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Huang, S. (2015). Hardness of Constraint Satisfaction and Hypergraph Coloring: Constructions of Probabilistically Checkable Proofs with Perfect Completeness. (Doctoral dissertation). Stockholm: KTH Royal Institute of Technology
Open this publication in new window or tab >>Hardness of Constraint Satisfaction and Hypergraph Coloring: Constructions of Probabilistically Checkable Proofs with Perfect Completeness
2015 (English)Doctoral thesis, monograph (Other academic)
Abstract [en]

A Probabilistically Checkable Proof (PCP) of a mathematical statement is a proof written in a special manner that allows for efficient probabilistic verification. The celebrated PCP Theorem states that for every family of statements in NP, there is a probabilistic verification procedure that checks the validity of a PCP proof by reading only 3 bits from it. This landmark theorem, and the works leading up to it, laid the foundation for many subsequent works in computational complexity theory, the most prominent among them being the study of inapproximability of combinatorial optimization problems.

This thesis focuses on a broad class of combinatorial optimization problems called Constraint Satisfaction Problems (CSPs). In an instance of a CSP problem of arity k, we are given a set of variables taking values from some finite domain, and a set of constraints each involving a subset of at most k variables. The goal is to find an assignment that simultaneously satisfies as many constraints as possible. An alternative formulation of the goal that is commonly used is Gap-CSP, where the goal is to decide whether a CSP instance is satisfiable or far from satisfiable, where the exact meaning of being far from satisfiable varies depending on the problems.We first study Boolean CSPs, where the domain of the variables is {0,1}. The main question we study is the hardness of distinguishing satisfiable Boolean CSP instances from those for which no assignment satisfies more than some epsilon fraction of the constraints. Intuitively, as the arity increases, the CSP gets more complex and thus the hardness parameter epsilon should decrease. We show that for Boolean CSPs of arity k, it is NP-hard to distinguish satisfiable instances from those that are at most 2^{~O(k^{1/3})}/2^k-satisfiable.

We also study coloring of graphs and hypergraphs. Given a graph or a hypergraph, a coloring is an assignment of colors to vertices, such that all edges or hyperedges are non-monochromatic. The gap problem is to distinguish instances that are colorable with a small number of colors, from those that require a large number of colors. For graphs, we prove that there exists a constant K_0>0, such that for any K >= K_0, it is NP-hard to distinguish K-colorable graphs from those that require 2^{Omega(K^{1/3})} colors. For hypergraphs, we prove that it is quasi-NP-hard to distinguish 2-colorable 8-uniform hypergraphs of size N from those that require 2^{(log N)^{1/4-o(1)}} colors.

In terms of techniques, all these results are based on constructions of PCPs with perfect completeness, that is, PCPs where the probabilistic proof verification procedure always accepts a correct proof. Not only is this a very natural property for proofs, but it can also be an essential requirement in many applications. It has always been particularly challenging to construct PCPs with perfect completeness for NP statements due to limitations in techniques. Our improved hardness results build on and extend many of the current approaches. Our Boolean CSP result and GraphColoring result were proved by adapting the Direct Sum of PCPs idea by Siu On Chan to the perfect completeness setting. Our proof for hypergraph coloring hardness improves and simplifies the recent work by Khot and Saket, in which they proposed the notion of superposition complexity of CSPs.

Abstract [sv]

Ett probabilistiskt verifierbart bevis (eng: Probabilistically Checkable Proof, PCP) av en matematisk sats är ett bevis skrivet på ett speciellt sätt vilket möjliggör en effektiv probabilistisk verifiering. Den berömda PCP-satsen säger att för varje familj av påståenden i NP finns det en probabilistisk verifierare som kontrollerar om en PCP bevis är giltigt genom att läsa endast 3 bitar från det. Denna banbrytande sats, och arbetena som ledde fram till det, lade grunden för många senare arbeten inom komplexitetsteorin, framförallt inom studiet av approximerbarhet av kombinatoriska optimeringsproblem.

