Questions tagged [cc.complexity-theory]
P versus NP and other resource-bounded computation.
2,656
questions
10
votes
1answer
330 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 ...
3
votes
1answer
259 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
1answer
56 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
293 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 ...
16
votes
0answers
435 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
95 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
156 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
288 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
115 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
58 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
456 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 ...
5
votes
1answer
250 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
77 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
245 views
Is $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?
Is it known that $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?
2
votes
0answers
483 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
254 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
118 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 ...
5
votes
2answers
192 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
78 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
45 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 $|A|=k$...
8
votes
1answer
224 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
155 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
208 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
121 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} [ (-...
5
votes
1answer
155 views
What does “hold uniformly” mean in the context of asymptotic analysis?
What does "hold uniformly" mean in the statement of Theorem 1.7 in
A Faster Subquadratic Algorithm for Finding Outlier Correlations? Here's the theorem text ("hold uniformly" is in the last line):
3
votes
0answers
90 views
Critical Assignments vs Read-Once Branching Programs - Reference Request
Straight to the point:
I'm looking for a reference for the fact that the complexity of a read-once branching program solving the search problem for an unsatisfiable formula $F$ is at least the ...
7
votes
0answers
189 views
Are there any known functions in NC2 but not in TC1?
Uniform TC1 is known to be in Catalytic Logspace. To improve the lower bound, we are looking for some functions known to be in NC2 but not in TC1.
Also, what are some of the natural functions known ...
8
votes
1answer
283 views
Nondeterminism is on average useless for circuits?
Savický and Woods (The Number of Boolean Functions Computed by Formulas of a Given Size) prove the following result.
Theorem[SW98]: For every constant $k>1$, almost all boolean functions with ...
1
vote
0answers
95 views
An explicit hard function for P/poly
It is known that $\textbf{MA}_{\textbf{EXP}} \not\subset \textbf{P/poly}$. Is it known any explicit language from $\textbf{MA}_{\textbf{EXP}}$ that does not belongs to $\textbf{P/poly}$?
(An example ...
-1
votes
1answer
61 views
How to make any graph 2-degenerate?
I have to show a PPT(polynomial time reduction) from 'Colorful graph Motif' to '2-Degenerate Steiner Tree'. As input graph should be 2-degenerate, but here is normal graph G (that is, basically an ...
0
votes
0answers
49 views
Complexity of extending $P_4 $-partition of cubic graphs
This is a question I posted on MathOverflow before but never got an answer. I am cross-posting it here.
Surprising phenomena occurs when we want to extend a partial solution of some easy problems. We ...
asked Jun 30 '18 at 16:47
Mohammad Al-Turkistany
19.5k4 gold badges51 silver badges134 bronze badges
6
votes
1answer
270 views
What are the consequences of solving XOR 3-SAT in Logspace?
XOR Formulas
Consider boolean formulas with connectives $\wedge$ (AND) and $\oplus$ (XOR). Such a boolean formula is a valid instance for XOR SAT if it is a conjunction of $\oplus$-clauses. An $\...
10
votes
0answers
242 views
Have people looked for parameterized algorithms for problems that are not in NP?
Are there problems that are not in NP (e.g., NEXP-complete problems) but admit FPT algorithms for a reasonable parameterization (and specifically, the standard parameterization of a problem -- the ...
10
votes
0answers
242 views
How would proof of the Lindelöf hypothesis improve our understanding of computational complexity classes?
A recent press release from the Viterbi School of Engineering at USC discussed the proof of the Lindelöf hypothesis by Athanassios Fokas, a visiting professor from the Department of Applied ...
-1
votes
1answer
96 views
Finding upper and lower bounds of a problem [closed]
We have n balls where 1 is a little heavier than the others and we want to find that heavier ball. We can only put some balls on one side of the scale and some on the other side and see if it leans ...
