Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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3
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2answers
332 views

Complexity of the Hamiltonian Subcycle problem

The problem is as follows: Given a graph $G$, find a (vertex) disjoint set of cycles $C$ on $G$ such that every vertex is visited by a cycle exactly once. My question is then: what is the ...
9
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1answer
226 views

closure properties of IP(2pfa) and AM(2pfa)

IP(2pfa) and AM(2pfa) are the classes of languages recognized with bounded error by private and public coin versions, respectively, of interactive proof systems with verifiers that are probabilistic ...
14
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1answer
277 views

Machine characterization of $SAC^i$

$SAC^i$ is the class of decision problems solvable by a family of $O({\log}^i{n})$ depth circuits with unbounded-fanin OR and bounded-fanin AND gates. Negations are only allowed at the input level. It ...
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2answers
936 views

NP-hard problems on expander graphs?

In a 2006 presentation titled EXPANDER GRAPHS - ARE THERE ANY MYSTERIES LEFT? , Nati Linial posed the following open problem: Which $NP$-hard computational problem on graph remain hard when ...
10
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1answer
401 views

What are the complexities of the following SAT subsets ?

Assume $P \neq NP$ Let use the following notation ${}^ia$ for tetration (ie. ${}^ia = \underbrace{a^{a^{\cdot^{\cdot^{\cdot^{a}}}}}}_{i \mbox{ times}}$). |x| is the size of the instance x. Let L be ...
6
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1answer
851 views

Oracle relative to which BPP = EXP

An oracle construction relative to which BPP = EXP is usually attributed to Heller (Mathematical System Theory Vol. 17, 1984). Unfortunately I don't have the paper available in my library. Could ...
26
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3answers
697 views

Intermediate problems between L and NL

It is well-known that directed st-connectivity is $NL$-complete. Reingold's breakthrough result showed that undirected st-connectivity is in $L$. Planar directed st-connectivity is known to be in $UL \...
5
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1answer
340 views

Complexity of finding vectors with optimal projection?

Input: a set $T$ of vectors $v_i=(x_i,y_i,z_i)$. Where $x_i,y_i,z_i$ are integers. Output: a subset of vectors $v_1,v_2,...,v_n$ with vector addition $m=\sum v_i$ such that the projection of $m$ on ...
23
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1answer
1k views

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 ...
10
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5answers
779 views

Complexity Classes for Cases Other Than “Worst Case”

Do we have complexity classes with respect to, say, average-case complexity? For instance, is there a (named) complexity class for problems which take expected polynomial time to decide? Another ...
15
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2answers
905 views

Barriers and Monotone Circuit Complexity

Natural proofs is a barrier towards proving lower bounds on the circuit complexity of boolean functions. They do not directly imply any such barrier in proving lower bounds on the $monotone$ circuit ...
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2answers
412 views

Exhausting Simulator of Zero-Knowledge Protocols in the Random Oracle Model

In a paper titled "On Deniability in the Common Reference String and Random Oracle Model," Rafael Pass writes: We note that when proving security according to the standard zero-knowledge definition ...
8
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2answers
449 views

Understanding QMA

This question comes out of an answer Joe Fitzsimons gave to a different question. Most natural complexity classes have a one-line "intuitive description" that helps characterize core problems in that ...
17
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2answers
655 views

Hardness of parameterized CLIQUE?

Let $0\le p\le 1$ and consider the decision problem CLIQUE$_p$ Input: integer $s$, graph $G$ with $t$ vertices and $\lceil p\binom{t}{2} \rceil$ edges Question: does $G$ contain a clique on at ...
8
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1answer
486 views

Algorithms and computational complexity of clique and biclique covers

I've been reading a paper by a mathematical chemist. He proposes some indices to measure the complexity of molecules. From here on in, instead of molecules, think undirected connected graphs: a ...
15
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10answers
3k views

Intractability of NP-complete problems as a principle of physics?

