# Tag Info

44

The short answer: the really minimum knowledge of math to understand the first half of the plan of GCT, once you've seen a little of groups, rings, and fields, is basically laid out in Chapter 3 of my thesis (shameless self plug). That chapter is, however, incomplete, in that I don't get to the representation theory part of things. The representation theory ...

41

The usual answer to "what could cause you to divide by a log?" is a combination of two things: a model of computation in which constant time arithmetic operations on word-sized integers are allowed, but in which you want to be conservative about how long the words are, so you assume $O(\log n)$ bits per word (because any fewer than that and you couldn't ...

37

I did a survey on Twitter about this a while back, results here. A few of my favorites: Parametric Polymorphism through Runtime Sealing, or, Theorems for Low, Low Prices! by Jacob Matthews and Amal Ahmed, ESOP 2008 DOI:10.1007/978-3-540-78739-6_2 F-ing Modules by Andreas Rossberg, Claudio Russo, and Derek Dreyer, TLDI 2010 Cons should not cons its arguments ...

35

The Rubik's Cube is a very natural (and to me, unexpected) example. An $n\times n\times n$ cube requires $\Theta(n^2/\log n)$ steps to solve. (Note that this is theta notation, so that's a tight upper and lower bound). This is shown in this paper [1]. It may be worth mentioning that the complexity of solving specific instances of the Rubik's cube is open,...

35

The Planar Separator Theorem states that in any planar $n$-vertex graph $G$ there exists a set of $O(\sqrt{n})$ vertices whose removal leaves the graph disconnected into at least two roughly balanced components. Moreover, such a set can be found in linear time. This (tight) result, proved by Lipton and Tarjan (improving on a previous result by Ungar) is a ...

30

I used to like quirky titles when I started out in computer science but got bored eventually. Some authors manage to write titles that are clever, memorable and relevant but most attempts at funny titles results in unnecessarily long, uninformative and kludgy phrases that I find difficult to remember and look up. There are papers like Pnueli's The Temporal ...

30

The Knot Equivalence Problem. Given two knots drawn in the plane, are they topologically the same? This problem is known to be decidable, and there do not seem to be any computational complexity obstructions to its being in P. The best upper bound currently known on its time complexity seems to be a tower of $2$s of height $c^n$, where $c = 10^{10^{6}}$, ...

29

The recent paper of László Babai showing that Graph Isomorphism is in Quasi-P is already a classic. Here is a more accessible exposition of the result published in the ICM 2018 proceedings.

28

This is really a false belief in math, but comes up often in TCS contexts: If random variables $X$ and $Y$ are independent, then conditioned on $Z$ they remain independent. (false even if $Z$ is independent of both $X$ and $Y$.)

27

Karp-Lipton Theorem states that $\mathsf{NP}$ cannot have polynomial-size boolean circuits unless the Polynomial hierarchy collapses to its second level. Two implications of this theorem for complexity theory: $\mathsf{NP}$ probably has no polynomial-size boolean circuits; proving lower bounds on circuit sizes is therefore a possible approach for ...

26

In a recent preprint, Harvey and Van Der Hoeven show how to compute Integer multiplication in time $O(n \log n)$ on a multi-tape Turing machine, culminating some 60 years of research (Karatsuba, Toom–Cook, Schönhage–Strassen, Fürer, Harvey–Van Der Hoeven–Lecerf). The paper has not yet been peer-reviewed, but prior work of the authors on this problem makes it ...

25

Random Self-Reducibility of the Permanent. Lipton showed that if there exist an algorithm that correctly computes the permanent of $1-1/(3n)$ fraction of all $\mathbb{F}^{n\times n}$, where $\mathbb{F}$ is a finite field of size at least $3n$, then this algorithm can be used as a black box to compute the permanent of any matrix with high probability. The ...

25

For a general audience you have to stick to things that they can see. As soon as you start theorizing they'll start up their mobile phones. Here are some ideas which could be worked out to complete examples: There is a surface which has only one side. A curve may fill an entire square. There are constant width curves other than a circle. It is possible to ...

24

Extending the example pointed out by Emil Jerabek in the comments, $\mathsf{EXPSPACE}$-complete problems arise naturally all over algebraic geometry. This started (I think) with the Ideal Membership Problem (Mayr–Meyer and Mayr) and hence the computation of Gröbner bases. This was then extended to the computation of syzygies (Bayer and Stillman). ...

24

Here's a version of the minimum circuit size problem (MCSP): given the $2^n$ bit truth table of a Boolean function, does it have a circuit of size at most $2^{n/2}$? Known to be not in $AC0$. Contained in $NP$. Generally believed to be $NP$-hard, but this is open. I believe it's not even known to be $AC0[2]$-hard. Indeed, recent work with Cody Murray (to ...

