Questions tagged [lower-bounds]
questions about lowerbounds on functions, usually the complexity of an algorithm or a problem
279
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Switching lemma for polynomials over $\mathbb{F}_2$
Suppose $f$ is in $\mathbb{F}_2[x_1,...,x_n]$ with total degree $d$.
Q. Is there any kind of switching lemma or restriction lemma in which by applying the lemma on $f$ we can reduce the total ...
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Big-O bounds on the k-th largest element of iid Gaussians
I'm interested in the following problem. Let $X_1, \dots, X_n$ be iid samples with a $N(0,1)$ distribution. Let $X_{[k]}$ be the $k$-th largest element of $\{X_1, \dots, X_n\}$, so e.g. $X_{[1]} = \...
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Lower bound for permutation generator
I'm interested in a problem akin to combinatorial circuits, but in terms of complexity. Apologies for missing the correct terminology, I'll appreciate any corrections.
Given $n$ inputs numbered $1 ......
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Can three stacks be implemented in one array, with O(1) push/pop time?
Two stacks can be efficiently implemented using one fixed sized array: stack #1 starts from the left end and grows to the right, and stack #2 starts from the right end and grows to the left. Is the ...
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47
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Arranging sets in a hierarchy
Suppose you have sets $S_1, \dots S_m$ such that $\sum_i |S_i| = n$. The goal is to arrange all the sets into a (possible unconnected) DAG such that $S_i$ is a parent (or ancestor) of $S_j$ iff $S_j \...
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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, ...
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The SQ argument in Balazs Szorenyi's paper
I am asking about the proof in Theorem 5 (page 6) of this paper,
http://www.inf.u-szeged.hu/~szorenyi/Cikkek/sq_d0_ext.pdf
Quite a few things about this short argument seem unclear to me,
Towards the ...
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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?
...
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Strong seeded randomness extractors with low entropy loss
I would like to implement a strong seeded randomness extractor for flat sources as a part of my project.
Most of the literature on seeded extractors is concentrated on minimizing seed length. ...
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47
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Optimally fair stable matching
There's a nice post by Gil Kalai which outlines the inherent bias in stable matching algorithms quantitatively.
In the traditional loyd shapeley algorithm for $n$ men and $n$ women, given randomly ...
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Why do most 0/1 matrices need linear arithmetic circuits of size $\Omega(n^2/\log(n))$?
I am reading Alon et al.'s paper Linear Circuits over $GF(2)$ and I am having trouble seeing the counting argument showing that most matrices need a circuit of size $\Omega(n^2/\log n)$. This result ...
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Lower bound for enumerating k closest pair of points
Consider 1 dimensional points $x_1, \dots x_n, x_i \in \mathbb{R}$ where the distance between any two points is defined as $d(x_i, x_j) = \vert x_i - x_j \vert$. The goal is to enumerate all $n^2$ ...
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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 ...
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Lower bound for reversing a list using queues
How do you prove (or disprove) that a list of length $n$ cannot be reversed in time $o(n \log n)$ using $O(1)$ queues?
Each queue is FIFO. Time refers to the number of operations on the queues.
...
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Inapproximability Results for APX-hard Geometric Optimization Problems
A lot of Geometric Optimization problems are NP-hard and APX-hard to approximate (No PTAS unless P=NP). One example of the geometric set cover problem can be found here. However, it is not easy to ...
user17918
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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 ...
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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 \...
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Lower bound for triangle-free graphs
I was reading a set of notes where it says
It can be shown that $\Omega(n^2)$ space is needed for one-pass algorithms to determine if an (unweighted, undirected) graph $G$ with $n$ nodes contains ...
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How fast can we find and disconnect roots in a forest?
Consider a forest of rooted trees. The problem is to support two operations:
disconnect(v): if v is the root of some tree in the forest, remove all edges of v;
findroot(v): find root of the tree ...
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Petri net termination
Termination is the following problem.
Input: a Petri Net with initial marking
Output: "yes" iff there exists an infinite firing sequence. The naive algorithm in the case of bounded nets for example ...
