All Questions

Filter by
Sorted by
Tagged with
59
votes
7answers
3k views

How to shoot down your proofs

What are general guidelines for checking your proofs? I believe this is important for graduate students like me. I already know what we need to do to prove something, but you always have to check ...
59
votes
4answers
3k views

Evidence that matrix multiplication can be done in quadratic time?

It is widely conjectured that $\omega$, the optimal exponent for matrix multiplication, is in fact equal to 2. My question is simple: What reasons do we have for believing that $\omega = 2$? I'm ...
59
votes
8answers
3k views

What is the complexity class most closely associated with what the human mind can accomplish quickly?

This question is something I've wondered about for a while. When people describe the P vs. NP problem, they often compare the class NP to creativity. They note that composing a Mozart-quality ...
58
votes
13answers
7k views

Open problems on the frontiers of TCS

In the thread Major unsolved problems in theoretical computer science?, Iddo Tzameret made the following excellent comment: I think we should distinguish between major open problems that are viewed ...
58
votes
15answers
2k views

Open access journals

With the advent of internet (and common sense) there is more and more demand for open-access research. Several researchers (including me) find it frustrating that published peer-reviewed research ...
58
votes
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 ...
57
votes
4answers
24k views

Why is 2SAT in P?

I've come across the polynomial algorithm that solves 2SAT. I've found it boggling that 2SAT is in P where all (or many others) of the SAT instances are NP-Complete. What makes this problem different? ...
56
votes
16answers
4k views

What tools do you use to write papers?

What tools do you use to write papers? From the little experience that I have, theoreticians spend a large amount of time writing and refining papers, besides actually being creative. That is, ...
56
votes
10answers
4k views

Provable statements about genetic algorithms

Genetic algorithms don't get much traction in the world of theory, but they are a reasonably well-used metaheuristic method (by metaheuristic I mean a technique that applies generically across many ...
56
votes
5answers
4k views

Overarching reasons why problems are in P or BPP

Recently, when talking to a physicist, I claimed that in my experience, when a problem that naively seems like it should take exponential time turns out nontrivially to be in P or BPP, an "overarching ...
55
votes
14answers
2k views

Where and how did computers help prove a theorem?

The purposes of this question is to collect examples from theoretical computer science where the systematic use of computers was helpful in building a conjecture that lead to a theorem, falsifying a ...
54
votes
13answers
4k views

Information Theory used to prove neat combinatorial statements?

What's your favorite examples where information theory is used to prove a neat combinatorial statement in a simple way ? Some examples I can think of are related to lower bounds for locally decodable ...
54
votes
9answers
23k views

Explain P = NP problem to 10 year old

It is my first question on this site. I am taking a master's course on theory of computation. How you would explain P = NP problem to a 10 year old child and why it has such a monetary reward on it? ...
54
votes
3answers
2k views

Surprising algorithms for counting problems

There are some counting problems which involve counting exponentially many things (relative to the size of the input), and yet have surprising polynomial-time exact, deterministic algorithms. Examples ...
54
votes
2answers
3k views

Can one amplify P=NP beyond P=PH?

In Descriptive Complexity, Immerman has Corollary 7.23. The following conditions are equivalent: 1. P = NP. 2. Over finite, ordered structures, FO(LFP) = SO. This can be thought of as "...
53
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 ...
53
votes
5answers
5k views

What kind of answer does TCS want to the question “Why do neural networks work so well?”

My Ph.D. is in pure mathematics, and I admit I don't know much (i.e. anything) about theoretical CS. However, I have started exploring non-academic options for my career and in introducing myself to ...
53
votes
5answers
29k views

What CS blogs should everyone read?

Many top notch computer science researchers and research groups) maintain active blogs that keep us updated on the latest research in the authors' fields of interest. In most cases, blog posts are ...
53
votes
1answer
2k views

Is there a gap amplification type of result for the Graph Isomorphism Problem?

Suppose $G_1$ and $G_2$ are two undirected graphs on vertex set $\{1, \dotsc, n\}$. The graphs are isomorphic if and only if there is a permutation $\Pi$ such that $G_1 = \Pi(G_2)$, or more formally, ...
52
votes
7answers
3k views

For which problems in P is it easier to verify the result than to find it?

For (search versions) of NP-complete problems, verifying a solution is clearly easier than finding it, since the verification can be done in polynomial time, while finding a witness takes (probably) ...
52
votes
1answer
1k views

A combinatorial version for the polynomial Hirsch conjecture

Consider $t$ disjoint families of subsets of {1,2,…,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ . Suppose that (*) For every $i \lt j \lt k$ and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$, ...
51
votes
21answers
4k views

Dinner-table description of theoretical computer science?

I'm often asked what a theoretical computer scientist does. It would be great to have some nice responses to this question. I tend to fall back to technical jargon and people's eyes usually glaze ...
51
votes
2answers
6k views

What are the outstanding questions in purely functional data structures?

This question is inspired by another question about what's new in PFDS since the publication of Okasaki's book in 1998. I'll start with two questions I have: Is there a purely functional set data ...
50
votes
18answers
13k views

Most memorable CS paper titles

Following a fruitful question in MO, I thought it would be worthwhile to discuss some notable paper names in CS. It is quite clear that most of us might be attracted to read (or at least glance at) a ...
50
votes
3answers
1k views

Rigorous security proof for Wiesner's quantum money?

In his celebrated paper "Conjugate Coding" (written around 1970), Stephen Wiesner proposed a scheme for quantum money that is unconditionally impossible to counterfeit, assuming that the issuing bank ...
49
votes
4answers
3k views

Why do we consider log-space as a model of efficient computation (instead of polylog-space) ?

