Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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32
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1answer
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Is Gap-3SAT NP-complete even for 3CNF formulas where no pair of variables appears in significantly more clauses than the average?

In this question, a 3CNF formula means a CNF formula where each clause involves exactly three distinct variables. For a constant 0<s<1, Gap-3SATs is the following promise problem: Gap-3SATs ...
31
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7answers
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Should we consider $\mathsf{P} \neq \mathsf{NP}$ a law of nature?

Many experts believe that the $\mathsf{P} \neq \mathsf{NP}$ conjecture is true and use it in their results. My concern is that the complexity strongly depends on the $\mathsf{P} \neq \mathsf{NP}$ ...
31
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6answers
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Is there a natural problem on the naturals that is NP-complete?

Any natural number can be regarded as a bit sequence, so inputting a natural number is the same as inputting a 0-1 sequence, so NP-complete problems with natural inputs obviously exist. But are there ...
31
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1answer
872 views

Treewidth and the NL vs L Problem

ST-Connectivity is the problem of determining whether there exists a directed path between two distinguished vertices $s$ and $t$ in a directed graph $G(V,E)$. Whether this problem can be solved in ...
31
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4answers
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Does a noisy version of Conway's game of life support universal computation?

Quoting Wikipedia, "[Conway's Game of Life] has the power of a universal Turing machine: that is, anything that can be computed algorithmically can be computed within Conway's Game of Life." Do such ...
31
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1answer
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Refereed games with uncorrelated semi-private coins

I was (and still am) really interested in the answer to this question, because this is an interesting variation on the complexity of games which hasn't been resolved, so I offered a bounty. I thought ...
31
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0answers
807 views

Is BPP= P known for ANY uniform model of computation?

Many believe that BPP $=$ P "should" hold for Turing machines. We even have some "witnesses" for this: otherwise some "strange" things would happen; see e.g. this paper by Implagliazzo and Wigderson. ...
30
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4answers
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Consequences of NP=PSPACE

What would be the nasty consequences of NP=PSPACE? I am surprised I did not found anything on this, given that these classes are among the most famous ones. In particular, would it have any ...
30
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6answers
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Why are so few natural candidates for NP-intermediate status?

It is well known by Ladner's Theorem that if ${\mathsf P}\neq \mathsf {NP}$, then there exist infinitely many $\mathsf {NP}$-intermediate ($\mathsf{NPI}$) problems. There are also natural candidates ...
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2answers
821 views

Is there an oracle such that SAT is not infinitely often in sub-exponential time?

Define $io$-$SUBEXP$ to be the class of languages $L$ such that there is a language $L' \in \cap_{\varepsilon > 0} TIME(2^{n^{\varepsilon}})$ and for infinitely many $n$, $L$ and $L'$ agree on all ...
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Hierarchies in NP (under the assumption that P != NP)

Assuming that P != NP, I believe it has been shown that there are problems which are not in P and not NP-Complete. Graph Isomorphism is conjectured to be such a problem. Is there any evidence of more ...
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3answers
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Justification of log f in DTIME hierarchy theorem

If we look at DTIME hierarchy theorem, we've got a log due to the overhead in simulation of a deterministic Turing Machine by a universal machine : $DTIME(\frac{f}{\log f}) \subsetneq DTIME(f)$ We ...
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2answers
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How hard is it to count the number of factors of an integer?

Given an integer $N$ of length $n$ bits, how hard is it to output the number of prime factors (or alternatively number of factors) of $N$? If we knew the prime factorization of $N$, then this would ...
30
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1answer
745 views

Can graph isomorphism be decided with square root bounded nondeterminism?

Bounded nondeterminism associates a function $g(n)$ with a class $C$ of languages accepted by resource-bounded deterministic Turing machines, to form a new class $g$-$C$. This class consists of those ...
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2answers
832 views

Is there a polynomial time algorithm to determine if the span of a set of matrices contains a permutation matrix?

