Pub Date : 2021-05-01DOI: 10.1007/s00037-021-00206-1
J. Torán, Florian Wörz
{"title":"Reversible Pebble Games and the Relation Between Tree-Like and General Resolution Space","authors":"J. Torán, Florian Wörz","doi":"10.1007/s00037-021-00206-1","DOIUrl":"https://doi.org/10.1007/s00037-021-00206-1","url":null,"abstract":"","PeriodicalId":51005,"journal":{"name":"Computational Complexity","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00037-021-00206-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47421044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-02DOI: 10.1007/s00037-020-00204-9
M. Giesbrecht, Armin Jamshidpey, É. Schost
{"title":"Subquadratic-Time Algorithms for Normal Bases","authors":"M. Giesbrecht, Armin Jamshidpey, É. Schost","doi":"10.1007/s00037-020-00204-9","DOIUrl":"https://doi.org/10.1007/s00037-020-00204-9","url":null,"abstract":"","PeriodicalId":51005,"journal":{"name":"Computational Complexity","volume":"21 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00037-020-00204-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51809933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-20DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.45
Ronen Shaltiel, Jad Silbak
A stochastic code is a pair of encoding and decoding procedures (Enc, Dec) where $${{rm Enc} : {0, 1}^{k} times {0, 1}^{d} rightarrow {0, 1}^{n}}$$ Enc : { 0 , 1 } k × { 0 , 1 } d → { 0 , 1 } n . The code is ( p, L )-list decodable against a class $$mathcal{C}$$ C of “channel functions” $$C : {0,1}^{n} rightarrow {0,1}^{n}$$ C : { 0 , 1 } n → { 0 , 1 } n if for every message $$m in {0,1}^{k}$$ m ∈ { 0 , 1 } k and every channel $$C in mathcal{C}$$ C ∈ C that induces at most pn errors, applying Dec on the “received word” C (Enc( m,S )) produces a list of at most L messages that contain m with high probability over the choice of uniform $$S leftarrow {0, 1}^{d}$$ S ← { 0 , 1 } d . Note that both the channel C and the decoding algorithm Dec do not receive the random variable S , when attempting to decode. The rate of a code is $$R = k/n$$ R = k / n , and a code is explicit if Enc, Dec run in time poly( n ). Guruswami and Smith (Journal of the ACM, 2016) showed that for every constants $$0 < p < frac{1}{2}, epsilon > 0$$ 0 < p < 1 2 , ϵ > 0 and $$c > 1$$ c > 1 there exist a constant L and a Monte Carlo explicit constructions of stochastic codes with rate $$R geq 1-H(p) - epsilon$$ R ≥ 1 - H ( p ) - ϵ that are ( p, L )-list decodable for size $$n^c$$ n c channels. Here, Monte Carlo means that the encoding and decoding need to share a public uniformly chosen $${rm poly}(n^c)$$ poly ( n c ) bit string Y , and the constructed stochastic code is ( p, L )-list decodable with high probability over the choice of Y . Guruswami and Smith pose an open problem to give fully explicit (that is not Monte Carlo) explicit codes with the same parameters, under hardness assumptions. In this paper, we resolve this open problem, using a minimal assumption: the existence of poly-time computable pseudorandom generators for small circuits, which follows from standard complexity assumptions by Impagliazzo and Wigderson (STOC 97). Guruswami and Smith also asked to give a fully explicit unconditional constructions with the same parameters against $$O(log n)$$ O ( log n ) -space online channels. (These are channels that have space $$O(log n)$$ O ( log n ) and are allowed to read the input codeword in one pass.) We also resolve this open problem. Finally, we consider a tighter notion of explicitness, in which the running time of encoding and list-decoding algorithms does not increase, when increasing the complexity of the channel. We give explicit constructions (with rate approaching $$1 - H(p)$$ 1 - H ( p ) for every $$p leq p_{0}$$ p ≤ p 0 for some $$p_{0} >0$$ p 0 > 0 ) for channels that are circuits of size $$2^{n^{Omega(1/d)}}$$ 2 n Ω ( 1 / d ) and depth d . Here, the running time of encoding and decoding is a polynomial that does not depend on the depth of the circuit. Our approach builds on the machinery developed by Guruswami and Smith, replacing some probabilistic arguments with explicit constructions. We also present a simplified and general approach that makes the redu