I'm always intrigued by the lack of numerical evidence from experimental mathematics for or against the P vs NP question. While the Riemann Hypothesis has some supporting evidence from numerical ...
10
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3answers
482 views

Is embedding a solution feasible for SAT?

I am interested in "hard" individual instances of NP-complete problems. Ryan Williams discussed the SAT0 problem at Richard Lipton's blog. SAT0 asks whether a SAT instance has the specific solution ...
15
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1answer
503 views

Do the proofs that permanent is not in uniform $\mathsf{TC^0}$ relativize?

This is a follow up to this question, and is related to this question of Shiva Kinali. It seems that the proofs in these papers (Allender, Caussinus-McKenzie-Therien-Vollmer, Koiran-Perifel) use ...
25
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4answers
2k views

Proofs, Barriers and P vs NP

It is well known that any proof resolving the P vs NP question must overcome relativization, natural proofs and algebrization barriers. The following diagram partitions the "proof space" into ...
15
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3answers
303 views

Are there any classes of functions which require provably different resources to compute versus computing their inverse?

Apologies in advance if this question is too simple. Basically, what I want to know is if there are any functions $f(x)$ with the following properties: Take $f_n(x)$ to be $f(x)$ when the domain and ...
9
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5answers
416 views

Results showing existence/non-existence of finite graphs with specific computable properties imply certain complexity results

Are there any known results showing that existence (or non-existence) of finite graphs with specific computable properties imply certain complexity results (such as P = NP)? Here's one completely ...
23
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1answer
833 views

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 ...
2
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1answer
252 views

Number of Vertex Covers and Permanent

Is there any relationship between the number of vertex covers of a graph $G$ and the permanent of $G$'s adjacency matrix?
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2answers
810 views

Problems between NC and P: How many have been resolved from this list?

In the paper "A Compendium of Problems Complete for P" by Greenlaw, Hoover and Ruzzo (PS) (PDF), there is a list of problems in P that are not known to be in NC and not known to be P-complete either. (...
27
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2answers
889 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 ...
7
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2answers
2k views

Monotone-2SAT and Vertex Cover

The following decision problem is called k-True-Monotone-2SAT: Given a 2-CNF boolean formula $F$ that does not contain any negated variables and given a positive integer $k$, can $F$ be satisfied ...
15
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1answer
675 views

LogDCFL-complete problems

LogCFL is the set of all languages that are logspace reducible to a context-free language. Similarly, LogDCFL is the set of all languages that are logspace reducible to a deterministic context-free ...
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4answers
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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 ...
1
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1answer
411 views

Complexity of two perfect matchings with minimum shared edges?

Perfect Matching problem is polynomial time solvable in general graphs. Given undirected simple graph, Is the problem of finding two perfect matching with minimum shared edges between them ...
4
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1answer
287 views

On Defining Probabilistic/Nondeterministic Circuits

Assume that we are interested in deterministic circuits of size $f(n)$. Here, $n$ represents the number of inputs to the circuit. The standard way of defining probabilistic/nondeterministic circuits ...
11
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1answer
399 views

Do people look at loop nestness in boolean circuits?

While an EE undergrad I attended some lectures that presented a nice characterization of boolean circuits in terms of how many nested loops they have. In complexity, boolean circuits are often thought ...
8
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1answer
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Reducing #SAT to #MONOTONE-2SAT

The problem #MONOTONE-2SAT is known to be #P-complete. This means that #SAT can be reduced to it. My question is: given a #SAT instance $F$, which is the transformation that converts $F$ to its ...
1
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1answer
1k views

Removing all but a few cycles in a graph

Let problem $S$ be defined as Given undirected graph $G$ and a set of cycles $C_1,C_2, \ldots, C_n$ in G, find minimum number of vertices that need to be deleted to remove all cycles in the ...
19
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1answer
413 views

Visualizing Unique Games

How would you design a picture to illustrate the unique games conjecture? This is for a "Current Events" presentation on unique games at the next AMS Joint Meeting and for the booklet that will be ...
34
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2answers
2k views

Semantic vs. Syntactic Complexity Classes

In his "Computational Complexity" book, Papadimitriou writes: RP is in some sense a new and unusual kind of complexity class. Not any polynomially bounded nondeterministic Turing machine can be the ...
38
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9answers
3k views

Optimal greedy algorithms for NP-hard problems

Greed, for lack of a better word, is good. One of the first algorithmic paradigms taught in introductory algorithms course is the greedy approach. Greedy approach results in simple and intuitive ...
45
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20answers
6k 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 ...
10
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2answers
1k views

NP-complete variants of undecidable problems?

