P versus NP and other resource-bounded computation.
2
votes
2answers
55 views
Is there a non-deterministic version of the complexity class PP?
From a quick skim of the literature (and complexity zoo), there doesn't seem to be a non-deterministic version of PP. Is there a reason for this (e.g. PP=non-deterministic PP?)
4
votes
1answer
73 views
Is algorithmic information theory still evolving?
I am currently looking for a subject for a thesis and encountered the field of algorithmic information theory.
The field seems very interesting for me, but it seems everything is the field was done ...
4
votes
0answers
48 views
Complexity of fractional SAT
Let $(a, k)$-SAT be $k$-SAT with the promise that if there is there is a satisfying assignment, then there is such an assignment that satisfies at least $a$ literals of every clause. Can 3-SAT with $...
6
votes
0answers
78 views
Relatively low ambitious frontiers
What are some of the current "relatively" low ambitious frontiers for MA/PhD thesis in complexity theory class separations/containment or quantum computing?
For example: In the draft version of Arora ...
4
votes
1answer
153 views
Boolean circuits which correspond to L/poly
Branching programs are usually used as a computation model for non-uniform logarithmic space $\mathsf{L}/\mathrm{poly}$.
Is there a reference about Boolean circuits corresponding to $\mathsf{L}/\...
6
votes
0answers
153 views
How many different proofs are there of parity is not in AC0?
The theorem that Parity is not in $\mathsf{AC}^0$ is one of the gemstones of complexity theory. I wonder how many different proofs there are of this result? What constitutes "different" is also a part ...
8
votes
0answers
208 views
L/quasipoly vs NL/poly
Savitch's theorem shows that NSPACE($S(n)$) $\subseteq$ SPACE($S(n)^2$), which means that nondeterminism can be replaced by more spaces in this situation. Is it known whether nondeterminism can be ...
0
votes
0answers
27 views
How should a reduction to the Cardinality Constrained Quadratic Knapsack Problem work?
in Polyhedral Study of the Cardinality Constrained Knapsack Problem the authors prove that the Cardinality Constrained Knapsack Problem is NP-Hard by reducing PARTITION to it.
Besides, it's easy to ...
7
votes
1answer
172 views
TIME(n) versus TIME(nlogn)
The time hierarchy theorem implies TIME($n$) is strictly contained in TIME($n\log^{1+ε}n$) for all ε>0. Is the relationship between TIME($n$) and TIME($nlogn$) known?
6
votes
0answers
88 views
Efficient quantum algorithm for CLASSICAL FFT
Is there a known improvement on the current O(n*log(n)) algorithm for CLASSICAL FFT using quantum computation? 'n' is the number of samples.
I need to find the amplitude and phase of the K dominating ...
3
votes
1answer
93 views
Does a non-constructive proof of bounds of a computable asymptotic complexity, with impossible fix, exist?
Does there exist an algorithm, about which a non-constructive $\omega$-consistent theory $A$ can prove that it has time complexity $O(f(n))$ where $n$ is some univariate function of the input, but ...
9
votes
4answers
232 views
Example where equivalence is easy but finding class representative is not
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 ...
-2
votes
0answers
68 views
Converting a propositional formula into an equisatisfiable CNF
For my homework I am asked the following question and determine whether it is true or false:
Converting a propositional formula into an equisatisfiable CNF formula in the worst case requires ...
7
votes
0answers
50 views
Complexity of deciding whether subspaces of Z_2^n cover every point 3*x times
When studying the complexity of checking identities in certain finite algebras, I came across the following decision problem:
Input: A positive integer $n \in N$ and a set of affine subspaces $H_1,...
0
votes
0answers
56 views
Proving that a problem is coNP-complete [duplicate]
Let $\le^{p}_{T}$ be the Turing or the Cook reduction and $\le_{m}^{p}$ the Karp-Levin reduction.
I know that to prove a problem $P_1$ is coNP-complete I just need to show that $P_1 \in coNP$ and ...
5
votes
0answers
74 views
Parsimonious Reduction from Unique-3SAT to NAE-3SAT
Using the result by Valiant and Vazirani, we know that Unique-3SAT (3SAT with a unique solution) is hard unless NP=RP. Also it is widely believed that the "Unique" version of any NP-complete problem ...
1
vote
0answers
37 views
How to efficiently verify if a semantic symmetry of a CNF formula is valid?
It is easy to verify that a syntactic symmetry of a CNF formula is correct.
Is it also possible to check in polynomial time that a semantic symmetry which is not a syntactic symmetry of a formula is ...