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How hard is APPROXIMATE-#SAT?
It is well known that the problem of counting the satisfying assignments of SAT, namely the problem #SAT, is #P-complete.
It is also suspected (somewhat less widely) that even deciding SAT should ...
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Which $SIZE$-$DEPTH(s, d)$ classes with $log(s(n))^{d(n) - 1} = o(n)$ can we not separate by known methods?
Define $SIZE$-$DEPTH(s, d)$ to be the functions which are computed by circuit families of size $O(s(n))$ and less than depth $d(n)$.
We know from Boppana's 1997 paper on the average sensitivity of ...
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Should GCT focus on $PSPACE\not\subseteq P/poly$?
GCT tries to show $P$ is not $NP$ by showing $NP$ is not in $P/poly$.
Could it be useful in showing $\Sigma_{i+1}\not\subseteq P^{\Sigma_i}/Poly$ at every $i>0$?
Suppose if it turns out that $\...
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Separations between testing and tolerant testing for (natural) classes of functions?
Note: I am considering here property testing in the query model, with regard to Hamming distance. (So, for instance, of Boolean functions—I'm phrasing it that way below.) I am in particular not ...
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Is $NEXP^{NP}$ known to not be contained in $NP/poly$?
To the best of my knowledge, it is known that $NEXP^{NP} \nsubseteq P/poly$, but it's still not known if $NEXP \nsubseteq P/poly$.
For more info, see "Superpolynomial circuits, almost sparse oracles ...
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What do stronger circuit lower bounds give in terms of derandomization?
We have $EXP\not\subseteq P/poly\implies BPP\subseteq io-DTIME(2^{n^\epsilon})$ at every $\epsilon>0$.
This is essentially $DTIME(2^{O(n)})\not\subseteq P/poly\implies BPP\subseteq io-DTIME(2^{n^\...
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Non-Uniform Lower Bounds for NSPACE
If I'm not mistaken it is not known whether $E^{NP} \subseteq {\rm SIZE}(n)$
where $E^{NP}$ is the class of problems solvable by a TM which works in time $2^{O(n)}$ and is allowed to make queries of ...
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Common terminology used for lower/upper bounds
Suppose you have developed an upper bound on the number of vertices of a particular graph. This bound is the best possible bound that can be found for any given instance. What do you call such a bound?...
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\alpha-path on Euclidean graphs
Consider the following problem:
Suppose we are given a G=(V, E) Euclidean Graph in the plane and a real $\alpha > 0$. For simplicity assume, there exists only one path whose summation of weights ...
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Problem dependent lower bound for stochastic bandits with full information
Suppose you have a $K$ armed stochastic bandit problem but with full information. There are $K$ arms with mean rewards $\mu_1,...,\mu_K$. At each step we have to select an arm, collect the reward from ...
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Reference for a circuit lower bound for slightly superexponential time
It is known that $EXP$ doesn't have circuits of size $n^k$. On the other hand proving $10 n$ lower bound on circuit size for $E$, $NE$ or even $E^{NP}$ is a known open problem.
My question is ...
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What are some problems in $P$ which have lower bounds assuming that $P \neq NP$ or the ETH?
Last year I had watched this talk online about problems in $P$ for which an algorithm which runs in some subclass of $P$, say in subquadratic time, would imply $P = NP$, or violate the ETH (...
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Lower bound for Yao's algorithm on general addition chains?
An addition chain of size $n$ for given integers $n_1,n_2\dots ,n_p$ is a sequence of integers $k_1,k_2\dots ,k_n$ such that
$k_1=1$,
for all $i$ $(2\le i\le m)$ we have $k_i=k_j+k_m$ for some $1\le ...
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Circuit's with gate computing only "simple" functions
I was curious if something like that is known or was studied.
Let's call a function simple if it is computable by $AC_0$ circuit of depth $\leq d$ and size $\leq n^k$ for fixed $k,d$.
Now let's ...