I denna avhandling fokuserar vi på en bred klass av optimeringsproblem i form av villkorsuppfyllningsproblem (engelska ``Constraint Satisfaction Problems'' CSPs). En instans av ett CSP av aritet k ges av en mängd variabler som tar värden från någon ändlig domän, och ett antal villkor som vart och ett beror på en delmängd av högst k variabler. Målet är att hitta ett tilldelning av variablerna som samtidigt uppfyller så många som möjligt av villkoren. En alternativ formulering av målet som ofta används är Gap-CSP, där målet är att avgöra om en CSP-instans är satisfierbar eller långt ifrån satisfierbar, där den exakta innebörden av att vara ``långt ifrån satisfierbar'' varierar beroende på problemet.Först studerar vi booleska CSPer, där domänen är {0,1}. Den fråga vi studerar är svårigheten av att särskilja satisfierbara boolesk CSP-instanser från instanser där den bästa tilldelningen satisfierar högst en andel epsilon av villkoren. Intuitivt, när ariten ökar blir CSP mer komplexa och därmed bör svårighetsparametern epsilon avta med ökande aritet. Detta visar sig vara sant och ett första resultat är att för booleska CSP av aritet k är det NP-svårt att särskilja satisfierbara instanser från dem som är högst 2^{~O(k^{1/3})}/2^k-satisfierbara.

Vidare studerar vi färgläggning av grafer och hypergrafer. Givet en graf eller en hypergraf, är en färgläggning en tilldelning av färger till noderna, så att ingen kant eller hyperkant är monokromatisk. Problemet vi analyserar är att särskilja instanser som är färgbara med ett litet antal färger från dem som behöver många färger. För grafer visar vi att det finns en konstant K_0>0, så att för alla K >= K_0 är det NP-svårt att särskilja grafer som är K-färgbara från dem som kräver minst 2^{Omega(K^{1/3})} färger. För hypergrafer visar vi att det är kvasi-NP-svårt att särskilja 2-färgbara 8-likformiga hypergrafer som har N noder från dem som kräv minst 2^{(log N)^{1/4-o(1)}} färger.

Samtliga dessa resultat bygger på konstruktioner av PCPer med perfekt fullständighet. Det vill säga PCPer där verifieraren alltid accepterar ett korrekt bevis. Inte bara är detta en mycket naturlig egenskap för PCPer, men det kan också vara ett nödvändigt krav för vissa tillämpningar. Konstruktionen av PCPer med perfekt fullständighet för NP-påståenden ger tekniska komplikationer och kräver delvis utvecklande av nya metoder. Vårt booleska CSPer resultat och vårt Färgläggning resultat bevisas genom att anpassa ``Direktsumman-metoden'' introducerad av Siu On Chan till fallet med perfekt fullständighet. Vårt bevis för hypergraffärgningssvårighet förbättrar och förenklar ett färskt resultat av Khot och Saket, där de föreslog begreppet superpositionskomplexitet av CSP.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2015. p. viii, 142
Series
TRITA-CSC-A, ISSN 1653-5723 ; 2015:13
Keywords
Combinatorial optimization, approximation, inapproximability, hardness, probabilistically checkable proofs, pcp, perfect completeness, boolean constraint satisfaction problem, csp, graph coloring, hypergraph coloring, direct sum, superposition, label cover
National Category
Computer Sciences
Research subject
Computer Science
Identifiers
urn:nbn:se:kth:diva-168576 (URN)978-91-7595-633-6 (ISBN)
Public defence
2015-10-09, F3, Lindstedtsvägen 26, KTH, Stockholm, 14:00 (English)
Opponent
Supervisors
Funder
EU, European Research Council, 226203Swedish Research Council
Note

