Questions tagged [lower-bounds]

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

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0answers
81 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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1answer
112 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 ...
5
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0answers
124 views

$\mathbf{AC}^0$ 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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0answers
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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1answer
481 views

Is the Kolmogorov complexity of the truth tables of the halting problem known asymptotically?

Let $HALT_n$ denote the string of length $2^n$ corresponding to the truth table of the halting problem for inputs of length $n$. If the sequence of Kolmogorov complexities $K(HALT_n)$ were $O(1)$, ...
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0answers
71 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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1answer
102 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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18answers
9k views

Examples of the price of abstraction?

Theoretical computer science has provided some examples of "the price of abstraction." The two most prominent are for Gaussian elimination and sorting. Namely: It is known that Gaussian ...
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0answers
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 ...
20
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1answer
868 views

How to prove that USTCONN requires logarithmic space?

USTCONN is the problem that requires deciding whether there is a path from the source vertex $s$ to the target vertex $t$ in a graph $G$, where these are all given as part of the input. Omer Reingold ...
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2answers
296 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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64 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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1answer
284 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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0answers
35 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 ...
3
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1answer
127 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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2answers
864 views

Lower bounds on #SAT?

The problem #SAT is the canonical #P-complete problem. It's a function problem rather than a decision problem. It asks, given a boolean formula $F$ in propositional logic, how many satisfying ...
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1answer
141 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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1answer
109 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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0answers
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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0answers
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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0answers
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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0answers
2k views

Monotone complexity of s-t connectivity

In the problem CONN, we obtain a directed $n$-vertex graph (encoded as a boolean string of $n^2$ bits, one for each potential edge), and want to decide whether there is a path between all $n^2$ pairs $...
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0answers
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 ...
14
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1answer
825 views

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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0answers
143 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 ...
2
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0answers
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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0answers
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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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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0answers
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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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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3answers
759 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 ...
3
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1answer
697 views

Time complexity of d-dimensional convex hull

Consider the convex hull problem in $\Re^d$: Input: a list of $n$ points $S$ in $\Re^d$, Output: the vertices of the convex hull of $S$. What is the best lower bound on the time complexity of ...
7
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1answer
158 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 $n^2$ ...
2
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2answers
189 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). ...
64
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5answers
2k views

Problems that can be used to show polynomial-time hardness results

When designing an algorithm for a new problem, if I can't find a polynomial time algorithm after a while, I might try to prove it is NP-hard instead. If I succeed, I've explained why I couldn't find ...
3
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2answers
125 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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3answers
842 views

Progress on generalized star-height problem?

The (generalized) star height of a language is the minimum nesting of Kleene stars required to represent the language by an extended regular expression. Recall that an extended regular expression over ...
2
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1answer
109 views

Lower bounds on simple hash table operations?

There are a variety of hash tables that support worst-case O(1)-time lookups and deletions and expected O(1)-time lookups. Is there a known lower-bound on hashing that says that there cannot be a hash ...
6
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1answer
267 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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0answers
95 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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0answers
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 ......
9
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2answers
849 views

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 ...
29
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1answer
2k views

Fourier coefficients Boolean Functions described by Bounded Depth Circuits with AND OR and XOR gates

Let $f$ be a Boolean function and let's think about f as a function from $\{-1,1\}^n$ to $\{ -1,1 \}$. In this language the Fourier expansion of f is simply the expansion of f in terms of square free ...
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0answers
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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0answers
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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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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0answers
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? ...
8
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3answers
794 views

One way randomised communication complexity of disjointness

I am looking for a reference for the (classical) one way randomised communication complexity of disjointness when the universe can be large. Say Alice and Bob both have sets of size $m$ chosen from a ...
4
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1answer
142 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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0answers
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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