24

The complexity of computing a bit (specified in binary) of an irrational algebraic number (such as $\sqrt{2}$) has the best known upper bound of $\mathsf{P^{{{PP}^{PP}}^{PP}}}$ via a reduction to the problem $\mathsf{BitSLP}$ which known to have this upper bound [ABD14]. On the other hand we do not even know if this problem is harder than computing the ...

23

There is no Zoo-style reference yet, but a recent automata-theoretic survey of Giovanni Pighizzini has been useful to me, especially the slides from his talk. Giovanni Pighizzini, Investigations on Automata and Languages over a Unary Alphabet, CIAA 2014. doi:10.1007/978-3-319-08846-4_3

22

It's the base rather than the exponent, but there's an $O(\varphi^k n^2)$ FPT time bound in "An Efficient Fixed Parameter Tractable Algorithm for 1-Sided Crossing Minimization", Vida Dujmovic, Sue Whitesides, Algorithmica 40:15–31, 2004. Also, it's a lower bound rather than an upper bound, but: "An $n^{1.618}$ lower bound on the time to simulate one queue ...

22

I know of many theoretical computer scientists whose first inspiration came from reading Godel, Escher, Bach Its becoming a bit dated at this point, but is still an excellent read.

22

Another natural topological problem, similar in spirit to Peter Shor's answer, is embeddability of 2-dimensional abstract simplicial complexes in $\mathbb{R}^3$. In general it's natural to ask when can we effectively/efficiently decide that an abstract $k$-dimensional simplicial complex can be embedded in $\mathbb{R}^d$. For $k=1$ and $d=2$ this is the graph ...

22

Importance is in the eyes of the beholder. However, I would say that the Feder–Vardi CSP dichotomy conjecture, proved independently by A. Bulatov and D. Zhuk, is a seminal result.

21

Are NP-completeness in the sense of Cook and NP-completeness in the sense of Karp different concepts, assuming P $\neq$ NP?

21

For a given graph $G$ and an integer $k\ge 1$, the $k$-th power of $G$, denoted by $G^k$, has the same vertex set such that two distinct vertices are adjacent in $G^k$ if their distance in $G$ is at most $k$. The $k$-th power of split graph problem asks if a given graph is the $k$-th power of a split graph. For $k=1$, the problem solvable in linear time....

20

Non-Uniform ACC Circuit Lower Bounds by Ryan Williams: https://people.csail.mit.edu/rrw/acc-lbs.pdf and Classical Verification of Quantum Computations by Urmila Mahadev: http://ieee-focs.org/FOCS-2018-Papers/pdfs/59f259.pdf seem like good candidates

19

Some graph classes allow polynomial-time algorithms for problems that are NP-hard for the class of all graphs. For instance, for perfect graphs, one can find a largest independent set in polynomial time (thanks to vzn in a comment for jogging my memory). Via a product construction, this also allows a unified explanation for several apparently quite ...

19

Lattice-basis reduction (LLL algorithm). This the basis for efficient integer polynomial factorization and some efficient cryptanalytic algorithms like breaking of linear-congruential generators and low-degree RSA. In some sense you can view the Euclidean algorithm as a special case.

19

Many problems that are PSPACE-complete become EXPSPACE-complete when the input is given "succinctly", i.e., via some encoding that lets you describe inputs that would normally be of exponential size. Here is an example on finite automata (equivalently, on directed graphs with labeled edges): deciding whether two automata accept the same language (have the ...

19

$SL = L$. $RL$ stands for randomized logspace and $RL=L$ is a smaller version of the problem $RP=P$. A major stepping stone was the proof of Reingold in '04 ("Undirected S-T Connectivity in Logspace") that $SL = L$, where $S$ stands for "symmetric" and $SL$ is an intermediate class between $RL$ and $L$. The idea is that you can think of a randomized ...

18

(from my comment above) The Fortnow and Melkebeek time/space lower-bound for SAT solvability ($n^{\phi - \epsilon}$ time and $n^{o(1)}$ space) contained the golden ratio in the exponent; but it has been improved later by Ryan Williams.

18

Algebraic geometry Noether's Normalization Lemma (NNL) for explicit varieties is currently only known to be in $\mathsf{EXPSPACE}$ (like general NNL), but is conjectured to be in $\mathsf{P}$ (and is in $\mathsf{P}$ assuming that PIT can be black-box derandomized). Update 4/18/18: It was recently shown that for the variety $\overline{\mathsf{VP}}$ it is in \$...

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