QC 20150916

Available from: 2015-06-11 Created: 2015-06-04 Last updated: 2018-01-11Bibliographically approved
Håstad, J., Huang, S., Manokaran, R., O’Donnell, R. & Wright, J. (2015). Improved NP-inapproximability for 2-variable linear equations. In: Leibniz International Proceedings in Informatics, LIPIcs: . Paper presented at 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015; Princeton UniversityPrinceton; United States (pp. 341-360). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 40
Open this publication in new window or tab >>Improved NP-inapproximability for 2-variable linear equations
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2015 (English)In: Leibniz International Proceedings in Informatics, LIPIcs, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH , 2015, Vol. 40, p. 341-360Conference paper, Published paper (Refereed)
Abstract [en]

An instance of the 2-Lin(2) problem is a system of equations of the form "xi + xj = b (mod 2)". Given such a system in which it’s possible to satisfy all but an C ε fraction of the equations, we show it is NP-hard to satisfy all but a Cε fraction of the equations, for any C < 11/8 = 1.375 (and any 0 < ε ≤ 1/8). The previous best result, standing for over 15 years, had 5/4 in place of 11/8. Our result provides the best known NP-hardness even for the Unique-Games problem, and it also holds for the special case of Max-Cut. The precise factor 11 8 is unlikely to be best possible; we also give a conjecture concerning analysis of Boolean functions which, if true, would yield a larger hardness factor of 3/2. Our proof is by a modified gadget reduction from a pairwise-independent predicate. We also show an inherent limitation to this type of gadget reduction. In particular, any such reduction can never establish a hardness factor C greater than 2.54. Previously, no such limitation on gadget reductions was known.

Place, publisher, year, edition, pages
Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2015
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-187142 (URN)10.4230/LIPIcs.APPROX-RANDOM.2015.341 (DOI)2-s2.0-84958547344 (Scopus ID)
Conference
18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015; Princeton UniversityPrinceton; United States
Note

QC 20160517

Available from: 2016-05-17 Created: 2016-05-17 Last updated: 2016-05-17Bibliographically approved
Huang, S. (2013). Approximation resistance on satisfiable instances for predicates with few accepting inputs. In: Dan Boneh, Tim Roughgarden, Joan Feigenbaum (Ed.), Symposium on Theory of Computing Conference: . Paper presented at Symposium on Theory of Computing Conference STOC '13, Palo Alto, CA, June, 1-4, 2013 (pp. 457-466).
Open this publication in new window or tab >>Approximation resistance on satisfiable instances for predicates with few accepting inputs
2013 (English)In: Symposium on Theory of Computing Conference / [ed] Dan Boneh, Tim Roughgarden, Joan Feigenbaum, 2013, p. 457-466Conference paper, Published paper (Refereed)
Abstract [en]

 For every integer k≥ 3, we prove that there is a predicate P on k Boolean variables with 2O(k1/3) accepting assignments that is approximation resistant even on satisfiable instances. That is, given a satisfiable CSP instance with constraint P, we cannot achieve better approximation ratio than simply picking random assignments. This improves the best previously known result by Hastad and Khot where the predicate has 2 O(k1/2) accepting assignments.

Our construction is inspired by several recent developments. One is the idea of using direct sums to improve soundness of PCPs, developed by Chan [5]. We also use techniques from Wenner [32] to construct PCPs with perfect completeness without relying on the d-to-1 Conjecture.

National Category
Computer Sciences
Identifiers
urn:nbn:se:kth:diva-124328 (URN)10.1145/2488608.2488666 (DOI)2-s2.0-84879803083 (Scopus ID)978-1-4503-2029-0 (ISBN)
Conference
Symposium on Theory of Computing Conference STOC '13, Palo Alto, CA, June, 1-4, 2013
Funder
EU, European Research Council, 6853
Note