Examples of bounded $NP$-complete variants of undecidable sets: Bounded Halting problem={ $(M, x, 1^t)$| NTM machine $M$ halts and accepts $x$ within $t$ steps} Bounded Tiling={ $(T, 1^t)$| there is ...
4
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1answer
423 views

Is this problem mappable to 3SAT or is it weaker than 3SAT?

Consider a variant of a satisifiability problem. Given n dimensions (n >= 3, n < 10,000 think of n as large but finite) The range of each dimension is either an interval over the integers or an ...
3
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2answers
505 views

Counting complexity of a scheduling problem. [closed]

Let $T={1,…,n}$ be a set of tasks. Each task $i$ has associated a non negative processing time $p_i$ and a deadline $d_i$. A feasible schedule of the tasks consists of a permutation of $n$ elements $\...
7
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2answers
396 views

On the class of the FNP version of the Hamiltonian Cycle problem

This post is linked to: FNP complexity class Many places say that the decision version of Hamiltonian Cycle is NP-Complete, and NP-Complete problems are those whose solution can be verified in ...
7
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2answers
549 views

SAT Solution Space - Definition of Cluster of Solutions

I'm looking for a formal definition of Cluster of Solutions. My current understanding is the following. Let $x$ be a boolean assignment on $n$ variables. Let $f: \{ 0,1 \} ^n \to \mathbb{N}$ be a ...
10
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1answer
417 views

Hardness of constrained star system problem?

A star system is a family $F$ of n subsets of n-elements set $S$. A star system is graphical if there is some graph $G(V,E)$ such that $F$ is the family of vertex neighborhoods in $G$. It is $NP$-...
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2answers
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Beginner's Guide to Derandomization

I found the book Pairwise Independence and Derandomization on the subject, but it's more research-oriented than tutorial oriented. I'm new to the subject of "Derandomization," and as such, I wanted ...
4
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3answers
503 views

FNP complexity class

Where can I find more information about the FNP complexity class? The only place I did find anything on FNP was http://en.wikipedia.org/wiki/FNP_(complexity) However, that isn't sufficient for me to ...
33
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2answers
2k views

NTIME(n^k) ≠ DTIME(n^k) ?

In "On determinism versus nondeterminism and related problems" (Proc. IEEE FOCS, pages 429–438, 1983), Paul, Pippenger, Szemerédi and Trotter proved that $\mathsf{NTIME}(n)\neq\mathsf{DTIME}(n)$. ...
14
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1answer
882 views

Integer relation detection for Subset Sum or NPP?

Is there a way to encode an instance of Subset Sum or the Number Partition Problem so that a (small) solution to an integer relation yields an answer? If not definitely, then in some probabilistic ...
7
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3answers
5k views

Polynomial Time Algorithm for Graph Isomorphism Testing [closed]

"Michael I. Trofimov" claims that he has found a poly-time algorithm for graph isomorphism, which works for all graphs. The paper is given in arXiv. The companion website gives a proof-of-concept ...
11
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1answer
617 views

Time complexity analysis for Reingold's UST-CONN algorithm

What is the exact time complexity of the undirected st-connectivity log-space algorithm by Omer Reingold ?
42
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8answers
6k views

Best Upper Bounds on SAT

In another thread, Joe Fitzsimons asked about "the best current lower bounds on 3SAT." I'd like to go the other way: What's the best current upper bounds on 3SAT? In other words, what is the time ...