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

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19
votes
3answers
926 views

Nontrivial algorithm for computing a sliding window median

I need to calculate the running median: Input: $n$, $k$, vector $(x_1, x_2, \dotsc, x_n)$. Output: vector $(y_1, y_2, \dotsc, y_{n-k+1})$, where $y_i$ is the median of $(x_i, x_{i+1}, \dotsc, ...
18
votes
1answer
180 views

Deterministic communication complexity vs partition number

Background: Consider the usual two-party model of communication complexity where Alice and Bob are given $n$-bit strings $x$ and $y$ and have to compute some Boolean function $f(x,y)$, where ...
1
vote
0answers
76 views

Does this meet the space requirements for the lower 3-SAT bounds?

According to "What are the best current lower bounds on 3SAT?", Ryan Williams has an answer that states that the (time * space) requirements for 3-SAT must meet or exceed $n^{2 \cos(\pi/7) - o(1)}$ ...
-1
votes
1answer
156 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 ...
20
votes
2answers
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 = ...
9
votes
0answers
101 views

Hardness of optimal sorting

For comparison-based sorting algorithms, asymptotically optimal algorithms in worst-case $\Theta(n\log n)$ comparisons are well known. From a purely theoretical perspective, however, exactly optimal ...
2
votes
0answers
85 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 ...
2
votes
1answer
113 views

Subset Sum bounds

Are there any bounds for the subset problem with respect to the number of the terms involved in the sum and the range of the possible values?For example looking for some 2 terms whose sum equals 3 you ...
18
votes
2answers
499 views

Bounds on the size of the smallest NFA for L_k-distinct

Consider the language $L_{k-distinct}$ consisting of all $k$-letter strings over $\Sigma$ such that no two letters are equal: $$ L_{k-distinct} :=\{w = \sigma_1\sigma_2...\sigma_k \mid \forall ...
8
votes
2answers
155 views

2DFA that requires many states in equivalent DFA?

Is there a 2DFA with $n$ states (where $n$ is nontrivial, say at least 4) that requires at least $2^n$ states to simulate using any DFA? A two-way DFA (2DFA) is a deterministic finite-state ...
3
votes
0answers
131 views

sketch of Razborovs 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 ...
7
votes
1answer
83 views

Lower bounds on alternative models of multiparty communication complexity

I'm a newcomer to communication complexity, and so far I've read the chapter in Arora-Barak and some papers giving lower bounds in various applications. A priori the definition of multiparty ...
3
votes
2answers
441 views

Top-k frequent items in data stream

Paper "HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm" introduce an efficient algorithm which can help to estimate distinct items in large data set. I am thinking about ...
4
votes
2answers
245 views

P/poly vs NP separation based on circuit trees instead of DAGs

there are various theorems that relate major complexity class separations to circuit family DAGs sizes, in particular for P/poly vs NP. in contrast, are there theorems/conjectures that relate ...
0
votes
1answer
46 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 ...
5
votes
1answer
172 views

Lower bounds for formulae sizes for addition

I am interested in the conversion of $\sum_{i=1}^n x_i = y$ to 3-CNF. Here $x_i$ is a binary 0/1 variable and $y$ is some positive integer. There are a number of practical methods for doing this, ...
8
votes
4answers
413 views

Lower bound for testing closeness in $L_2$ norm?

I was wondering if there was any lower bound (in terms of sample complexity) known for the following problem: Given sample oracle access to two unknown distributions $D_1$, $D_2$ on $\{1,\dots,n\}$, ...
7
votes
1answer
94 views

Is it known whether hypergraph minimal covers are P-enumerable?

Is it known or unknown whether hypergraph minimal covers are P-enumerable? I would be most happy with lower bounds. I'd also like to hear about conditional results, which assume some conjecture is ...
2
votes
0answers
122 views

Natural Proofs and methods for polylog depth circuit lower bounds

I have a question about the following question and its answer. Status on circuit lower bounds for polylog-bounded depth circuits In the above question, it is asked about methods to prove lower ...
10
votes
2answers
233 views

Is there an explanation for the difficulty of proving quadratic lower bounds for interesting NP problems?

This is a follow up to my previous question: Best known deterministic time complexity lower bound for a natural problem in NP I find it bewildering that we haven't been able to prove any quadratic ...
8
votes
0answers
156 views

Grigoriev-Karpinski for the permanent

Grigoriev and Karpinski (ps.Z) showed that any depth-3 circuit over a fixed finite field computing $\mathrm{Det}_n$ requires $2^{\Omega(n)}$ size. I had the misconception(?) until recently that the ...
3
votes
0answers
175 views

Streaming Algorithm Lower Bounds by Communication Complexity

I am learning the methods for proving lower bounds on streaming algorithms using communication complexity. My question is about a basic technique to prove lower bounds on streaming models using the ...
4
votes
0answers
211 views

The latest concerning Valiant-Vazirani

I'm wondering what the best method obtained for the Valiant-Vazirani theorem is. We can state the following criteria: (1) First and foremost, has anyone been able to derandomize it? (2) If not, ...
9
votes
2answers
231 views

Lower bound on number of oracle calls for solving $n$ instances of the halting problem

I encountered the following question, which is an easy exercise (spoiler below). We are given $n$ instances of the halting problem (i.e. TMs $M_1,...,M_n$), and we need to decide exactly which of ...
1
vote
1answer
195 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 ...
7
votes
0answers
121 views

Size complexity of probabilistic two-way automata for a Boolean function

I'm interested in computing Boolean functions $f:\{0,1\}^n\rightarrow\{0,1\}$ with two-way finite automata and I will measure the complexity of a Boolean function by the number of states for the ...
11
votes
2answers
325 views

What's the most efficient algorithm for Divisibility?