QC 20130719

Available from: 2013-06-28 Created: 2013-06-28 Last updated: 2018-01-11Bibliographically approved
Huang, S. (2013). Improved Hardness of Approximating Chromatic Number. In: Prasad Raghavendra, Sofya Raskhodnikova, Klaus Jansen, José D. P. Rolim (Ed.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: 16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21-23, 2013. Proceedings. Paper presented at 16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2013 and the 17th International Workshop on Randomization and Computation, RANDOM 2013; Berkeley, CA; United States; 21 August 2013 through 23 August 2013 (pp. 233-243). Springer
Open this publication in new window or tab >>Improved Hardness of Approximating Chromatic Number
2013 (English)In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: 16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21-23, 2013. Proceedings / [ed] Prasad Raghavendra, Sofya Raskhodnikova, Klaus Jansen, José D. P. Rolim, Springer, 2013, p. 233-243Conference paper, Published paper (Refereed)
Abstract [en]

We prove that for sufficiently large K, it is NP-hard to color K-colorable graphs with less than 2Ω(K 1/3) colors. This improves the previous result of K versus K1/25 log K in Khot [1].

Place, publisher, year, edition, pages
Springer, 2013
Series
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), ISSN 0302-9743 ; 8096
Keywords
Chromatic number, K-colorable graphs, NP-hard
National Category
Computer Sciences
Identifiers
urn:nbn:se:kth:diva-136350 (URN)10.1007/978-3-642-40328-6_17 (DOI)2-s2.0-84885224075 (Scopus ID)978-364240327-9 (ISBN)
Conference
16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2013 and the 17th International Workshop on Randomization and Computation, RANDOM 2013; Berkeley, CA; United States; 21 August 2013 through 23 August 2013
Funder
EU, European Research Council, 6853
Note

QC 20131206

Available from: 2013-12-04 Created: 2013-12-04 Last updated: 2018-01-11Bibliographically approved
Huang, S. & Lu, P. (2012). A Dichotomy for Real Weighted Holant Problems. In: 2012 IEEE 27th Annual Conference On Computational Complexity (CCC). Paper presented at 27th Annual IEEE Conference on Computational Complexity (CCC), JUN 26-29, 2012, Porto, Portugal (pp. 96-106). IEEE
Open this publication in new window or tab >>A Dichotomy for Real Weighted Holant Problems
2012 (English)In: 2012 IEEE 27th Annual Conference On Computational Complexity (CCC), IEEE , 2012, p. 96-106Conference paper, Published paper (Refereed)
Abstract [en]

Holant is a framework of counting characterized by local constraints. It is closely related to other well-studied frameworks such as #CSP and Graph Homomorphism. An effective dichotomy for such frameworks can immediately settle the complexity of all combinatorial problems expressible in that framework. Both #CSP and Graph Homomorphism can be viewed as sub-families of Holant with the additional assumption that the equality constraints are always available. Other subfamilies of Holant such as Holant* and Holant(c) problems, in which we assume some specific sets of constraints to be freely available, were also studied. The Holant framework becomes more expressive and contains more interesting tractable cases with less or no freely available constraint functions, while, on the other hand, it also becomes more challenging to obtain a complete characterization of its time complexity. Recently, complexity dichotomy for a variety of sub-families of Holant such as #CSP, Graph Homomorphism, Holant* and Holant(c) were proved. The dichotomy for the general Holant framework, which is the most desirable, still remains open. In this paper, we prove a dichotomy for the general Holant framework where all the constraints are real symmetric functions. This setting already captures most of the interesting combinatorial problems defined by local constraints, such as (perfect) matching, independent set, vertex cover and so on. This is the first time a dichotomy is obtained for general Holant Problems without any auxiliary functions. One benefit of working with Holant framework is some powerful new reduction techniques such as Holographic reduction. Along the proof of our dichotomy, we introduce a new reduction technique, namely realizing a constraint function by approximating it. This new technique is employed in our proof in a situation where it seems that all previous reduction techniques fail, thus this new idea of reduction might also be of independent interest. Besides proving dichotomy and developing new technique, we also obtained some interesting by-products. We prove a dichotomy for #CSP restricting to instances where each variable appears a multiple of d times for any d. We also prove that counting the number of Eulerian-Orientations on 2k-regular graphs is #P-hard for any k >= 2.