8
votes
2answers
236 views
Non-Orthogonal Vectors Problem
Consider the following problems:
Orthogonal Vectors Problem
Input: A set $S$ of $n$ Boolean vectors each of length $d$.
Question: Do there exist distinct vectors $v_1$ and $v_2 \in S$ ...
4
votes
3answers
117 views
Complexity of isotopy of embedded graphs
I am looking for previous work on the following problem: given two graphs embedded in the plane without crossing, determine if they are isotopic. By isotopic I mean that there is a continuous ...
4
votes
0answers
81 views
Asymptotic complexity of mass production
For a function $f:\{0,1\}^n \rightarrow \{0,1\}^m$, let $C(f)$ be the circuit complexity (for concreteness, constants and NOT gates are free, while 2-input AND gates cost 1).
Let $k{\times}f : \{0,1\}...
10
votes
1answer
185 views
Fast classical simulation of quantum algorithms
Are there examples of cases where the classical simulation of a quantum algorithm for a problem outperforms the best previously known classical algorithm for this problem? "Outperforms" doesn't have ...
1
vote
0answers
54 views
Reasoning about NP hardness of optimization problems with closed form functions as input
(This may not be a research level question per se. I can delete this question if the community thinks this way too)
I am trying to understand how to reason about hardness of optimization problems ...
10
votes
1answer
281 views
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, but ...
0
votes
0answers
29 views
Minimum cost circulation problem with bounded number of edges
Note: the question was taken from https://cs.stackexchange.com/questions/95479/minimum-cost-circulation-problem-with-bounded-number-of-edges
Since there was no answer in that forum (even after ...
3
votes
1answer
165 views
Conesequences of $\forall k\in \mathbb N \space NP\not\subseteq TISP(poly(n),n^k)$
Does $\forall k\in \mathbb N \space NP\not\subseteq TISP(poly(n),n^k)$ has any separation of classes or consequences?
My main question is can use this to show that $P \neq NP$ or some thing useful ...
0
votes
0answers
49 views
Modular and multiplication operations
Does
modular operation reduce to integer multiplication
integer multiplication reduce to modular operation
in fully linear time and/or in logarithmic time on linear number of processors?
0
votes
1answer
42 views
Understanding the definition of a “restriction of a resolution derivation”
I am reading the paper An Introduction to Lower Bounds on Resolution Proof Systems. In its subsection 2.3 the author gives an inductive definition of the restriction of a resolution refutation $\pi$.
...
-2
votes
1answer
215 views
Why is it a mystery if PSPACE ?= EXPTIME?
It seems obvious to me that $PSPACE \neq EXPTIME$. I, however, do not believe that my seemingly obvious logic would not be picked up by more intelligent people if it was so simple, so I'm assuming ...
15
votes
0answers
400 views
An algebra of complexity classes
A key feature of unrelativized computation is its composability out of smaller fragments, and to partially capture the composability, I came up with an algebra of fine-grained complexity classes.
For ...
3
votes
1answer
62 views
NP completeness of Hamiltonian cycle for the family of *dual graphs* to plane, cubic, triply connected graphs?
It is well known that the Hamiltonian cycle problem is NP-complete on the family of planar, cubic and triply connected graphs: https://epubs.siam.org/doi/abs/10.1137/0205049
For a problem I'm ...
6
votes
1answer
135 views
Does there exist a polynomial time algorithm to determine whether an equation is a consequence of (x+y)*(y+z)=(x*y)+(y*z)?
Consider the identity $C$
$$(x+y)\times(y+z)=(x\times y)+(y\times z).$$
Is the problem of determining whether an identity is a consequence of $C$ in universal algebra a polynomial time problem? This ...
10
votes
1answer
182 views
Is this game EXPSPACE-complete?
Let $M$ be a polynomial-time deterministic machine that can ask questions to some oracle $A$. Initially $A$ is empty but this is can be changed after a game that will be described below. Let $x$ be ...
5
votes
1answer
92 views
Is Bayes optimal RL of a finite set of DFAs feasible?
Let $Q$ be a finite set of states, $\Sigma$ a finite alphabet, $q_0\in Q$ the start state and $F\subseteq Q$ the set of accepting sets. Let $\{\delta_k:Q\times\Sigma\rightarrow Q\}_{k=1}^n$ be a set ...