I would like to find a polynomial time algorithm that determines if the span of a given set of matrices contains a permutation matrix. If any one knows if this problem is of a different complexity ...
29
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7answers
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Proving lower bounds by proving upper bounds

The recent breakthrough circuit complexity lower-bound result of Ryan Williams provides a proof technique that uses upper-bound result to prove complexity lower-bounds. Suresh Venkat in his answer to ...
29
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3answers
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A decision problem which is not known to be in PH but will be in P if P=NP

Edit: As Ravi Boppana correctly pointed out in his answer and Scott Aaronson also added another example in his answer, the answer to this question turned out to be “yes” in a way which I had not ...
29
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2answers
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Derandomizing Valiant-Vazirani?

The Valiant-Vazirani theorem says that if there is a polynomial time algorithm (deterministic or randomized) for distinguishing between a SAT formula that has exactly one satisfying assignment, and an ...
29
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2answers
903 views

Polynomial method for complexity results

Polynomial methods, say Combinatorial Nullstellensatz and Chevalley–Warning theorem are powerful tools in additive combinatorics. By representing a problem with proper polynomials, they can guarantee ...
29
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4answers
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If P = NP were true, would quantum computers be useful?

Suppose that P = NP is true. Would there then be any practical application to building a quantum computer such as solving certain problems faster, or would any such improvement be irrelevant based on ...
29
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2answers
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Hierarchy for BPP vs derandomization

In one sentence: would the existence of a hierarchy for $\mathsf{BPTIME}$ imply any derandomization results? A related but vaguer question is: does the existence of a hierarchy for $\mathsf{BPTIME}$ ...
29
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1answer
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Are there canonical non-relativizing techniques?

In a lot of domains, there are canonical techniques which everybody working in the field should master. For example, for logspace reductions, the "bit trick" for composition consisting of not ...
29
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1answer
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Fourier coefficients Boolean Functions described by Bounded Depth Circuits with AND OR and XOR gates

Let $f$ be a Boolean function and let's think about f as a function from $\{-1,1\}^n$ to $\{ -1,1 \}$. In this language the Fourier expansion of f is simply the expansion of f in terms of square free ...
29
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3answers
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coNP certificate for Graph Isomorphism

It is easy to see that graph isomorphism(GI) is in NP. It is a major open problem whether GI is in coNP. Are there any potential candidates of properties of graphs that can be used as coNP ...
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2answers
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Can you identify the sum of two permutations in polynomial time?

There were two questions asked recently on cs.se which were either related to or had a special case equivalent to the following question: Suppose you have a sequence $a_1, a_2, \ldots a_n$ of $n$ ...
29
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3answers
999 views

Is NPI contained in P/poly?

It is conjectured that $\mathsf{NP} \nsubseteq \mathsf{P}/\text{poly}$ since the converse would imply $\mathsf{PH} = \Sigma_2$. Ladner's theorem establishes that if $\mathsf{P} \ne \mathsf{NP}$ then $\...
29
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0answers
951 views

Does $EXP\neq ZPP$ imply sub-exponential simulation of BPP or NP?

By simulation I mean in the Impaglazzio-Widgerson [IW98] sense, i.e. sub-exponential deterministic simulation which appears correct i.o to every efficient adversary. I think this is a proof: if $EXP\...
28
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16answers
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Hard-looking algorithmic problems made easy by theorems

I am looking for nice examples, where the following phenomenon occurs: (1) An algorithmic problem looks hard, if you want to solve it working from the definitions and using standard results only. (2) ...
28
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6answers
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Alternative proofs of Schwartz–Zippel lemma

I'm only aware of two proofs of Schwartz–Zippel lemma. The first (more common) proof is described in the wikipedia entry. The second proof was discovered by Dana Moshkovitz. Are there any other ...
28
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6answers
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Which SAT problems are easy?

What are "easy regions" for satisfiability? In other words, sufficient conditions for some SAT solver to be able to find a satisfying assignment, assuming it exists. One example is when each clause ...
28
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2answers
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NP-intermediate problems with efficient quantum solutions

Peter Shor showed that two of the most important NP-intermediate problems, factoring and the discrete log problem, are in BQP. In contrast, the best known quantum algorithm for SAT (Grover's search) ...
28
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2answers
824 views

Tight Lower bounds on Savitch's theorem

First of all, I apologize in advance for any stupidity. I am by no means an expert on complexity theory (far from it! I am an undergraduate taking my first class in complexity theory) Here's my ...
28
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3answers
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How many instances of 3-SAT are satisfiable?