Place, publisher, year, edition, pages
IEEE, 2012
Series
IEEE Conference on Computational Complexity. Proceedings, ISSN 1093-0159
Keywords
Holant problem, #CSP, complexity dichotomy, reduction technique
National Category
Computer Sciences
Identifiers
urn:nbn:se:kth:diva-104718 (URN)10.1109/CCC.2012.16 (DOI)000308976600010 ()2-s2.0-84866508430 (Scopus ID)978-0-7695-4708-4 (ISBN)
Conference
27th Annual IEEE Conference on Computational Complexity (CCC), JUN 26-29, 2012, Porto, Portugal
Note

QC 20121109

Available from: 2012-11-09 Created: 2012-11-09 Last updated: 2018-01-12Bibliographically approved
Huang, S. (2012). Approximation Resistance on Satisfiable Instances for Predicates Strictly Dominating Parity.
Open this publication in new window or tab >>Approximation Resistance on Satisfiable Instances for Predicates Strictly Dominating Parity
2012 (English)Report (Other academic)
Abstract [en]

In this paper, we study the approximability of Max CSP(P) where P is a Boolean predicate. We prove that assuming Khot’s d-to-1 Conjecture, if the set of accepting inputs of P strictly contains all inputs with even (or odd) parity, then it is NP-hard to approximate MaxCSP(P) better than the simple random assignment algorithm even on satisfiable instances.This is a generalization of a work by O’Donnell and Wu which proved that it is NP-hard to approximate satisfiable instances of Max CSP(NTW) beyond 5/8 + epsilon for any epsilon > 0 based on Khot’s d-to-1 Conjecture, where NTW is the “Not Two” predicate of size 3.

Publisher
p. 21
Series
Electronic Colloquium on Computational Complexity (ECCC), ISSN 1433-8092 ; 40
Keywords
pcp, approximation, d-to-1, perfect completeness
National Category
Computer Sciences
Identifiers
urn:nbn:se:kth:diva-108165 (URN)
Funder
EU, European Research Council, 6853
Note

QC 20130109

Available from: 2013-01-09 Created: 2012-12-19 Last updated: 2018-01-11Bibliographically approved
Cai, J.-Y., Huang, S. & Lu, P. (2012). From Holant to #CSP and Back: Dichotomy for Holant (c) Problems. Algorithmica, 64(3), 511-533
Open this publication in new window or tab >>From Holant to #CSP and Back: Dichotomy for Holant (c) Problems
2012 (English)In: Algorithmica, ISSN 0178-4617, E-ISSN 1432-0541, Vol. 64, no 3, p. 511-533Article in journal (Refereed) Published
Abstract [en]

We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted H-colorings. We consider these problems for general complex valued functions that take boolean inputs. We show that results from one framework can be used to derive results in another, and this happens in both directions. Holographic reductions discover an underlying unity, which is only revealed when these counting problems are investigated in the complex domain a",. We prove three complexity dichotomy theorems, leading to a general theorem for Holant (c) problems. This is the natural class of Holant problems where one can assign constants 0 or 1. More specifically, given any signature grid on G=(V,E) over a set of symmetric functions, we completely classify the complexity to be in P or #P-hard, according to F , of Sigma Pi f(v)(sigma vertical bar E(v)), sigma:E ->{0,1}v epsilon V where f(v) epsilon F boolean OR {0, 1} (0, 1 are the unary constant 0, 1 functions). Not only is holographic reduction the main tool, but also the final dichotomy can be only naturally stated in the language of holographic transformations. The proof goes through another dichotomy theorem on Boolean complex weighted #CSP.

