# All Questions

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### Parameterized complexity from P to NP-hard and back again

I'm looking for examples of problems parametrized by a number $k \in \mathbb{N}$, where the problem's hardness is non-monotonic in $k$. Most problems (in my experience) have a single phase transition, ...
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 ...
8k 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 ...
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 ...
9k views

### How to get a job

I'm new to the site. On mathoverflow this would be community wiki, but I don't see how to set that here. Not a research question, but hopefully of interest to professional theoretical computer ...
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 ...
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 ...
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 ...
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? ...
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 ...
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, ...
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 ...
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### 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 ...
31k 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 ...
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 ...
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 ...
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 ...
14k 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 ...
4k 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) ...
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 ...
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 "...
23k 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?
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### 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, ...
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 ...
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 ...
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 $T \in {\cal F}_k$, ...
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 ...
2k 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 ...
5k 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 ...
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 ...
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 ...
4k views

### If you could rename dynamic programming…

If you could rename dynamic programming, what would you call it?
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### Theoretical explanations for practical success of SAT solvers?

What theoretical explanations are there for the practical success of SAT solvers, and can someone give a "wikipedia-style" overview and explanation tying them all together? By analogy, the smoothed ...
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?
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### 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 ...
8k 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 ...
5k 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 ...
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 ...