# Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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### What is known about the complexity of finding minimum circuits for SAT?

What is known about the complexity of finding minimal circuits that compute SAT up to length $n$? More formally: what is the complexity of a function which, given $1^{n}$ as input outputs a minimal ...
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### Is the 2016 implementation of Shor's algorithm really scalable?

In the 2016 Science paper "Realization of a scalable Shor algorithm" [1], the authors factor 15 with only 5 qubits, which is fewer than the 8 qubits "required" according to Table 1 of [2] and Table 5 ...
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### What is the complexity of this covering problem?

Edit: I first misformulated my constraint (2), it is now corrected. I also added more information and examples. With some colleagues, studying some other algorithmic question, we were able to reduce ...
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### Approximate degree of $\textrm{AC}^0$

EDIT (v2): Added a section at the end on what I know about the problem. EDIT (v3): Added discussion on threshold degree at the end. Question This question is mainly a reference request. I don't ...
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### What are consequences of the collapse of CH?

I don't grasp the full complexity of the counting hierarchy $CH$. I understand $CH$ is in $PSPACE$, and contains $PH$ within its second level, due to the Toda's theorem. But, what would be important ...
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### Counting Isomorphism Types of Graphs

Polya's counting theorem leads to an algorithm for counting (precisely) the number of isomorphism types of graphs with $n$ vertices in $\exp (\sqrt n )$ steps. From Polya theorem you get a formula ...
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### Is “Experimental Complexity Theory” being used to solve open problems?

Scott Aaronson proposed an interesting challange: can we use supercomputers today to help solve CS problems in the same way that physicists use large particle colliders? More concretely, my ...
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### What evidence do we have for (and against) Unique Games Conjecture?

Subhash Khot's Unique Games Conjecture is one of active research areas in complexity theory. What evidence do we have for it? What evidence do we have against it?
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### How much would a SAT oracle help speeding up polynomial time algorithms?

Access to a $SAT$ oracle would provide a major, super-polynomial speed-up for everything in ${\bf NP}-{\bf P}$ (assuming the set is not empty). It is less clear, however, how much would $\bf P$ ...
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### For which k is PLANAR NAE k-SAT in P?

The Not All Equal $k$-SAT problem (NAE $k$-SAT), given a set $C$ of clauses over a set $X$ of boolean variables such that each clause contains at most $k$ literals, asks whether there exists a truth ...
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### What's the status of Babai's Graph isomorphism result?

It's been over a year since his January 2017 retraction and correction. Is there news? If not is this normal for validation to take this long? I would expect it would get plenty of attention. Has ...
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### Computational complexity of the 3-partition problem with distinct numbers

This question is related to an answer I posted in response to another question. The 3-partition problem is the following problem: Instance: Positive integers a1, …, an, where n=3m and the sum of the ...
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### Minimum Unsatisfiable 3-CNF Formulae

I am currently interested in obtaining (or constructing) and studying 3-CNF formulae which are unsatisfiable, and are of minimum size. That is, they must consist of as few clauses (m = 8 preferably) ...
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### Logspace algorithms on graphs with bounded tree width

Tree width measures how close a graph is to a tree. It is NP-hard to compute tree width. The best known approximation algorithm achieves $O(\sqrt{{\log}n})$ factor. Courcelle's theorem states that ...
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### I want an easy Gadget to prove Planar Hamiltonian Cycle NP-Complete (from Hamiltonian Cycle)

It is known that Hamiltonian (Ham for short) Cycle is NP-complete and that Planar Ham Cycle is NP-Complete. The proof for Planar Ham Cycle is not from Ham Cycle. Is there a nice gadget that will, ...
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### Sampling satisfiable 3-SAT formulas

Consider the following computational task: We want to sample a 3-SAT formula of $n$ variables (a variant: $n$ variables $m$ clauses) with respect to the uniform probability distribution, conditioned ...
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### Testing whether letters can be scheduled to achieve a word in a regular language

I fix a regular language $L$ on an alphabet $\Sigma$, and I consider the following problem that I call letter scheduling for $L$. Informally, the input gives me $n$ letters and an interval for each ...
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### Why is HAMILTONIAN CYCLE so different from PERMANENT?

A polynomial $f(x_1,\ldots,x_n)$ is a monotone projection of a polynomial $g(y_1,\ldots,y_m)$ if $m$ = poly$(n)$, and there is an assignment $\pi:\{y_1,\ldots,y_m\}\to\{x_1,\ldots,x_n, 0,1\}$ such ...
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### Knot Recognition as a Proof of Work

Currently bitcoin has a proof of work (PoW) system using SHA256. Other hash functions use a proof of work system use graphs, partial hash function inversion. Is it possible to use a Decision problem ...
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### Is there a natural problem in quasi-polynomial time, but not in polynomial time?

