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

learn more… | top users | synonyms

88
votes
16answers
5k 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 ...
48
votes
4answers
1k 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 ...
40
votes
4answers
2k views

What are the best current lower bounds on 3SAT?

What are the best current lower bounds for time and circuit depth for 3SAT?
37
votes
0answers
842 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 ...
35
votes
3answers
2k views

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 ...
33
votes
2answers
1k views

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 ...
30
votes
2answers
981 views

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 ...
27
votes
8answers
2k views

Optimal greedy algorithms for NP-hard problems

Greed, for lack of a better word, is good. One of the first algorithmic paradigms taught in introductory algorithms course is the greedy approach. Greedy approach results in simple and intuitive ...
26
votes
1answer
1k 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 ...
25
votes
7answers
1k views

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 ...
24
votes
0answers
1k 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 $\{0,1,\ldots,n+1\}$. ...
23
votes
3answers
2k 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, ...
22
votes
2answers
932 views

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, ...
22
votes
2answers
954 views

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 ...
22
votes
2answers
485 views

Best current space lower bound for SAT?

Following on from a previous question, what are the best current space lower bounds for SAT? With a space lower bound I here mean the number of worktape cells used by a Turing machine which uses ...
21
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 = ...
21
votes
2answers
934 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 ...
20
votes
4answers
953 views

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 ...
20
votes
2answers
1k views

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 ...
20
votes
2answers
850 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 ...
20
votes
5answers
377 views

Reducing space usage of st-connectivity with multiple passes?

Suppose a graph $G$ with $n$ vertices is presented as a stream of $m$ edges, but multiple passes are allowed over the stream. Monika Rauch Henzinger, Prabhakar Raghavan, and Sridar Rajagopalan ...
19
votes
1answer
364 views

Explain Gurvits's tensor-rank interpretation of Deolalikar's paper

[Note: I believe this question in no way hinges on the correctness or incorrectness of Deolalikar's paper.] On Scott Aaronson's blog Shtetl Optimized, in the discussion about Deolalikar's recent ...
19
votes
1answer
260 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 ...
18
votes
1answer
637 views

Most efficient way to convert an $\text{AC}^0$ circuit to a circuit (of any depth) with gate fanout 1

EDIT (Aug 22, 2011): I am further simplifying the question and putting a bounty on the question. Perhaps this simpler question will have an easy answer. I'm also going to strikethrough all the parts ...
18
votes
2answers
600 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 ...
18
votes
0answers
393 views

Why is HAMILTONIAN CYCLE so different from PERMANENT?

A polynomial $f(x_1,\ldots,x_n)$ is a monotone projection of a polynomial $g(y_1,\ldots,y_m)$ if $m$ = poly$(n)$, and there is an assignment $\pi:\{y_1,\ldots,y_m\}\to\{x_1,\ldots,x_n, 0,1\}$ such ...
17
votes
6answers
1k views

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 ...
17
votes
4answers
683 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 ...
17
votes
2answers
442 views

Lower bounds on Gaussian complexity

Define the Gaussian complexity of an $n \times n$ matrix to be the minimal number of elementary row and column operations required to bring the matrix into upper-triangular form. This is a quantity ...
17
votes
2answers
432 views

Lower bounds for constant-depth formulae?

We know a lot about the limitations of (polynomial size) constant-depth circuits. Since (polynomial size) constant-depth formulae are an even more restricted model of computation, all problems known ...
16
votes
2answers
483 views

Number of binary gates needed to compute AND and OR of n input bits simultaneously

What is the minimal number of binary gates needed to compute AND and OR of $n$ input bits simultaneously? The trivial upper bound is $2n-2$. I believe that this is optimal, but how to prove this? The ...
15
votes
2answers
802 views

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 ...
15
votes
2answers
416 views

Complexity lower bound: the gap between decision trees and RAMs

I recently discovered a quadratic lower bound on the complexity of a problem in the decision tree model, and I wonder whether this result could be partially generalized to the random access machine ...
15
votes
1answer
426 views

Any polynomial which is hard to count but easy to decide?

Every monotone arithmetic circuit, i.e. a $\{+,\times\}$-circuit, computes some multivariate polynomial $F(x_1,\ldots,x_n)$ with nonnegative integer coefficients. Given a polynomial ...
14
votes
2answers
539 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 ...
14
votes
2answers
543 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 $p$ ...
14
votes
1answer
384 views

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 ...
14
votes
1answer
273 views

Best communication complexity lower bound of disjointness

It is well known that no deterministic two-party protocol can solve disjointness problem (DISJ) on $n$-bit inputs without sending $n+1$ bits in the worst case (see, e.g., the book by Kushilevitz and ...
14
votes
2answers
593 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 ...
14
votes
1answer
324 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 ...
13
votes
3answers
781 views

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 ...
13
votes
1answer
262 views

Reference to lower bound on separator in a grid?

It is easy to verify that given the d dimensional grid of the integer points $\{1,\ldots,n\}^d$, with the regular adjacency, one can find a separator of size $n^{d-1}$ (just pick any middle ...
13
votes
2answers
425 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 ...
12
votes
2answers
410 views

Which problems in computational geometry or graph theory are believed to be $\Omega(n^3)$?

This is intended as a follow up question to Robin Kothari's previous post on polynomial time hardness results. Specifically, I'm interested in seeing some hardness proofs for problems that are ...
12
votes
1answer
381 views

Problems to reduce from to prove an $\Omega(n\log n)$ lower bound

What are the standard problems we can reduce from to prove $\Omega(n\log n)$ lower bounds? Of course, state problems other than sorting and element distinctness.
12
votes
3answers
533 views

How can I show a Gap-P problem is outside #P

There are a number of problems in combinatorial representation theory and algebraic geometry for which no positive formula is known. There are several examples I am thinking of, but let me take ...
12
votes
2answers
1k views

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 ...
12
votes
1answer
253 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 ...
11
votes
2answers
1k views

Simple proof of Ω(n lg n) worst-case bound for uniqueness/distinctness?

There are several proofs for the loglinear lower bound for the element uniqueness/distinctness problem (based on algebraic computation trees or adversarial arguments), but I'm looking for one that's ...
11
votes
2answers
241 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 ...