{"title":"Explicit List-Decodable Codes with Optimal Rate for Computationally Bounded Channels","authors":"Ronen Shaltiel, Jad Silbak","doi":"10.4230/LIPIcs.APPROX-RANDOM.2016.45","DOIUrl":"https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.45","url":null,"abstract":"A stochastic code is a pair of encoding and decoding procedures (Enc, Dec) where $${{rm Enc} : {0, 1}^{k} times {0, 1}^{d} rightarrow {0, 1}^{n}}$$ Enc : { 0 , 1 } k × { 0 , 1 } d → { 0 , 1 } n . The code is ( p, L )-list decodable against a class $$mathcal{C}$$ C of “channel functions” $$C : {0,1}^{n} rightarrow {0,1}^{n}$$ C : { 0 , 1 } n → { 0 , 1 } n if for every message $$m in {0,1}^{k}$$ m ∈ { 0 , 1 } k and every channel $$C in mathcal{C}$$ C ∈ C that induces at most pn errors, applying Dec on the “received word” C (Enc( m,S )) produces a list of at most L messages that contain m with high probability over the choice of uniform $$S leftarrow {0, 1}^{d}$$ S ← { 0 , 1 } d . Note that both the channel C and the decoding algorithm Dec do not receive the random variable S , when attempting to decode. The rate of a code is $$R = k/n$$ R = k / n , and a code is explicit if Enc, Dec run in time poly( n ). Guruswami and Smith (Journal of the ACM, 2016) showed that for every constants $$0 < p < frac{1}{2}, epsilon > 0$$ 0 < p < 1 2 , ϵ > 0 and $$c > 1$$ c > 1 there exist a constant L and a Monte Carlo explicit constructions of stochastic codes with rate $$R geq 1-H(p) - epsilon$$ R ≥ 1 - H ( p ) - ϵ that are ( p, L )-list decodable for size $$n^c$$ n c channels. Here, Monte Carlo means that the encoding and decoding need to share a public uniformly chosen $${rm poly}(n^c)$$ poly ( n c ) bit string Y , and the constructed stochastic code is ( p, L )-list decodable with high probability over the choice of Y . Guruswami and Smith pose an open problem to give fully explicit (that is not Monte Carlo) explicit codes with the same parameters, under hardness assumptions. In this paper, we resolve this open problem, using a minimal assumption: the existence of poly-time computable pseudorandom generators for small circuits, which follows from standard complexity assumptions by Impagliazzo and Wigderson (STOC 97). Guruswami and Smith also asked to give a fully explicit unconditional constructions with the same parameters against $$O(log n)$$ O ( log n ) -space online channels. (These are channels that have space $$O(log n)$$ O ( log n ) and are allowed to read the input codeword in one pass.) We also resolve this open problem. Finally, we consider a tighter notion of explicitness, in which the running time of encoding and list-decoding algorithms does not increase, when increasing the complexity of the channel. We give explicit constructions (with rate approaching $$1 - H(p)$$ 1 - H ( p ) for every $$p leq p_{0}$$ p ≤ p 0 for some $$p_{0} >0$$ p 0 > 0 ) for channels that are circuits of size $$2^{n^{Omega(1/d)}}$$ 2 n Ω ( 1 / d ) and depth d . Here, the running time of encoding and decoding is a polynomial that does not depend on the depth of the circuit. Our approach builds on the machinery developed by Guruswami and Smith, replacing some probabilistic arguments with explicit constructions. We also present a simplified and general approach that makes the redu","PeriodicalId":51005,"journal":{"name":"Computational Complexity","volume":"30 1","pages":"1-70"},"PeriodicalIF":1.4,"publicationDate":"2021-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47687978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-06DOI: 10.4230/LIPIcs.ITCS.2020.2
Orr Paradise
Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs: $$circ quad$$ ∘ A PCP is strong if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim. $$circ quad$$ ∘ A PCP is smooth if each location in a proof is queried with equal probability. We prove that all sets in $$mathcal{NP}$$ NP have PCPs that are both smooth and strong, are of polynomial length and can be verified based on a constant number of queries. This is achieved by following the proof of the PCP theorem of Arora et al. (JACM 45(3):501–555, 1998), providing a stronger analysis of the Hadamard and Reed–Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in $$mathcal{NP}$$ NP has a smooth strong canonical PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of $$mathcal{NP}$$ NP witnesses to correct proofs. This improves on the recent construction of Dinur et al. (in: Blum (ed) 10th innovations in theoretical computer science conference, ITCS, San Diego, 2019) of PCPPs that are strong canonical but inherently non-smooth. Our result implies the hardness of approximating the satisfiability of “ stable ” 3CNF formulae with bounded variable occurrence , where stable means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric). This proves a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (in: Chan (ed) Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, SODA, San Diego, 2019), suggesting a connection between the hardness of these instances and other stable optimization problems.