Consider the 3-SAT problem on n variables. The number of possible distinct clauses is: $$C = 2n \times 2(n-1) \times 2(n -2) / 3! = 4 n(n-1)(n-2)/3 \text.$$ The number of problem instances is the ...
28
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2answers
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A category of NP-complete problems?

Does it make sense to consider a category of all NP-complete problems, with morphisms as poly-time reductions between different instances? Has anyone ever published a paper about this, and if so, ...
28
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6answers
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Natural NP-complete problems with “large” witnesses

The question on cstheory "What is NP restricted to linear size witnesses?" asks about the class NP restricted to linear size $O(n)$ witnesses, but Are there natural NP-complete problems in which (...
28
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2answers
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Kolmogorov's conjecture that $P$ has linear-size circuits

In his book, Boolean Function Complexity, Stasys Jukna mentions (page 564) that Kolmogorov believed that every language in P has circuits of linear size. No reference is mentioned and I couldn't find ...
28
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3answers
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Is there a candidate for a natural problem in $P/poly - P$?

I want to know if non-uniformity helps computing functions in practice. It is easy to show that there are functions in $P/poly - P$, take any uncomputable function $f$ and consider the language {$0^{f(...
28
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1answer
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Toy examples for barriers to $P \ne NP$

Are there any toy examples that provide 'essential' insights into understanding the three known barriers to $P = NP$ problem - relativization, natural proofs and algebrization?
28
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2answers
929 views

Approximate counting problem capturing BQP

In the black-box model, the problem of determining the output of a BPP machine $M(x,r)$ on input $x$ is the approximate counting problem of determining $E_r M(x,r)$ with additive error 1/3 (say). Is ...
28
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4answers
800 views

Complexity of minimising polynomial formula size

Let $f(x_1,\dots,x_n)$ be a degree $d$ polynomial in $n$ variables over $\mathbb{F}_2$, where $d$ is constant (say 2 or 3). I would like to find the smallest formula for $f$, where "formula" and "...
28
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1answer
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Functions that are Not Efficiently Computable but Learnable

We know that (see, e.g., Theorems 1 and 3 of [1]), roughly speaking, under suitable conditions, functions that can be efficiently computed by Turing machine in polynomial time ("efficiently computable"...
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2answers
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Can we not output the Kolmogorov complexity?

Let us fix a prefix-free encoding of Turing-machines and a universal Turing-machine $U$ that on input $(T,x)$ (encoded as the prefix-free code of $T$ followed by $x$) outputs whatever $T$ outputs on ...
28
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2answers
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How many DFAs accept two given strings?

Fix an integer $n$ and alphabet $\Sigma=\{0,1\}$. Define $DFA(n)$ to be the collection of all finite-state automata on $n$ states with starting state 1. We are considering all DFAs (not just connected,...
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5answers
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Is it a rule that discrete problems are NP-hard and continuous problems are not?

In my computer science education, I increasingly notice that most discrete problems are NP-complete (at least), whereas optimizing continuous problems is almost always easily achievable, usually ...
27
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6answers
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How are real numbers specified in computation?

This may be a basic question, but I've been reading and trying to understand papers on such subjects as Nash equilibrium computation and linear degeneracy testing and have been unsure of how real ...
27
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2answers
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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 ...
27
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4answers
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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 $k = 4$ ...
27
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2answers
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Ladner's Theorem vs. Schaefer's Theorem

While reading the article "Is it Time to Declare Victory in Counting Complexity?" over at the "Godel's Lost Letter and P=NP" blog, they mentioned the dichotomy for CSP's. After some link following, ...
27
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3answers
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Succinct Problems in $\mathsf{P}$

The study of Succinct representation of graphs was initiated by Galperin and Wigderson in a paper from 1983, where they prove that for many simple problems like finding a triangle in a graph, the ...
27
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3answers
989 views

Natural problems in $NP \cap coNP$ not in $UP \cap coUP$?

Are there any natural problems in $NP \cap coNP$ that are not (known to be/thought to be) in $UP \cap coUP$? Obviously the big one everyone knows about in $NP \cap coNP$ is the decision version of ...

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