Questions tagged [lower-bounds]

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

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4
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1answer
145 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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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 ...
3
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101 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 ...
3
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1answer
275 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 ...
3
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0answers
84 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. ...
5
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55 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 ...
4
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1answer
131 views

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 ...
31
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0answers
5k views

Combinatorics of Bellman-Ford or how to make cyclic graphs acyclic?

Roughly speaking, my question is: How costly is to make a cyclic graph acyclic while preserving all simple $s$-$t$ paths? Let $K_n$ be a complete undirected graph on vertices $\{0,1,\ldots,n+1\}$. (...
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1answer
115 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 ...
5
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141 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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1answer
180 views

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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3answers
1k views

Representing OR with polynomials

I know that trivially the OR function on $n$ variables $x_1,\ldots, x_n$ can be represented exactly by the polynomial $p(x_1,\ldots,x_n)$ as such: $p(x_1,\ldots,x_n) = 1-\prod_{i = 1}^n\left(1-x_i\...
5
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1answer
221 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 ...
7
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1answer
249 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 ...
3
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1answer
167 views

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 ...
11
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1answer
581 views

Lower bounds for learning in the membership query and counterexample model

Dana Angluin (1987; pdf) defines a learning model with membership queries and theory queries (counterexamples to a proposed function). She shows that a regular language that is represented by a ...
6
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238 views

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 ...
-5
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1answer
191 views

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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211 views

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 ...
17
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3answers
875 views

Succinct data structures survey?

Fischer's paper this month reminded me how little I know about the art of succinct data structures, and algorithms to use them. For those that don't know about succinct data structures: Given a ...
0
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1answer
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^\...
2
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1answer
107 views

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
677 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?...
12
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1answer
572 views

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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62 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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109 views

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 ...
7
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1answer
238 views

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 (...
6
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0answers
165 views

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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165 views

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 ...
7
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1answer
525 views

Razborov's Approximation methods

The approximation mothods is used for deriving lower bounds on the monotone circuit size of k-cliue and perfect matching problem. People in parameterized complexity theory strongly believe that k-...
9
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0answers
172 views

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 ...
2
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1answer
147 views

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 ...
3
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1answer
136 views

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 ...
5
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0answers
284 views

Sketch of Razborov's paper "On the method of approximations"

(The following question has bothered me for many years.) Razborov seems to have obtained some of the strongest/award winning lower bounds on circuits found in the field over many years, largely ...
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0answers
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 $1$ ...
2
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1answer
95 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 ...
3
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2answers
129 views

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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0answers
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 ...
5
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1answer
133 views

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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0answers
513 views

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" ...
1
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1answer
570 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 ...
5
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92 views

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 ...
9
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2answers
488 views

What's the “smallest” complexity class for which an $\omega \hspace{.02 in}(n)$ circuit lower bound is known?

I believe the answers to this question give classes such that for all polynomials $p$, there is a problem in the class which does not have circuits of size $p(n)$. However, I'm asking about circuit ...
5
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0answers
167 views

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^\...
8
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1answer
200 views

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 ...
4
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1answer
171 views

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 $|H|^{\theta(|G|)}$ ...
15
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0answers
372 views

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 ...
4
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0answers
94 views

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 ...
6
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1answer
340 views

How much memory is needed for counting distinct elements in a stream exactly with high probability

Assume we know a parameter $n\in\mathbb N$, and then get to observe a sequence of elements $x_1,\ldots, x_n$, one at a time. Our goal is to count the number of distinct elements in $x_1,\ldots, x_n$, ...