All Questions

52
votes
13answers
3k views

For which algorithms is there a large gap between the theoretical analysis and reality?

Two ways of analyzing the efficiency of an algorithm are to put an asymptotic upper bound on its runtime, and to run it and collect experimental data. I wonder if there are known cases where there ...
43
votes
10answers
3k views

Kolmogorov complexity applications in computational complexity

Informally speaking, Kolmogorov complexity of a string $x$ is a length of a shortest program that outputs $x$. We can define a notion of 'random string' using it ($x$ is random if $K(x) \geq 0.99 |x|$)...
14
votes
2answers
766 views

Is APX contained in NP?

A problem P is said to be in APX if there exists some constant c > 0 such that a polynomial-time approximation algorithm exists for P with approximation factor 1 + c. APX contains PTAS (seen by ...
33
votes
9answers
2k views

Surprising Results in Complexity (Not on the Complexity Blog List)

What were the most surprising results in complexity? I think it would be useful to have a list of unexpected/surprising results. This includes both results that were surprising and came out of ...
14
votes
5answers
2k views

History of recursion

Who introduced the idea of recursion? Can someone explain where it came from and how it impacted computer science?
10
votes
2answers
297 views

Is there a definitive reference for Turing machines with multiple oracle tapes?

Most of the literature seems to be concerned with machines with single oracles for specific problems, however there appear to be a few papers that consider machines with multiple oracles. Is there a ...
25
votes
2answers
3k views

Are lambda calculus and combinatory logic the same?

I am currently reading "Lambda-Calculus and Combinators" by Hindley and Seldin. I'm not an expert, but have always taken an interest in lambda calculus because of involvement with functional ...
14
votes
3answers
535 views

How hard is it to reduce termination to partial correctness?

If you are familiar with program verification you are likely to prefer reading the Question before the Background. If you are not familiar with program verification then you may still be able to ...
16
votes
2answers
340 views

Finding small sets of integers in which every element is a sum of two others

This is a follow-up to this question on math.stackexchange. Let us say that a non-empty set S ⊆ ℤ is self-supporting if for every a ∈ S, there exist distinct ...
20
votes
3answers
387 views

Property testing in other metrics?

There is a large literature on "property testing" -- the problem of making a small number of black box queries to a function $f\colon\{0,1\}^n \to R$ to distinguish between two cases: $f$ is a ...
10
votes
2answers
225 views

Shuffling of tokens on a graph using local swaps

Let $G= (V, E)$ be a non-regular connected graph whose degree is bounded. Suppose that each node contain a unique token. I want to uniformly shuffle the tokens amongst the graph using only local ...
16
votes
1answer
392 views

Extensions of beta-theory of lambda calculus

The beta-eta-theory of the lambda-calculus is Post-complete. Can additional rules be added to extend the beta-theory of the lambda-calculus to get confluent theories other than the beta-eta theory? ...
37
votes
9answers
4k views

References for TCS proof techniques

Are there any references (online or in book form) that organize and discuss TCS theorems by proof technique? Garey and Johnson do this for the various kinds of widget constructions needed for NP-...
24
votes
2answers
525 views

Parallel Dynamic Search

Is there a natural parallel analog to red-black trees with similar or even not-terribly-worse properties for updates while being reasonably work-efficient ? More generally, what's the best we can do ...
13
votes
1answer
970 views

Chernoff bound for weighted sums

Consider $X = \sum_i \lambda_i Y_i^2$, where lambda_i > 0 and Y_i is distributed as a standard normal. What kind of concentration bounds can one prove on X, as a function of the (fixed) coefficients ...
20
votes
1answer
403 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 ...
8
votes
5answers
534 views

Algorithm for inverting a bijective function.

Does there exist a generalized algorithm for finding the inverse function of an arbitrary bijective function? In order for this algorithm to be useful, it must eventually halt once the correct answer ...
11
votes
1answer
237 views

Can someone suggest a recent survey on product form Markov chains?

I'm especially interested in their use in model checking applications. I have Open, Closed and Mixed Networks of Queues with Different Classes of Customers by Baskett et al. Any other suggestions ...
18
votes
2answers
587 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 ...
18
votes
1answer
565 views

Best known joint containments for/by NP and Parity-P?

Parity-P is the set of languages recognized by a non-deterministic Turing machine which can only distinguish between an even number or odd number of "acceptance" paths (rather than a zero or non-zero ...
26
votes
3answers
1k views

Translating SAT to HornSAT

Is it possible to translate a boolean formula B into an equivalent conjunction of Horn clauses? The Wikipedia article about HornSAT seems to imply that it is, but I have not been able to chase down ...
15
votes
1answer
329 views

Compared growth of the number of syntactic classes and Nerode classes.

For a language L ⊆ Σ^*, define the syntactic congruence ≡ of L as the least congruence on Σ^* that saturates L, i.e. : u ≡ v ⇔ (∀ x, y)[xuy ∈ L ↔ xvy ∈ L]. Now define the Nerode equivalence as the ...
24
votes
4answers
2k views

Bounded-cardinality bounded-frequency set cover: hardness of approximation

Consider the minimum set cover problem with the following restrictions: each set contains at most $k$ elements and each element of the universe occurs in at most $f$ sets. Example: the case $k = 4$ ...
349
votes
93answers
108k views

Algorithms from the Book.