{"title":"Smooth and Strong PCPs","authors":"Orr Paradise","doi":"10.4230/LIPIcs.ITCS.2020.2","DOIUrl":"https://doi.org/10.4230/LIPIcs.ITCS.2020.2","url":null,"abstract":"Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs: $$circ quad$$ ∘ A PCP is strong if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim. $$circ quad$$ ∘ A PCP is smooth if each location in a proof is queried with equal probability. We prove that all sets in $$mathcal{NP}$$ NP have PCPs that are both smooth and strong, are of polynomial length and can be verified based on a constant number of queries. This is achieved by following the proof of the PCP theorem of Arora et al. (JACM 45(3):501–555, 1998), providing a stronger analysis of the Hadamard and Reed–Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in $$mathcal{NP}$$ NP has a smooth strong canonical PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of $$mathcal{NP}$$ NP witnesses to correct proofs. This improves on the recent construction of Dinur et al. (in: Blum (ed) 10th innovations in theoretical computer science conference, ITCS, San Diego, 2019) of PCPPs that are strong canonical but inherently non-smooth. Our result implies the hardness of approximating the satisfiability of “ stable ” 3CNF formulae with bounded variable occurrence , where stable means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric). This proves a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (in: Chan (ed) Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, SODA, San Diego, 2019), suggesting a connection between the hardness of these instances and other stable optimization problems.","PeriodicalId":51005,"journal":{"name":"Computational Complexity","volume":"30 1","pages":"1-77"},"PeriodicalIF":1.4,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42613339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-26DOI: 10.1007/s00037-022-00227-4
H. Hirai, M. Ikeda
{"title":"A cost-scaling algorithm for computing the degree of determinants","authors":"H. Hirai, M. Ikeda","doi":"10.1007/s00037-022-00227-4","DOIUrl":"https://doi.org/10.1007/s00037-022-00227-4","url":null,"abstract":"","PeriodicalId":51005,"journal":{"name":"Computational Complexity","volume":"31 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43271512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-28DOI: 10.1007/s00037-021-00215-0
Amit Sinhababu, T. Thierauf
{"title":"Factorization of Polynomials Given by Arithmetic Branching Programs","authors":"Amit Sinhababu, T. Thierauf","doi":"10.1007/s00037-021-00215-0","DOIUrl":"https://doi.org/10.1007/s00037-021-00215-0","url":null,"abstract":"","PeriodicalId":51005,"journal":{"name":"Computational Complexity","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46408042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-01DOI: 10.1007/s00037-020-00194-8
Or Meir
{"title":"Toward Better Depth Lower Bounds: Two Results on the Multiplexor Relation","authors":"Or Meir","doi":"10.1007/s00037-020-00194-8","DOIUrl":"https://doi.org/10.1007/s00037-020-00194-8","url":null,"abstract":"","PeriodicalId":51005,"journal":{"name":"Computational Complexity","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00037-020-00194-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44365439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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