Masalah dengan kesenjangan kompleksitas terbuka yang besar


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Pertanyaan ini adalah tentang masalah yang ada celah kompleksitas terbuka yang besar antara batas bawah yang diketahui dan batas atas, tetapi bukan karena masalah terbuka pada kelas kompleksitas itu sendiri.

Untuk lebih tepatnya, katakanlah masalah memiliki kelas gap A,B (dengan AB , tidak didefinisikan secara unik) jika A adalah kelas maksimal yang dapat kita buktikan sebagai , dan adalah batas atas minimal yang diketahui. , Yaitu kami memiliki algoritma dalam memecahkan masalah. Ini berarti jika kita akhirnya menemukan bahwa masalahnya adalah -complete dengan , itu tidak akan mempengaruhi teori kompleksitas secara umum, yang bertentangan dengan menemukan algoritma untuk masalah lengkap.BABBCACBPNP

Saya tidak tertarik dengan masalah dengan dan , karena sudah menjadi objek pertanyaan ini .APB=NP

Saya mencari contoh masalah dengan kelas gap yang sejauh mungkin. Untuk membatasi ruang lingkup dan tepat pertanyaan, saya terutama tertarik pada masalah dengan dan , yang berarti keanggotaan dalam dan - yang koheren dengan pengetahuan saat ini, tanpa membuat kelas diketahui runtuh (misalnya kelas dari daftar ini ).APBEXPTIMEPEXPTIME


Apa yang Anda maksud dengan kelas masalah? Asumsikan masalahnya adalah SAT, bagaimana Anda mendefinisikan kelas?
RB

SAT adalah NP-lengkap sehingga kita dapat mengambil dan tidak ada celah di sini, karena kompleksitas SAT cocok persis dengan kelas yang sudah terkenal. Menampilkan setiap hasil baru pada kompleksitas SAT (yaitu milik kelas yang lebih kecil) akan menjadi terobosan dalam teori kompleksitas. Memang pertanyaan itu tidak sepenuhnya terdefinisi dengan baik, karena tergantung pada kompleksitas kelas mana yang dianggap "mainstream", dan A , B tidak didefinisikan secara unik. Namun pertanyaan spesifiknya didefinisikan dengan baik: contoh-contoh bahasa yang koheren dengan pengetahuan saat ini bahwa mereka ada dalam P atau EXPTIME-complete. A=B=NPA,B
Denis

sebenarnya masih belum sepenuhnya terdefinisi dengan baik karena "non-collapsing", jadi itu bergantung pada gagasan "kelas terkenal". Jelas masalah PSPACE-complete tidak sesuai dengan persyaratan, meskipun berada di P atau EXPTIME-complete koheren dengan pengetahuan saat ini. Misalnya daftar ini dapat digunakan sebagai referensi untuk apa yang merupakan kelas "terkenal": en.wikipedia.org/wiki/List_of_complexity_classes
Denis

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Itu tidak cukup sesuai dengan tagihan pertanyaan spesifik Anda tetapi untuk semua penampilan teori eksistensial real keras kepala menolak setiap klasifikasi lebih lanjut di luar NP-hard dan dalam PSPACE (yang terakhir per 1988 hasil JF Canny). en.wikipedia.org/wiki/Existential_theory_of_the_reals
anemone

Jawaban:


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The Knot Kesetaraan Masalah .

Dengan dua simpul yang ditarik di pesawat, apakah secara topologi sama? Masalah ini diketahui dapat diputuskan, dan tampaknya tidak ada penghalang kompleksitas komputasi untuk keberadaannya dalam P. Batas atas terbaik yang saat ini diketahui pada kompleksitas waktunya tampaknya adalah menara dengan ketinggian detik c n , di mana c = 10 10 6 , dan n adalah jumlah penyilangan dalam diagram simpul. Ini berasal dari ikatan oleh Coward dan Lackenby pada jumlah gerakan Reidemeister yang diperlukan untuk mengambil satu simpul ke yang setara. Lihat kertas Lackenby yang lebih baru2cnc=10106n untuk beberapa hasil terkait yang lebih baru dan untuk bentuk eksplisit dari ikatan yang saya berikan di atas (halaman 16).


Thank you for your answer. Do you know the current bounds? Can you point to a reference stating the current state of the art? I am having trouble finding a clear one.
Denis

I've been trying go find something more recent than the 1998 paper of Hass, Lagarias, and Pippenger here. This states that the knot equivalence problem is known to be decidable. I wouldn't be surprised if somebody had shown that it was in EXPTIME since then, but I don't believe anything better than that is known, and it certainly isn't clear that it's not in P. I am fairly sure that none of the results showing that deciding whether something is knotted is in NP extend to this more general problem.
Peter Shor

This MO question is related: mathoverflow.net/questions/77786/… In particular, using recent results announced by Lackenby in people.maths.ox.ac.uk/lackenby/ekt11214.pdf , one obtains that for any knot type K, determining if a given knot is equivalent to K is in NP (note that this does not improve on the Knot Equivalence Problem)
Arnaud

@Arnaud: in fact, it looks to me like these results prove that for two diagrams with at most n crossings, the Knot Equivalence Problem can be solved in time at most a tower of 2's of height cn, where c is an enormous constant. I should check this and edit my answer.
Peter Shor

@PeterShor Yes indeed. I was focusing on the more recent result because it may lead to an improved bound when it is published, if the actual polynomial is explicited.
Arnaud

23

Here's a version of the minimum circuit size problem (MCSP): given the 2n bit truth table of a Boolean function, does it have a circuit of size at most 2n/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 appear in CCC'15) shows there's no uniform NC0 reduction from PARITY to MCSP.


