Questions tagged [lower-bounds]

questions about lowerbounds on functions, usually the complexity of an algorithm or a problem

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Sample complexity lower bound to learn the mode (the value with the highest probability) of a distribution with finite support

Say we have a black-box access to a distribution $\mathcal{D}$ with finite support $\{1,2,...,n\}$ with probability mass function $i \mapsto p_i$. How many samples of $\mathcal{D}$ are needed to learn ...
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Is the center of a BFS tree a good approximation of the graphs center?

Given a graph $G=(V,E)$, a center is a vertex $v\in V$ with minimal eccentricity (i.e., $v\in\text{argmin}_v\max_u d(u,v)$). Finding the center of the graph can easily be done using all-pairs-shortest-...
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Online assignment lower bound results

I am reading the following paper which presents a $(1-\epsilon)$-competitive online algorithm for the MaxMin (similar to the makespan) problem, defined as follows: a set of requests are arriving in an ...
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8 votes
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144 views

What is the least compressible probability distribution? (under entropy constraint, for an expected squared error metric)

Consider a distribution $\mathcal D$ over the reals, a real parameter $H\in\mathbb R^+$, and an integer parameter $k\in\mathbb N$. The Entropy-Constrained Quantization problem asks what is the best ...
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6 votes
1 answer
208 views

Lower bound for the OR problem

Let us have booleans $x_1, \cdots, x_n$. Any algorithm that determines $\bigvee_1^n x_i$ with probability at least $2/3$ requires $\Omega(n)$ time. It is not too difficult to prove this, but the proof ...
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2 votes
1 answer
153 views

Divide and Conquer Algorithm for 1-Median Problem

Let $P_1$ and $P_2$ be two disjoint point sets in $\mathbb{R}^d$ and $n = \vert P_1\vert = \vert P_2\vert$ and $P = P_1\cup P_2$. Let $c^\star$ be the optimal 1-median for $P$ and $opt^\star$ is the ...
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1 vote
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Decision tree vs. pebble game lower bounds

This question concerns two types of lower bounds. In a pebbling lower bound, we are concerned with the complexity of constructing the output from the input. For example, if the only way we could ...
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$\mathbf{AC}^0$ lower bounds for $\mathsf{Gap}\text{-}\mathsf{Max3SAT}$

Various gapped maximization problems are known not to be $\mathbf{NP}$-hard under $\mathbf{AC}^0$ reductions, e.g., $\mathsf{Gap}_{1,\epsilon}\text{-}\mathsf{Max3SAT}$ (see, e.g., Proposition 4 of ...
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  • 161
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Low-Treewidth Sorting Networks

It was previously asked if there exist Boolean circuits of treewidth $O(\log n)$ that compute the majority function $\text{MAJ}_n$ on $n$ inputs. While a construction using online algorithms and the ...
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1 answer
113 views

Complexity for universal Counter Machine with {0,1}-valued registers

Consider a universal $\{0,1\}$-$k$-counter machine where each of the $k$ registers has a value in $\{0,1\}$ (as opposed to any non-negative integer in the usual formulation), and there are states $q_1,...
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Triangle detection hardness in regular graphs

Consider a tripartite graph over $n^{1-\epsilon}$ vertices each in sets $I, J, K$. Suppose we impose a constraint that every vertex has degree $n^\epsilon/c$ for some constant $\epsilon > 0$ and ...
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2 votes
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Obtaining a lower bound of a matrix norm

I was wondering (on a setting where $\vec X_i \sim \mathcal{N}(\vec\mu, \mathbb{I})$ are $n$ random $d$-dimensional multivariate normal vectors with unknown mean $\vec\mu$) how I could obtain a lower ...
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What is known about the stabilizer rank of this simple state?

Consider the uniform superposition of all length-$n$ bit-strings of Hammming weight $w$, $$ |\phi_w\rangle =\sum_{x\in \{0,1\}^n,|x|=w} |x\rangle$$ What is known or conjectured about the stabilizer ...
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3 votes
1 answer
303 views

ETH based lower bound for $k$-COLORING of bounded degree graph

It is known that there is no $2^{o(n)}$-time algorithm for 3-COLORABILITY of graphs of maximum degree four, unless ETH fails [1]. Is a there a similar result for $k$-COLORABILITY assuming only ETH (...
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3 votes
1 answer
128 views

What is the best simulation of majority utilizing $\bmod\{2,3,\dots,p\}$ gates?

It is known $AC^0[2]$ cannot get majority function. Is there a literature on simulation of $MAJ$ function utilizing $AC^0[2,3,\dots,p]$ gates for a finite fixed set of primes $2$ to $p=O(1)$? What is ...
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Comparing SAT to MCSP reduction class separations and faster SAT class separations?