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Compressing information about the halting problem for oracle Turing machines
The halting problem is well-known to be uncomputable. However, it is possible to exponentially "compress" information about the halting problem, so that decompressing it is computable.
More precisely,...
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Application of weak determinantal identities to GCT?
In general determinants have many identities. Would it help the $GCT$ program by invoking the paradigm of identities such as to state that if the permanent is converted to determinant then it has to ...
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Fourth(?) moment method for minimum value
I would like to lower bound the quantity $\Pr[X\ge t, Y\ge t]=\Pr[\min(X,Y)\ge t]$ using the small moments of $X$, $Y$ and $XY$.
In particular I am interested in the case where $E[X]=E[Y]=0$, but $E[X ...
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On $\Delta_i^P$
We know $P\subseteq NP\cap coNP\subseteq\Delta_i^P=P^{\Sigma_{i-1}^P}\subseteq \Sigma_i^P\cap\Pi_i^P=NP^{\Sigma_{i-1}^P}\cap coNP^{\Sigma_{i-1}^P}$.
If $P=BPP$ is there a 'higher' randomized class ...
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Tribes function vs (modified) polynomial threshold circuits
Are there lowerbounds known for representing the Tribes function by a circuit consisting of a single layer of polynomial threshold gates feeding into maybe a trivial summing gate? (Even for degree $1$ ...
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Counting xyz-graphs in $\mathbb{Z}_n^3$
This is a followup question to: Lower bound on the largest restrained cubic subset
How many distinct xyz-graphs exist in $\mathbb{Z}_n^3$? We denote this number as $C(n)$
This question may be seen ...
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Lower bound on the largest restrained cubic subset
Consider an $n \times n \times n$ cube. I would like to consider subsets of points in the cube with the two following constraints:
Each row in the cube (in any of the three directions) has exactly 2 ...
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Reference request: complexity of $k$-partite $k$-SAT
Let's consider following variation of $k$-SAT that I will call $k$-partite $k$-SAT:
given $n$ variables that are divided into $k$ groups and a $k$-SAT formula $\phi$ such that each clause has literal ...
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Information theoretic lower-bound on object graph serialization
This might be a daft quesstion, but here comes. I became intriqued about data serialization formats and tried to look for research on what could be the information theoric lower bound on encoding ...
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How to prove "obvious" facts?
The title is somewhat "arrogant": say, most of us treat $P\neq NP$ as an "obvious" fact, albeit no proof is in sight. But my question is at a much, much lower level, is about a fact which "should be" ...
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Using epsilon biased sets for circuit lower bounds
I have seen instances of how the technique of epsilon biased sets can be used to construct hard functions against a circuit class - like how in the recent paper of Kane-Williams this was used to ...
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Converse to natural proofs theorem?
Natural proofs paper shows 'if there is a natural property not possessed by any function in P/poly then there is no $2^{n^\epsilon}$-hard PRG'.
Is it easy to see the converse 'if there is no $2^{n^\...
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Where can I find examples of error correcting codes of the following types?
First, apologies if this question is in appropriate or trivial for this site. I'm a physicist looking for some help outside his comfort zone.
In PRL 87 167902 (2001) it is claimed that
"...for an ...
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Hardness of Subgraph isomorphism problem for sparse pattern graph
Subgraph isomorphism problem is a well studied problem: given graphs $G$ and $H$, one needs to answer if $H$ contains $G$ as a subgraph. It was proven that this problem requires $|H|^{\theta(|G|)}$ ...
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Set Intersection lower bounds
Consider $S_1, ...,S_n \subseteq [U]$ where size of $U$ is polylogarithmic in $n$. We allow infinite time to pre-process these sets and then ask queries of the form $S_i \cap S_j$ is empty or not. We ...
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Sampling Functions Efficiently vs Pseudorandom Generators
Let $X$ be a set of $n$-bit Boolean functions of the form $f:\{0,1\}^n\rightarrow \{0,1\}$. For instance, $X$ could be the set of $n$-bit monotone Boolean functions, or the set of $n$-bit functions ...
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