Keywords
Holant problem, #CSP, Holographic reduction, Dichotomy
National Category
Computer Sciences Mathematics
Identifiers
urn:nbn:se:kth:diva-103121 (URN)10.1007/s00453-012-9626-6 (DOI)000308228400009 ()2-s2.0-84867229589 (Scopus ID)
Note

QC 20121004

Available from: 2012-10-04 Created: 2012-10-04 Last updated: 2018-01-12Bibliographically approved
Guo, H., Huang, S. & Lu, P. (2011). The Complexity of Weighted Boolean #CSP Modulo k. In: Thomas Schwentick and Christoph Dürr (Ed.), 28th International Symposium on Theoretical Aspects of Computer Science, STACS 2011. Paper presented at STACS 2011 (pp. 249-260). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik
Open this publication in new window or tab >>The Complexity of Weighted Boolean #CSP Modulo k
2011 (English)In: 28th International Symposium on Theoretical Aspects of Computer Science, STACS 2011 / [ed] Thomas Schwentick and Christoph Dürr, Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik , 2011, p. 249-260Conference paper, Published paper (Refereed)
Abstract [en]

We prove a complexity dichotomy theorem for counting weighted Boolean CSP modulo k  for any positive integer k1 . This generalizes a theorem by Faben for the unweighted setting. In the weighted setting, there are new interesting tractable problems. We first prove a dichotomy theorem for the finite field case where k  is a prime. It turns out that the dichotomy theorem for the finite field is very similar to the one for the complex weighted Boolean #CSP, found by [Cai, Lu and Xia, STOC 2009]. Then we further extend the result to an arbitrary integer k .

Place, publisher, year, edition, pages
Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2011
Series
Leibniz International Proceedings in Informatics (LIPIcs), ISSN 1868-8969 ; 9
Keywords
#CSP, dichotomy theorem, counting problems, computational complexity
National Category
Computer Sciences
Identifiers
urn:nbn:se:kth:diva-42721 (URN)10.4230/LIPIcs.STACS.2011.249 (DOI)2-s2.0-79960009639 (Scopus ID)978-3-939897-25-5 (ISBN)
Conference
STACS 2011
Funder
EU, European Research Council, 6853
Note
QC 20111013Available from: 2011-10-13 Created: 2011-10-12 Last updated: 2018-01-12Bibliographically approved
Cai, J.-Y., Huang, S. & Lu, P. (2010). From Holant To #CSP And Back: Dichotomy For Holant^c Problems. In: Otfried Cheong and Kyung-Yong Chwa and Kunsoo Park (Ed.), Algorithms and Computation - 21st International Symposium. Paper presented at ISAAC 2010 (pp. 253-265). Springer
Open this publication in new window or tab >>From Holant To #CSP And Back: Dichotomy For Holant^c Problems
2010 (English)In: Algorithms and Computation - 21st International Symposium / [ed] Otfried Cheong and Kyung-Yong Chwa and Kunsoo Park, Springer , 2010, p. 253-265Conference paper, Published paper (Refereed)
Abstract [en]

We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted H-colorings. We consider these problems for general complex valued functions that take Boolean inputs. We show that results from one framework can be used to derive results in another, and this happens in both directions. Holographic reductions discover an underlying unity, which is only revealed when these counting problems are investigated in the complex domain . We prove three complexity dichotomy theorems, leading to a general theorem for Holantc problems. This is the natural class of Holant problems where one can assign constants 0 or 1. More specifically, given any signature grid on G=(VE) over a set  of symmetric functions, we completely classify the complexity to be in P or #P-hard, according to , of:E01vVfv(E(v))where fv01 (0, 1 are the unary constant 0, 1 functions). Not only is holographic reduction the main tool, but also the final dichotomy can be only naturally stated in the language of holographic transformations. The proof goes through another dichotomy theorem on Boolean complex weighted #CSP.

Place, publisher, year, edition, pages
Springer, 2010
Series
Lecture Notes in Computer Science, ISSN 0302-9743 ; 6506
National Category
Computer Sciences
Identifiers
urn:nbn:se:kth:diva-42822 (URN)10.1007/978-3-642-17517-6 (DOI)2-s2.0-78650905937 (Scopus ID)978-3-642-17516-9 (ISBN)
Conference
ISAAC 2010
Funder
EU, European Research Council, 6853
Note
QC 20111013Available from: 2011-10-13 Created: 2011-10-12 Last updated: 2018-01-12Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-1600-5290

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