Assume $SAT$ is in $QuasiP$. We immediately infer $NQuasiP=QuasiP$ and $EXP=NEXP$. From https://people.csail.mit.edu/rrw/easy-witness-nqp.pdf we infer $NQuasiP$ is not in $P/poly$ which implies $...
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1 vote
1 answer
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Average-case randomized communication complexity in the small-advantage regime

Let $f\colon \{0, 1\}^n \times \{0, 1\}^n \to \{0, 1\}$. I'm interested in randomized communication protocols $\pi$ that compute $f$ in the weak sense that $$ \Pr_{x, y}\left[\Pr_r[\pi(x, y, r) = f(x, ...
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1 vote
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Complexity Lower Bounds for 3D Sparse Gaussian Elimination

I'm interested in lower bounds on the complexity in the real-RAM model of solving systems of linear equations which have the sparsity pattern of a three-dimensional cubic mesh. Specifically, consider ...
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1 vote
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215 views

Is there a known lower-bound on what the exponent could be, even if it turned out that P=NP?

Underlying motivation for the question: if someone showed that $\text{P}=\text{NP}$ but the algorithm thus produced for, e.g., $3\text{-SAT}$, runs in time $\Omega(n^G)$ where $G$ is Graham's number, ...
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0 answers
38 views

Maximum resistor with sublinear number of measurements

Consider a set $X = \{x_1, \dots, x_n\}$ of positive real numbers (or natural numbers, if you like) to be a set of resistors. For any subset $S \subset X$, we can build resistive circuits and measure ...
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14 votes
1 answer
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Are there languages decidable in linear time by RAM machines that have superlinear time complexity lower bounds for Multitape Turing machines?

Question: Are there languages decidable in linear time by RAM machines that have superlinear time complexity lower bounds for Multitape Turing machines? Background: I recently stumbled upon the ...
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6 votes
0 answers
158 views

Computing and maintaining the minimum of a set $S$ of integers while allowing updates on $S$

This question is about computing and maintaining the minimum of a set $S$ of integers while allowing updates on $S$. The computation model we are considering is the unit-cost RAM machine with linear ...
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Proof that quantum computers can't easily invert permutations

Let's say I am given a permutation $\sigma$ that maps $n$ bit strings to $n$ bit strings. I want to output $1$ if $\sigma^{-1}(1)$ is even and $0$ if $\sigma^{-1}(1)$ is odd. This is the famous ...
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2 votes
0 answers
162 views

Time complexity of Succinct-CVP

I want to know what is the best known lower time complexity of Succinct-CVP? The succinct version of many P-complete problems are EXP-complete and Succinct-CVP is EXP-complete too (It is because of ...
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0 votes
1 answer
139 views

Application of Yao's Minmax Principle for Adaptive Randomized Algorithms

Reference Request: I am interested in references where Yao's Minimax Principle is applied for adaptive randomized algorithms if any. More generally, I am interested in minimax lower bound results for ...
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2 votes
0 answers
163 views

Why is it difficult for $GCT$ to prove super quadratic lower bound?

We have a quadratic lower bound for the Permanent versus Determinant problem. Why is it difficult for $GCT$ to improve it?
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5 votes
2 answers
343 views

Are there problems in $DTIME(n^k) - DTIME(n^{k-1})$ that are not hard for $DTIME(n^{k-1})$ under nearly linear time reductions?

Background It can be challenging to find computational problems that are solvable in $DTIME(n^k) - DTIME(n^{k-1})$ where $k \geq 2$. Although some natural problems are known to exist, many of them ...
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2 votes
2 answers
200 views

Problem in deterministic time $n^p$ and not lower

I'm looking for any language $L$ candiate to be in $DTIME(n^p) -DTIME(n^{p-1})$ (it takes at least $n^{p-1}$ steps to determine if an input is in L with a 2-tape $TM$, but L is polynomially solvable). ...
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10 votes
3 answers
882 views

Maximum shortest word accepted by pushdown automata

Given a fixed alphabet, consider all deterministic pushdown automata with $n$ states that accept a nonempty language. What is the maximum length of the shortest word accepted by a deterministic ...
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3 votes
2 answers
126 views

Understanding non-equivalence of proof lengths according to proof systems

Here, in section 4.3, Fortnow says: But to prove P != NP we would need to show that tautologies cannot have short proofs in an arbitrary proof system. I am ...
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2 votes
0 answers
100 views

Lower bounds for list/set data structures without delete

I'm interested in lower bounds on the amortized time cost for either of the following dynamic data structure problems, in the cell probe or RAM model, or any model that lets us do operations on words ...
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1 vote
0 answers
100 views

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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6 votes
1 answer
287 views

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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0 votes
0 answers
87 views

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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9 votes
2 answers
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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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0 votes
0 answers
45 views

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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1 vote
0 answers
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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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1 vote
1 answer
84 views

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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  • 1,443
1 vote
0 answers
268 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? ...
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4 votes
1 answer
147 views

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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  • 141
1 vote
0 answers
43 views

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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3 votes
0 answers
103 views

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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7 votes
1 answer
162 views

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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  • 1,035
3 votes
1 answer
277 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 ...
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3 votes
0 answers
86 views

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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5 votes
0 answers
57 views

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 ...
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-1 votes
1 answer
119 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 ...
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  • 111
5 votes
0 answers
142 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 \...
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5 votes
1 answer
236 views

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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7 votes
1 answer
257 views

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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