23

The complexity of computing a bit (specified in binary) of an irrational algebraic number (such as 2) has the best known upper bound of PPPPPPP via a reduction to the problem 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 parity of n bits - for all we know this problem could be in AC0. Notice however that we know that no finite automaton can compute the bits of an irrational algebraic number [AB07]


21

Another natural topological problem, similar in spirit to Peter Shor's answer, is embeddability of 2-dimensional abstract simplicial complexes in R3. In general it's natural to ask when can we effectively/efficiently decide that an abstract k-dimensional simplicial complex can be embedded in Rd. For k=1 and d=2 this is the graph planarity problem and has a linear-time algorithm. For k=2 and d=2 there is also a linear time algorithm. The k=2, d=3 case was open until last year, when it was shown to be decidable by Matousek, Sedgwick, Tancer, and Wagner. They say that their algorithm has a primitive recursive time bound, but larger than a tower of exponentials. On the other hand they speculate that it might be possible to put the problem in NP, but going beyond that would be challenging. However, there doesn't seem to be any strong evidence that a polytime algorithm is impossible.

The latter paper has many references for further reading.


16

Multicounter automata (MCAs) are finite automata equipped with counters that can be incremented and decremented within one step but only take integers >=0 as numbers. Unlike Minsky machines (aka counter automata), MCAs are not allowed to test whether a counter is zero.

One of the algorithmic problems with a huge gap related to MSCs is the Reachability problem. E.g., whether the automaton can reach, from a configuration with the initial state and all counters zero, a configuration with an accepting state, and all counters zero again.

The problem is hard for EXPTIME (as shown by Richard Lipton in 1976), decidable (Ernst Mayr, 1981) and solvable in Fω3 (thanks, Sylvain, for pointing this out). A huge gap.


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Hi Thomas, there is a claim of an explicit (and most likely not tight) complexity upper bound in a recent arXiv paper: arxiv.org/abs/1503.00745. The proposed upper bound in Fω3 is however way beyond the complexity classes the original poster was interested in.
Sylvain

@Sylvain Cool! Thanks for sharing this. :)
Michael Wehar

@Sylvain Is EXPTIME the best known lower bound?
Michael Wehar

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@Michael: the best lower bound on the decision problem is actually EXPSPACE (Lipton, 1976, cpsc.yale.edu/sites/default/files/files/tr63.pdf). However, the algorithm by Mayr (1981, dx.doi.org/10.1145/800076.802477), Kosaraju (1982, dx.doi.org/10.1145/800070.802201), and Lambert (1992, dx.doi.org/10.1016/0304-3975(92)90173-D) analysed in the mentioned arXiv paper is known to require at least Ackermannian (i.e., Fω) time.
Sylvain

@Sylvain Thank you very much for all of the additional information. I really appreciate it. :)
Michael Wehar

11

QMA(2) (Quantum Merlin-Arthur with two unentangled provers): certainly QMA-hard, but only known to be in NEXP.


9

The computational problem associated to Noether's Normalization Lemma for explicit varieties ("explicit" in the sense of this paper [freely available full version]). Best known upper bound is EXPSPACE (note, SPACE, not TIME!) but it is conjectured to be in P (and indeed, its being in P is essentially equivalent to derandomizing PIT).


Can you provide more info on this in an explicit form? looks like some kind of bpp-complete problem?

@Arul: Neither PIT nor this problem is BPP-complete in any sense that I am aware of. (In fact, showing that BPP-complete problems exist is still open, and requires non-relativizing techniques - a result going back to Sipser.) However, derandomizing either has a hardness-randomness trade-off, in that their derandomization is essentially equivalent to lower bounds. Aside from the paper linked in the answer ("GCT 5"), lookup hardness-randomness and Kabanets-Impagliazzo.
Joshua Grochow

I will do that but I was interested in this phrase 'and indeed, its being in P is essentially equivalent to derandomizing PIT' which seems to say PIT is some kind of proxy complete problem

@Arul: Yes, to see why PIT is such a "proxy complete problem," see the things I referred to in my previous comment.
Joshua Grochow

why does he use 'Dedicated to Sri Ramakrishna' in many of his works?

6

The Skolem problem (given a linear recurrence with integer base cases and integer coefficients, does it ever reach the value 0) is known to be NP-hard and not known to be decidable. As far as I know anything in between would be consistent with our current knowledge without any collapses of standard complexity classes.

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