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Questions tagged [lower-bounds]

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

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What are the best current lower bounds on 3SAT?

What are the best current lower bounds for time and circuit depth for 3SAT?
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2answers
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Are the problems PRIMES, FACTORING known to be P-hard?

Let PRIMES (a.k.a. primality testing) be the problem: Given a natural number $n$, is $n$ a prime number? Let FACTORING be the problem: Given natural numbers $n$, $m$ with $1 \leq m \leq n$, ...
29
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7answers
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Proving lower bounds by proving upper bounds

The recent breakthrough circuit complexity lower-bound result of Ryan Williams provides a proof technique that uses upper-bound result to prove complexity lower-bounds. Suresh Venkat in his answer to ...
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2answers
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How does the Mulmuley-Sohoni geometric approach to producing lower bounds avoid producing natural proofs (in the Razborov-Rudich sense)?

The exact phrasing of the title is due to Anand Kulkarni (who proposed this site be created). This question was asked as an example question, but I’m insanely curious. I know very little about ...
18
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2answers
701 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 i\in[k]...
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2answers
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(0,1)-vector XOR problem

this is a rewrite of another recent question of mine [1] that wasnt stated well (it had a semi obvious simplification, mea culpa) but I think theres still a nontrivial question at the heart of it. ...
57
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4answers
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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 ...
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4answers
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Separating Logspace from Polynomial time

It is clear that any problem that is decidable in deterministic logspace ($L$) runs in at most polynomial time ($P$). There is a wealth of complexity classes between $L$ and $P$. Examples include $NL$,...
18
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2answers
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Hierarchy theorem for circuit size

I think that a size hierarchy theorem for circuit complexity can be a major breakthrough in the area. Is it an interesting approach to class separation? The motivation for the question is that we ...
14
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1answer
353 views

Lower bounds on the size of CFGs for specific finite languages

Consider the following natural question: Given a finite language $L$, what is the smallest context-free grammar generating $L$? We can make the question more interesting by specifying a sequence of ...
110
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17answers
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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 elimination ...
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0answers
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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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2answers
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Cohomological approach to boolean complexity

A few years ago, there was some work by Joel Friedman relating lower circuit bounds to Grothendieck cohomology (see papers: http://arxiv.org/abs/cs/0512008, http://arxiv.org/abs/cs/0604024). Has this ...
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0answers
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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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2answers
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Formula size lower bounds for AC0 functions

Question: What is the best known formula size lower bound for an explicit function in AC0? Is there an explicit function with an $\Omega(n^2)$ lower bound? Background: Like most lower bounds, ...
19
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4answers
855 views

Parity and $AC^0$

Parity and $AC^0$ are like inseparable twins. Or so it has seemed for the last 30 years. In the light of Ryan's result, there will be renewed interest in the small classes. Furst Saxe Sipser to Yao ...
25
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3answers
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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, x_{i+k-...
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2answers
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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 ...
12
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2answers
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Reversing a list using two queues

This question is inspired by an existing question about whether a stack can be simulated using two queues in amortized $O(1)$ time per stack operation. The answer seems to be unknown. Here is a more ...
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4answers
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What are the best known upper bounds and lower bounds for computing O(log n)-Clique?

Input: a graph with n nodes, Output: A clique of size $O(\log n)$, Providing links to references would be great
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2answers
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Communication lower bounds for partial boolean functions

There are well known techniques for proving lower bounds on the communication complexity of boolean functions, like fooling sets, the rank of the communication matrix, and discepancy. 1) How do we ...
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3answers
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Circuit lower bounds over arbitrary sets of gates

In the 1980s, Razborov famously showed that there are explicit monotone Boolean functions (such as the CLIQUE function) that require exponentially many AND and OR gates to compute. However, the basis ...
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3answers
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Lower Bounds for Data Structures

Are results known which rule out the existence of "too-good-to-be-true" data structures? For example: can one add $Split$ and $Join$ functionality to an order maintenance data structure (see Dietz ...
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6answers
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References on Circuit Lower Bounds

Preamble Interactive proof systems and Arthur-Merlin protocols were introduced by Goldwasser, Micali and Rackoff and Babai back in 1985. At first, it was thought that the former is more powerful than ...
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2answers
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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 $p$ ...
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2answers
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Protocol partition number and deterministic communication complexity

Besides (deterministic) communication complexity $cc(R)$ of a relation $R$, another basic measure for the amount of communication needed is the protocol partition number $pp(R)$. The relation between ...
15
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1answer
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Do the proofs that permanent is not in uniform $\mathsf{TC^0}$ relativize?

This is a follow up to this question, and is related to this question of Shiva Kinali. It seems that the proofs in these papers (Allender, Caussinus-McKenzie-Therien-Vollmer, Koiran-Perifel) use ...
11
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4answers
531 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 $\{1,\dots,n\}$, ...
8
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1answer
346 views

Lower bound for NFA accepting 3 letter language

Related to a recent question (Bounds on the size of the smallest NFA for L_k-distinct) Noam Nisan asked for a method to give a better lower bound for the size of an NFA than what we get from ...
21
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2answers
987 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 ...
17
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2answers
571 views

Better lower bounds than 3n for non-boolean functions?

Blum's $3n-o(n)$ lower bound is the best known circuit lower bound over the complete basis for an explicit function $f : \{0,1\}^n \to \{0,1\}$, cf. Jukna's answer to this question for related results....
15
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0answers
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Lower bounds on single-source shortest paths in directed graphs

Are there any non-trivial lower bounds on the complexity of single-source shortest paths (SSSP) in a directed graph, where all edges have non-negative edge weights? Can we rule out the possibility of ...
11
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1answer
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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 ...
11
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1answer
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Lower bounds for Nondeterministic Multiparty Communication

This is a continuation of my previous question on communication lower bounds for partial boolean functions. Can someone suggest any reference on lower bounds for nondeterministic multiparty ...
9
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2answers
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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 be ...
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1answer
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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 ...
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2answers
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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 $f:\{0,1\...
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2answers
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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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0answers
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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 ...
8
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1answer
334 views

Are there more polynomial time problems with complexity lower bounds?

I'm looking for more problems in $P$ with classical time complexity lower bounds. Some people might wonder how you could prove such a lower bound. See below. Exponential Lower Bounds: Claim: If ...
5
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1answer
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Stronger Lower Bounds on Nondeterministic Multiparty Communication

This is a continuation of my previous question on Lower bounds for Nondeterministic Multiparty Communication. From the answer, the $\mu^\infty$ norm lower bounds nondeterministic multiparty ...
5
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0answers
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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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One-way randomized complexity of (variants of) Gap-Hamming-Distance?

The $\textsf{GapHammingDistance}$ problem over $\{0,1\}^n$ is defined as follows: Alice (resp. Bob) is given an input $x\in\{0,1\}^n$ (resp, $y\in\{0,1\}^n$), under the promise that their Hamming ...
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2answers
433 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 P/...
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0answers
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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 ...
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0answers
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What is the strongest known lower bound against SIZE(n)?

What is the best known lower bound against (nonuniform) circuits of size $O(n)$? I understand that we don't know of any explicit functions that need circuits of size more than something like $5n$. But ...
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1answer
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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 ...
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1answer
389 views

Nondeterministic communication complexity of set disjointness?

In the two-party setting, bounds of $\Theta(n)$ bits are known for deterministic and bounded-error randomized protocols for $\text{DISJ}_n$. (Here $\text{DISJ}_n$ is the $n$-element set disjointness ...
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2answers
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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})} ...
3
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2answers
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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 ...