# Questions tagged [lower-bounds]

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

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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 ...
4answers
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 ...
4answers
4k views

### What are the best current lower bounds on 3SAT?

What are the best current lower bounds for time and circuit depth for 3SAT?
0answers
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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, ...
3answers
1k views

1answer
579 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 ...
2answers
2k 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 ...
2answers
611 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 ...
1answer
849 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 ...
2answers
704 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]...
2answers
769 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 $p$ ...
2answers
581 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....
3answers
762 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 ...
2answers
527 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 ...
0answers
443 views

### Can short-distance connectivity be harder than connectivity?

Has anybody seen the following (or similar) question being considered: Can it be easier to determine the presence/absence of $s$-$t$ paths than to determine the presence/absence of short $s$-$t$ ...
1answer
504 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 ...
2answers
504 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 ...
1answer
490 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 $f(x_1,\ldots,x_n)$...
1answer
590 views

### Is Dynamic Programming never weaker than Greedy?

In the circuit complexity, we have separations between powers of various circuit models. In the proof complexity, we have separations between powers of various proof systems. But in the algorithmic,...
0answers
311 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 ...
0answers
579 views

### 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 ...
1answer
394 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 (...
2answers
674 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 ...
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 ...
2answers
652 views

### How many negations do we need to compute monotone functions?

Razborov proved that the monotone function matching is not in mP. But can we compute matching using a polynomial size circuit with a few negations? Is there a P/poly circuit with $O(n^\epsilon)$ ...
1answer
395 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 ...
3answers
672 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 ...
1answer
356 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 ...