence problem for regular expressions with squaring requires exponential space. Regular expressions with squaring are like ordinary ...
9
votes
1answer
264 views

Depth reduction for Boolean circuits

This result by Tavenas, Koiran and others show that any polynomial computed by a circuit of size $s$ is computed by a depth-4 homogenous circuit of size $s^{\sqrt{d}}$. Are there any similar results ...
6
votes
1answer
245 views

Example problem that is not in $2^{o(n)}$ but could be solved in $O(2^{cn})$ for any $c > 0$ (suggested by wording of ETH)

In the wikipedia article on Time Complexity it is written that: The exponential time hypothesis (ETH) is that 3SAT, the satisfiability problem of Boolean formulas in conjunctive normal form with, ...
1
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0answers
252 views

On $BPP$ in $P^{NP}$ and $SETH$

It is believed showing $BPP$ in $P$ involves good $PRG$s and faces lower bound barriers. Does showing $BPP$ in $P^{NP}$ which would mean $BPP\neq EXP^{NP}$ face similar $PRG$ and give lower bounds? ...
14
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4answers
2k views

Counting the number of vertex covers: when is it hard?

Consider the #P-complete problem of counting the number of vertex covers of a given graph $G = (V, E)$. I'd like to know if there is any result showing how the hardness of such problem varies with ...
13
votes
2answers
543 views

Does PPAD really capture the notion of finding another unbalanced vertex?

Complexity class PPAD was invented by Christos Papadimitriou in his seminal 1994 paper. The class is designed to capture the complexity of search problems where the existence of a solution is ...
67
votes
7answers
3k views

Which interesting theorems in TCS rely on the Axiom of Choice? (Or alternatively, the Axiom of Determinacy?)

Mathematicians sometimes worry about the Axiom of Choice (AC) and Axiom of Determinancy (AD). Axiom of Choice: Given any collection ${\cal C}$ of nonempty sets, there is a function $f$ that, given a ...
7
votes
0answers
124 views

Can relativization technique be applied to natural NP-complete languages?

Levin [1] defined distNP is the distributional problem (L,D), where L ∈ NP, and D is an ensemble of efficiently samplable distributions over problem instances. We say that a distNP problem (L,D) is ...
12
votes
3answers
895 views

Edge-partitioning cubic graphs into claws and paths

Again an edge-partitioning problem whose complexity I'm curious about, motivated by a previous question of mine. Input: a cubic graph $G=(V,E)$ Question: is there a partition of $E$ into $E_1, E_2, \...
9
votes
1answer
275 views

What is the reference for the proof Gödel's first incompleteness theorem based on the undecidability of the halting problem?

A weaker form of Gödel's First Incompleteness Theorem, direct proofs of which in Gödel's manner are lengthy, involved and at some place rather counter-intuitive, has a simple and intuitive proof based ...
3
votes
1answer
113 views

Computational hardness for sampling a uniform matching

A famous result of Jerrum, Sinclair, and Vigoda shows that there exists a polynomial-time algorithm which takes a bipartite graph $G$ and produces a random perfect matching $M$ of $G$ (assuming one ...

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