What is the most efficient (in time complexity) algorithm known nowadays for the Divisibity Decision Problem: given two integers, say $a$ and $b$, does $a$ divide $b$? Let it be clear that what I ask ...
11
votes
2answers
446 views

Status on circuit lower bounds for polylog-bounded depth circuits

Bounded depth circuit complexity is one of the main areas of research within circuit complexity theory. This topic has origins in results like "the parity function is not in $AC^{0}$" and "the mod $p$ ...
19
votes
2answers
717 views

Lower bound for determinant and permanent

In light of the recent chasm at depth-3 result (which among other things yields a $2^{\sqrt{n}\log{n}}$ depth-3 arithmetic circuit for the $n \times n $ determinant over $\mathbb{C}$), I have the ...
8
votes
1answer
323 views

Can we get a sorted list from a sorted matrix in $O(n^2)$

I'm confused. I want to prove that that the problem of sorting a $n$ by $n$ matrix i.e. the rows and columns are in ascending order is $\Omega(n^2\log n)$. I proceed by assuming that it can be done ...
1
vote
0answers
128 views

Is there a tight lower bound on the complexity of SSSP on a graph?

I'm an undergrad and I'm not sure if this is the right way to ask this question. I want to know the lower bound on single-source shortest path computation in a general graph. The graph is allowed to ...
2
votes
0answers
95 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?
10
votes
1answer
550 views

How many disjoint edge-cuts a DAG must have?

The following question is related to the optimality of the Bellman-Ford $s$-$t$ shortest path dynamic programming algorithm (see this post for a connection). Also, a positive answer would imply that ...
4
votes
1answer
268 views

Probabilistic circuit complexity or size of probabilistic 2-way automata for Boolean functions

If we consider circuits with arbitrary binary logic gates one can prove by a counting argument that there exists a Boolean function on $n$ variables that require a circuit of size $ \Theta \left( ...
4
votes
2answers
252 views

lower bound of majority function?

If a circuit ({AND OR NOT} circuit) with depth d computes the majority function, what's the best lower bound for majority function? I know the lower bound for parity function is $ 2^{\Omega (n^{1/d})} ...
24
votes
0answers
831 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 $\{0,1,\ldots,n+1\}$. ...
6
votes
1answer
141 views

Lower bounds for 3SUM with a free cache

Consider the 3SUM problem: given a set $S$ of $n$ numbers identify $x$,$y$,$z$, s.t $x + y = z$. It is believed that the simple $O(n^2)$ algorithm is the best possible; reductions from 3SUM have been ...
7
votes
0answers
256 views

On optimality of Grover algorithm with high success probability

It is well-known that bounded error quantum query complexity of the function $OR(x_1,x_2,\ldots, x_n)$ is $\Theta(\sqrt{n})$. Now the question is what if we want our quantum algorithm to succeed for ...
1
vote
0answers
36 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 ...
6
votes
0answers
171 views

Data structures lower bounds on Turing machines

Have there been any results on lower bounds for implementing data structures on Turing machines, e.g. stacks, queues, etc ? I guess that people are mostly interested in models with random access, but ...
36
votes
0answers
641 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 ...
14
votes
1answer
303 views

Characterization of read-once formulae over the full binary basis

Background A read-once formula over a set of gates (also called a basis) is a formula in which each input variable appears once. Read-once formulas are commonly studied over the De Morgan basis ...
20
votes
2answers
915 views

Can addition be carried out in less than depth 5?

Using carry look ahead algorithm we can compute addition using a polynomial size depth 5 (or 4?) $AC^0$ circuit family. Is it possible to reduce the depth? Can we compute the addition of two binary ...
3
votes
1answer
162 views

Exponential blowup in Simple Proof of a theorem of Statman by Mairson

I'm studying "A simple proof of a theorem of Statman" by H.G. Mairson. At page 4, he encodes set/type theory in lambda calculus. In particular, note che "op" trick in the definition of $eq_{k+1}$. ...
11
votes
1answer
220 views

Is it possible to use random restrictions to obtain a lower-bound for $\mathsf{TC^0}$?

There are several well-known $\mathsf{AC^0}$ circuit size lower-bound results based on random restrictions and the Switching Lemma. Can we develop a Switching Lemma result to prove a size lower-bound ...
-2
votes
1answer
137 views

Lower bounds on $Q_{\epsilon}(IP)$

I want to show that $Q_{\epsilon}(IP) \geq (1-O(\epsilon))n$, where $IP:\{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ is the usual mod 2 inner product. I have Nayak's lower bound, but I am not sure ...
6
votes
1answer
361 views

How to generate a permutation uniformly by repeating using an one-bit uniform random generator?

If I have an one-bit uniform random generator, how can I use it to generate a permutation uniformly for the sequence {1, 2, ..., n}. I have a solution: run the one-bit random generator n*n times to ...
4
votes
0answers
229 views

Lower Bounds on Running time of Graph Algorithms

Are there any non-trivial lower bounds on the running time of graph algorithms in RAM/PRAM/ models of computation ? I am not looking for the NP-Hardness results here. Following is a result that I ...
8
votes
0answers
247 views

Existence of “colouring matrices” — a generalisation

This is a generalisation of the following post: Existence of "colouring matrices". As the base case turned out to be fairly straightforward (in essence, precisely equal to the existence of Sperner ...
9
votes
2answers
174 views

Existence of “colouring matrices”

Edit: there is now a follow-up question related to this post. Definitions Let $c$ and $k$ be integers. We use the notation $[i] = \{1,2,...,i\}$. A $c \times c$ matrix $M = (m_{i,j})$ is said to ...