3
votes
0answers
50 views
Pseudodeterministically choosing elements from efficiently samplable distributions (or, the plausibility of a weak choice principle)
Suppose we have a poly-time samplable family of distribution. I.e., a family of distributions $D_n \subseteq \{0, 1\}^{\mathsf{poly}(n)}$ and an algorithm $S$ for which $D_n = (r \leftarrow^\$ \{0,1\}^...
5
votes
3answers
436 views
Sorting a programs instructions until it works
Lets say I have a computer program below.
(define (factorial x)
(if (= x 0)
1
(else (* x (factorial (- x 1)))))
I then take each line of the ...
0
votes
0answers
65 views
Formal definition of alternative logarithic space turing machine
In barak and arora and other references I found the formal definition for alternative machines and NL but there is no formal definition for AL.
NL definition : there is a read only tape for input and ...
5
votes
1answer
197 views
Distributions which are intractable to sample from?
I'm looking for an interesting family of probability distributions $P$ that is intractable to efficiently sample from.
I'm not sure what the right notion of intractable is, though I know the notion ...
1
vote
1answer
56 views
Can we replace deterministic part of alternative turing machine with some other equivalent machines?
I'm sorry if it is a low level question but I am so confusing.
If $DTime(n)\subseteq \Sigma_2Time(n^{0.2})$ then $DTime(n) \subseteq \Sigma_2DTime(n^{0.2})$
Is this true that $\Sigma_2DTime(n^{0.2})...
4
votes
1answer
218 views
Is $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?
Is it known that $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?
1
vote
0answers
163 views
Relationship of the P=PSPACE problem to alternating complexity classes
From a very naive point of view of someone who haven't studied complexity theory in depth, I am wondering whether the theorems that APSPACE=EXPTIME and AP=PSPACE have any derived results on separating ...
3
votes
1answer
207 views
Can we efficiently distinguish between P and BPP?
Let's say algorithm $D$ distinguishes $BPP$ from $P$ if
there exists a language $L \in BPP$ such that for all $A \in PTM$,
$$D(\langle A\rangle) \in L \leftrightarrow D(\langle A \rangle) \notin L_A$$...
4
votes
0answers
108 views
A variant of hitting set: finding a matching to hit all edge sets
In general, the hitting set problem is given a family $\cal S$ of sub-sets, $\{S_1, \cdots, S_h\}$, and a universal set $U = \bigcup_{i\in [1,h]} S_i$. It asks for a minimum set $H \subseteq U$ ...
14
votes
1answer
1k views
Deciding whether an interval contains a prime number
What is the complexity of deciding whether an interval of the natural numbers contains a prime? A variant of the Sieve of Eratosthenes gives an $\tilde O(L)$ algorithm, where $L$ is the length of the ...
4
votes
2answers
119 views
Max cut problem between two connected subgraphs
Let $G$ be a connected graph.
Consider the problem of finding a partition $G = A \cup B$ into connected subgraphs, so that the cut between $A$ and $B$ is maximized. Is there anything which is known ...
-1
votes
1answer
59 views
Restricted Universe Exact Cover
Apologies for a simple question - I am a beginning graduate student in TCS.
Consider the following $\mathrm{ExactCover}$ problem: Given a collection $\mathcal{S}$ of subsets of a universe set $U$ and ...
2
votes
0answers
37 views
Min cut problem on unbalanced partitions for planar graphs with unit capacity edges
The question is: given a planar graph $G$ with unit capacity edge weights and a fixed positive integer $k$, what is an approximation algorithm for finding the minimum size of a cut $(A,B)$ with $|A|=k$...
6
votes
1answer
134 views
How “hard” is it to maximize a polynomial function subject to linear constraints?
General Problem
Suppose we have a multivariate polynomial function $f(\mathbf{x})$, and several linear functions $\ell_i(\mathbf{x})$. What is known about the complexity of solving the following ...
4
votes
0answers
116 views
Simulate a heap in linear time
Is there anything in the literature on the following problem?:
Take a sequence of operations of Insert(element) and PopMin and ...
3
votes
0answers
161 views
What is consequence of $PH\subseteq NSPACE((\log n)^2)$?
What is consequences of $PH\subseteq NSPACE((\log n)^2)$?
We don't even know PH is equals to L or not. I am wondering what will be happened when $PH\subseteq NSPACE((\log n)^2)$?
6
votes
1answer
108 views
Average-case analogue of Small-bias Spaces
Recall that an $\epsilon$-biased space is a set $S \subset \{0,1\}^n$ such that for every non-zero linear test $\alpha \in \{0,1\}^n \setminus \{0\}^n$, the expected bias
$$| \mathbb{E}_{x \in S} [ (-...