Paul Erdos talked about the "Book" where God keeps the most elegant proof of each mathematical theorem. This even inspired a book (which I believe is now in its 4th edition): Proofs from the Book. If ...
25
votes
1answer
4k views

What's the difference between the Actor Model of Concurrency and Communicating Sequential Processes

I'm trying to wrap my head around what the real differences between the Actor Model of concurrency and Communicating Sequential Processes (CSP) model of concurrency. So far the best that I have ...
37
votes
3answers
1k views

Parameterized complexity of Hitting Set in finite VC-dimension

I'm interested in the parameterized complexity of what I'll call the d-Dimensional Hitting Set problem: given a range space (i.e. a set system / hypergraph) S = (X,R) having VC-dimension at most d and ...
25
votes
2answers
794 views

Approximating the sign rank of a matrix

The sign rank of a matrix A with +1,-1 entries is the least rank (over the reals) of a matrix B which has the same sign pattern as A (i.e., $A_{ij}B_{ij}>0$ for all $i,j$). This notion is important in ...
27
votes
2answers
1k views

What are the consequences of Parity-L = P?

Parity-L is the set of languages recognized by a non-deterministic Turing machine which can only distinguish between an even number or odd number of "acceptance" paths (rather than a zero or non-zero ...
212
votes
59answers
84k views

Major unsolved problems in theoretical computer science?

Wikipedia only lists two problems under "unsolved problems in computer science": P = NP? The existence of one-way functions What are other major problems that should be added to this list? Rules: ...
22
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 ...
25
votes
2answers
3k views
15
votes
3answers
713 views

Can it be determined if language L lies in NP?

Given a language L defined by a Turing Machine that decides it, is it possible to determine algorithmically whether L lies in NP?
4
votes
1answer
19k views

Is integer factorization an NP-complete problem? [duplicate]

Possible Duplicate: What are the consequences of factoring being NP-complete? What notable reference works have covered this?
13
votes
1answer
305 views

Finding odd holes in circulant Paley graphs

The Paley graphs Pq are those whose vertex-set is given by the finite field GF(q), for prime powers q≡1 (mod 4), and where two vertices are adjacent if and only if they differ by a2 ...
12
votes
2answers
317 views

Computational query complexity of SQ-learning

It is known that for PAC learning, there are natural concept classes (e.g. subsets of decision lists) for which there are polynomial gaps between the sample complexity needed for information theoretic ...
25
votes
4answers
2k views

DFA intersection in subquadratic space?

The intersection of two (minimal) DFAs with n states can be computed using O(n2) time and space. This is optimal in general, since the resulting (minimal) DFA may have n2 states. However, if the ...
60
votes
17answers
4k views

Applications of TCS to classical mathematics?

We in TCS often use powerful results and ideas from classical mathematics (algebra, topology, analysis, geometry, etc.). What are some examples of when it has gone the other way around? Here ...
-2
votes
2answers
379 views

Can every non-deterministic finite automate be changed into one with only one acceptance state? [closed]

How can an arbitrary non-deterministic finite automate be converted into one with only one accept stage? If so, what is the proof that this can always be done?
-6
votes
1answer
5k views

What is the advantage of red/black trees in comparison with unbalanced trees? [closed]

In which situations would I use a red/black tree instead of an unbalanced tree?
38
votes
7answers
5k views

Many-one reductions vs. Turing reductions to define NPC

Why do most people prefer to use many-one reductions to define NP-completeness instead of, for instance, Turing reductions?
23
votes
1answer
340 views

Cliquewidth of Almost Cographs

(I posted this question to MathOverflow two weeks ago, but so far without a rigorous answer) I have a question about graph width measures of undirected simple graphs. It is well-known that cographs (...
39
votes
13answers
2k views

Using error-correcting codes in theory

What are applications of error-correcting codes in theory besides error correction itself? I am aware of three applications: Goldreich-Levin theorem about hard core bit, Trevisan's construction of ...
21
votes
2answers
513 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 ...
20
votes
3answers
488 views

Are recursive forms of Godel's statement possible?

The self-referentiality of the P/NP problem has sometimes been highlighted as a barrier to its resolution, see, for instance, Scott Aaronson's paper, is P vs. NP formally independent? One of the many ...
36
votes
5answers
1k views

Complexity of testing for a value versus computing a function

In general we know that the complexity of testing whether a function takes a particular value at a given input is easier than evaluating the function at that input. For example: Evaluating the ...
10
votes
2answers
267 views

Is there an official name for a notion of “reusably universal”?

There are several different (probably inequivalent) notions of computational universality (see for example the last couple pages of http://www.dna.caltech.edu/~woods/download/WoodsNearyTCS07-DRAFT.pdf)...
10
votes
3answers
922 views

A more intuitive proof of the Zone theorem ?

The Zone theorem says that if we stab an arrangement of n lines with another line, the total complexity of its zone, the set of all 0-, 1-, and 2-faces adjacent to it, is O(n). The actual constant is ...
10
votes
5answers
3k views

What are good references on understanding online learning?

Specifically, I'm asking for resources to learn about machine learning systems that can update their respective belief networks (or equivalent) during operation. I've even run across a few, though I ...
23
votes
3answers
1k views

What is known about solutions to sparse integer linear programming problems?

If I have a set of linear constraints in which each constraint has at most (say) 4 variables (all nonnegative and with {0,1} coefficients except for one variable that can have a -1 coefficient), what ...
5
votes
0answers
250 views

When designing an explicitly parallel language, what built in functions should be parallelized? [closed]

As stated by the title. Some examples that I would include would be map and conditionals. What other functions should be built in already parallel for users to expand on it?

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