This might be a subjective question rather than one with a concrete answer, but anyway. In complexity theory we study the notion of efficient computations. There are classes like $\mathsf{P}$ stands ...
49
votes
2answers
7k views

Realizability theory: difference in power between Lambda calculus and Turing Machines

I have three related subquestions, which are highlighted by bullet points below (no, they could not be split, if you are wondering). Andrej Bauer wrote, here, that some functions are realizable ...
49
votes
5answers
20k views

Relationship between Turing Machine and Lambda calculus?

Is there a relationship between the Turing Machine and the Lambda calculus - or did they just happen to arise about the same time?
49
votes
3answers
2k views

Is there a sensible notion of an approximation algorithm for an undecidable problem?

Certain problems are known to be undecidable, but it is nevertheless possible to make some progress on solving them. For example, the halting problem is undecidable, but practical progress can be ...
48
votes
12answers
3k views

What is the theoretical basis of imperative programming?

Functional programming has a theoretical basis in lambda calculus and combinatory logic. As someone involved with statistical computing, I find these concepts to be very useful for modeling. Is ...
48
votes
11answers
3k views

What is the most efficient way to generate a random permutation from probabilistic pairwise swaps?

The question I am interested in is related to generating random permutations. Given a probabilistic pairwise swap gate as the basic building block, what is the most efficient way to produce a ...
48
votes
3answers
4k views

Shallow versus Deep Embeddings

When encoding a logic into a proof assistant such as Coq or Isabelle, a choice needs to be made between using a shallow and a deep embedding. In a shallow embedding logical formulas are written ...
47
votes
20answers
7k views

NP-hard problems on trees

Several optimization problems that are known to be NP-hard on general graphs are trivially solvable in polynomial time (some even in linear time) when the input graph is a tree. Examples include ...
47
votes
8answers
4k views

Are there non-constructive algorithm existence proofs?

I remember I might have encountered references to problems that have been proven to be solvable with a particular complexity, but with no known algorithm to actually reach this complexity. I struggle ...
47
votes
6answers
2k views

Ways for a mathematician to stay informed of current research in complexity theory

Complexity theory is a strong secondary interest of mine but it's not my primary research interest, so there is no hope for me to attend all the conferences, read all the blogs, and ensure that the "...
47
votes
4answers
1k views

What are the consequences of $\mathsf{L}^2 \subseteq \mathsf{P}$?

We know that $\mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{P}$ and that $\mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{L}^2 \subseteq $ $\mathsf{polyL}$, where $\mathsf{L}^2 = \mathsf{...
46
votes
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?
46
votes
5answers
5k views

What is the most intuitive dependent type theory I could learn?

I am interested in getting a really solid grasp on dependent typing. I've read most of TaPL and read (if not fully absorbed) 'Dependent Types' in ATTaPL. I've also read and skimmed a bunch of articles ...
46
votes
6answers
3k views

Good examples for how to write well in TCS

I was editing a student manuscript. The student remarked that it would be nice to see examples of quality writing in published work, and I realized that I couldn't really come up with good examples ...
46
votes
5answers
1k views

Are there Conservation Laws in Complexity Theory?

Let me start with some examples. Why is it so trivial to show CVP is in P but so hard to show LP is in P; while both are P-complete problems. Or take primality. It is easier to show composites in NP ...
45
votes
7answers
6k views

What constitutes denotational semantics?

On a different thread, Andrej Bauer defined denotational semantics as: the meaning of a program is a function of the meanings of its parts. What bothers me about this definition is that it doesn't ...
45
votes
5answers
6k views

Is the Chomsky-hierarchy outdated?

The Chomsky(–Schützenberger) hierarchy is used in textbooks of theoretical computer science, but it obviously only covers a very small fraction of formal languages (REG, CFL, CSL, RE) compared to the ...
45
votes
4answers
12k views

Approximation algorithms for Metric TSP

It is known that metric TSP can be approximated within $1.5$ and cannot be approximated better than $123\over 122$ in polynomial time. Is anything known about finding approximation solutions in ...
45
votes
5answers
2k views

Historical reasons for adoption of Turing Machine as primary model of computation.

It's my understanding that Turing's model has come to be the "standard" when describing computation. I'm interested to know why this is the case -- that is, why has the TM model become more widely-...
45
votes
4answers
3k views

Generalized Ladner's Theorem

Ladner's Theorem states that if P ≠ NP, then there is an infinite hierarchy of complexity classes strictly containing P and strictly contained in NP. The proof uses the completeness of SAT under many-...
45
votes
3answers
3k views

An NP-complete variant of factoring.

Arora and Barak's book presents factoring as the following problem: $\text{FACTORING} = \{\langle L, U, N \rangle \;|\; (\exists \text{ a prime } p \in \{L, \ldots, U\})[p | N]\}$ They add, further ...
45
votes
5answers
2k views

Positive topological ordering

Suppose I have a directed acyclic graph with real-number weights on its vertices. I want to find a topological ordering of the DAG in which, for every prefix of the topological ordering, the sum of ...
45
votes
0answers
1k 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 $...
44
votes
8answers
6k views

The importance of Integrality Gap

I always had trouble in understanding the importance of the Integrality Gap (IG) and bounds on it. IG is the ratio of (the quality of) an optimal integer answer to (the quality of) an optimal real ...
44
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 $K(x) \geq 0.99 |x|$)...

15 30 50 per page