László Babai recently proved that the Graph Isomorphism problem is in quasipolynomial time. See also his talk at University of Chicago, note from the talks by Jeremy Kun GLL post 1, GLL post 2, GLL ...
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### Problems that are easy on unweighted graphs, but hard for weighted graphs

Many algorithmic graph problems can be solved in polynomial time both on unweighted and weighted graphs. Some examples are shortest path, min spanning tree, longest path (in directed acyclic graphs), ...
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### Curious about computer-assisted NP-completeness proofs

In the paper "THE COMPLEXITY OF SATISFIABILITY PROBLEMS" by Thomas J. Schaefer, the author has mentioned that ...
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### Monotone arithmetic circuits

The state of our knowledge about general arithmetic circuits seems to be similar to the state of our knowledge about Boolean circuits, i.e. we don't have good lower-bounds. On the other hand we have ...
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### Problems outside of P that are not P-hard

While reading an answer by Peter Shor and an earlier question by Adam Crume I realized that I have some misconceptions about what it means to be $\mathsf{P}$-hard. A problem is $\mathsf{P}$-hard if ...
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### Statements that imply $\mathbf{P}\neq \mathbf{NP}$

This is sort of an open-ended question - for which I apologize in advance. Are there examples of statements that (seemingly) don't have anything to do with complexity or Turing machines but the ...
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### 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 ...
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### Does NP-hardness imply P-hardness?

If a problem is NP-hard (using polynomial time reductions), does that imply that it is P-hard (using log space or NC reductions)? It seems intuitive that if it is as hard as any problem in NP that it ...
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### Papers on relation between computational complexity and algebraic geometry/topology?

I was wondering what papers I should read to understand this question A unexpected connection to other areas of mathematics such as algebraic geometry or higher cohomology. Perhaps even an area of ...
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### Adding integers represented by their factorization is as hard as factoring? Reference request

I'm looking for a reference for the following result: Adding two integers in the factored representation is as hard as factoring two integers in the usual binary representation. (I'm pretty sure ...
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### NP complete graph problems about structural properties

(This question is a bit of a "survey".) I'm currently working on a problem where I'm trying to partition the edges of a tournament into two sets, both of which are required to fulfill some structural ...
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### How does the BosonSampling paper avoid easy classes of complex matrices?

In The computational complexity of linear optics (ECCC TR10-170), Scott Aaronson and Alex Arkhipov argue that if quantum computers can be efficiently simulated by classical computers then the ...
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Today in New York and all over the world Christos Papadimitriou's birthday is celebrated. This is a good opportunity to ask about the relations between Christos' complexity class PPAD (and his other ...
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### Tardos Function Counterexample to Blum's $P\neq NP$ Claim

In this thread, Norbet Blum's attempted $P \neq NP$ proof is succinctly disproved by noting that the Tardos function is a counterexample to Theorem 6. Theorem 6: Let $f \in \mathcal{B}_n$ be any ...
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### How much computational power fits into a cubic centimeter?

This question is a followup on the question about DNA algorithms asked by Aadita Mehra. In comments there, Joe Fitzsimmons said, in part: [T]he radius of the system must scale proportionately to ...
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### What is UG-hardness, and how is it different from NP-hardness based on the unique games conjecture?

There are many inapproximability results which rely on the unique games conjecture. For example, Assuming the unique games conjecture, it is NP-hard to approximate the maximum cut problem within a ...
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### Are there natural separations in the nondeterministic time hierarchy?

The original Nondeterministic Time Hierarchy Theorem is due to Cook (the link is to S. Cook, A hierarchy for nondeterministic time complexity, JCSS 7 343–353, 1973). The theorem states that for any ...
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### Multiplicative version of 3-SUM

What is known about the time complexity of the following problem, which we call 3-MUL? Given a set $S$ of $n$ integers, are there elements $a,b,c\in S$ such that $ab=c$? This problem is similar to ...
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### Circuit lower bounds and kolmogorov complexity

Consider the following reasoning: Let $K(x)$ denote the Kolmogorov complexity of the string $x$. Chaitin's incompleteness theorem says that for any consistent and sufficiently strong formal ...
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### Generating a tower defense maze, aka Finding the K most vital nodes (“nodewise interdiction”) in an unweighted grid-graph

In a tower defense game, you have an NxM grid with a start, a finish, and a number of walls. Enemies take the shortest path from start to finish without passing through any walls (they aren't usually ...
In graph property testing, an algorithm queries a target graph for the presence or absence of edges and needs to determine whether the target either has a certain property or is $\epsilon$-far from ...