Questions tagged [lower-bounds]

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

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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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1answer
360 views

Lower bound for finding repeated elements in sorted array

This is inspired by [1] (which still needs answers). What is the tight lower bound (or optimal algorithms) for the "finding repeated elements" problem: Given a sorted integer array of size $n$, how ...
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1answer
94 views

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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65 views

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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157 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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162 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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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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96 views

Is this question $NP_R$ hard?

Consider $n$ variables $x_1, \cdots, x_n$ and $f=\sum a_i x_1^{d_{i1}}\cdots x_n^{d_{in}}$ such that for each $i$, $d_{i1}+\cdots+d_{in}=d$ for some fixed $d$ and $a_i\geq 0$. I am interested in the ...
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115 views

Correlation bounds

Is there any survey of results other than viola's on correlation bounds and its applications? Also is there any lecture notes on gowers uniformity and its applications in proving correlation bounds?
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38 views

Lower bounds for minimum variance estimators in limited space

Cramer-Rao, Rao-Blackwell and Lehmann-Scheffé, all give you ways to prove that a statistical estimator has the lowest variance possible. Is there any CS related work on the minimum variance ...
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602 views

Major conjectures used to prove complexity lower bounds?

Complexity theory uses a large number of unproven conjectures. There are several hardness conjectures in David Johnson's NP-Completeness Column 25. What are the other major conjectures not mentioned ...
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1answer
561 views

equivalent way(s) of expressing P=?NP problem in linear programming?

the paper "In defense of the Simplex Algorithm's worst-case behavior" Disser/Skutella [1] was recently cited on this tcs.se site by saeed on another interesting question. the paper introduces the idea ...
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362 views

Questions about computing matrix rigidity

Matrix rigidity was introduced by Valiant in 1977: The rigidity $Rig_M(r)$ of boolean matrix $M$ over GF(2) is the smallest number of entries of $M$ that must be changed in order to reduce its rank ...
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1answer
82 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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1answer
310 views

"Send Once"-One way Multiparty Communication Complexity

There are plenty results on multiparty communication complexity, and one way protocol which anyone playing communicatin games is able to send one person, is a basic setting. I want to consider more ...
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1answer
115 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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1answer
110 views

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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1answer
670 views

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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47 views

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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139 views

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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36 views

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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39 views

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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210 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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151 views

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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96 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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98 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, ...
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262 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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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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61 views

\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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55 views

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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91 views

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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134 views

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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136 views

Consequences of the existence of the following algorithm: does it imply any complexity class separation / collapse?

Let $G$ be a $3$-regular graph. Let $O$ be the number of vertex covers of $G$ having odd cardinality, and let $E$ be the number of vertex covers of $G$ having even cardinality. Let $\Delta = O - E$. ...
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84 views

Implication of lower bounds in Boolean circuit to other models of computation

Suppose that one can prove that some hard function $f$ with $n$ bit input does not admit any Boolean circuit of size at most $n^t$. Then, how strong can we say about how hard $f$ is in other models? ...
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193 views

Circuit lower bound for NAND based arbitrary deterministic computation [closed]

What is the lowest known lower-bound on the number of NAND gates needed in order to perform an arbitrary deterministic computation with a fixed length input and output?
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192 views

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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103 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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142 views

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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2answers
144 views

Mergeable Exact Order Statistics Data Structure

Given $n$ sets of integers $S_1, S_2, \cdots, S_n$, it is guaranteed that $$ x < y, \text{ for } \forall x \in S_i \text{ and } \forall y \in S_{i+1} $$ and let's denote this relationship as $S_i &...
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1answer
165 views

What do these lower bounds really mean?

I was reading an abstract for a paper that gives a minimum number M of good agents that guarantees f-Byzantine gathering, if there are f Byzantine agents. It gives a lower bound of f+1 for strongly ...
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1answer
156 views

Functions and Counting Problems in Streaming Computation

I have read a stream computation paper in STOC07(Paul Beame, T. S. Jayram, and Atri Rudra. Lower bounds for randomized read/write stream algorithms.) and FOCS08(Paul Beame and Trinh Huynh. On the ...
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1answer
127 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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150 views

Lowerbound for the minimum distance between points in a "nice" triangulation

Suppose we have a unit square $S$ that contains $n$ points. Assume we always have a point at each of the four corners. No we triangulate $S$ by adding non-intersecting segments between the points. ...
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131 views

Lower bounds on counting functions

I have a question about counting problems on arbitrary (not necessarily polynomial time) functions. Let $F_n = \{f : \{0,1\}^n \to \{0,1\}\}$ be the set of all boolean functions with $n$ inputs (...
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72 views

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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73 views

What is the best reduction we know from flavors of $SAT$ to $MCSP$?

Consider $\mathcal P(n,m)$ to be a class from the set $\{k\mbox{-}\mathsf{SAT}(n,m),\mathsf{CIRCUITSAT}(n,m)\}$ where $k$ is fixed, $n$ is number of variables and $m$ is number of clauses. Denote $\...
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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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71 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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44 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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To find OR of $\sqrt{n}$ numbers each of $n$ bits?

Given $\sqrt{n}$ numbers of $n$ bits each. I need to find its OR and store it at another number RESULT of $n$ bits. Trivially it can be done in $\mathcal{O}(n \sqrt{n})$ time, or $\mathcal{O}(n \sqrt{...