1
vote
0answers
107 views
Proof of SAT is complete for NP via first-order reductions
So I have been reading this: https://people.cs.umass.edu/~immerman/book/ch7.pdf
I do not understand the proof of theorem 7.16, which says that SAT is complete for NP via first-order reductions. My ...
2
votes
1answer
529 views
What is best known space requrement for solving SATISFIABILITY problem in exp time
I searched a lot for finding best space requirement algorithm for SATISFIABILITY problem but I didn't find any thing better than brute force that is in DSPACE(n). is there exists better bound? and ...
2
votes
0answers
44 views
Arithmetic circuits with restrictions on occurrence of pairs of variables
I am curious if the following model was studied or has some obvious lower bounds:
We want to compute a polynomial $P(x_1,x_2, \dots , x_n)$. Suppose we have a graph G on $n$ nodes that we are going ...
2
votes
3answers
138 views
Reference request: Arithmetic circuit complexity
I am completely new to this field. I want to read the prelims and unfortunately, I don't see any book written for a beginner in this area. Can anyone give me some basic starting points/references(text/...
3
votes
2answers
168 views
What is the best approximation and exact algorithm for vertex cover on cubic graphs?
"Best" = best performing in terms of run-time for exact algorithm and approximation ratio for an approximation algorithm.
2
votes
0answers
870 views
BQNC and Abelian Hidden Subgroup Problem
We know integer factorization is in $BPP^{BQNC}$ from Cleve and Watrous.
Is Abelian Hidden Subgroup Problem also in $BPP^{BQNC}$?
In particular is Discrete Logarithm in $BQNC$ or at least in $BPP^{...
2
votes
0answers
201 views
Primality in $NC$ hierarchy?
AKS primality testing solves whether a given integer is prime in $P$. AKS algorithm is following:
Input: integer n > 1.
Check if $n$ is a perfect power: if $n = a^b$ for integers $a > 1$ and $b &...
0
votes
1answer
109 views
Is Prime Bounded Quadratic Congruence NP-complete?
Bounded Quadratic Congruence:
Instance: Three positive integers $a$, $b$ and $c$.
Question: Is there a positive integer $x<c$ such that $x^{2} \equiv a \, (mod \ b)$?
Bounded Quadratic ...
5
votes
0answers
140 views
How hard is it to generate a set of relatively prime numbers between two given bounds?
Informal Question
How hard is it to generate a set of relatively prime numbers between two given bounds?
Decision Problem
Given $a$, $b$, and $k \in \mathbb{N}$. Does there exist a set $S \...
-1
votes
1answer
53 views
NP-Complete graph problems where a special vertex is given as input?
I am currently working on a graph theory problem where the instance includes a graph and a special vertex in the graph. I am trying to prove the NP-completeness of the problem as well as explore ...
0
votes
0answers
114 views
What definition for $FPT$ algorithm for $KSUM$ gives $W[P]=FPT\implies KSUM$ is $FPT$?
In the definition on $KSUM$ problem we are given $n$ input integers and we have to decide if $K$ of them sum to $0$.
$KSUM$ is $FPT$ if there is a $O(f(K)poly(n))$ algorithm for it. However Downey ...
3
votes
1answer
52 views
Complexity of distributively verifying that the diameter is small
Consider a graph $G=(V,E)$ and an integer parameter $k$.
I'm interested in the round complexity, in the CONGEST model, of checking if the diameter of the graph is "much larger" or "much smaller" than ...
5
votes
2answers
120 views
A stronger multiplexing rule for soft linear logic?
In (intuitionistic) linear logic the usual rules for the storage modality $!$ are promotion, dereliction, contraction, and weakening:
$$\frac{!\Gamma\vdash A}{!\Gamma\vdash !A}(prom) \qquad \frac{\...
4
votes
0answers
77 views
Circuits computing functions of inputs smaller than $n$
The usual circuit complexity concerns circuits where circuit $C_n$ computes function $f_n$. I am interested in circuits such that $C_n$ can compute $f_i$ for all $i \leq n$. I am assuming that the ...
