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  1. Home
  2. Books
  3. Complexity Dichotomies for Counting Problems
  • Cited by 11
  • Volume 1: Boolean Domain
  • Jin-Yi Cai, University of Wisconsin, Madison, Xi Chen, Columbia University, New York
Publisher:
Cambridge University Press
Online publication date:
November 2017
Print publication year:
2017
Online ISBN:
9781107477063
DOI:
https://doi.org/10.1017/9781107477063
Subjects:
Mathematics, Algorithmics, Complexity, Computer Algebra, Computational Geometry, Logic, Categories and Sets, Computer Science
Information

Book description

Complexity theory aims to understand and classify computational problems, especially decision problems, according to their inherent complexity. This book uses new techniques to expand the theory for use with counting problems. The authors present dichotomy classifications for broad classes of counting problems in the realm of P and NP. Classifications are proved for partition functions of spin systems, graph homomorphisms, constraint satisfaction problems, and Holant problems. The book assumes minimal prior knowledge of computational complexity theory, developing proof techniques as needed and gradually increasing the generality and abstraction of the theory. This volume presents the theory on the Boolean domain, and includes a thorough presentation of holographic algorithms, culminating in classifications of computational problems studied in exactly solvable models from statistical mechanics.

Reviews

‘This remarkable volume presents persuasive evidence that computer applications obey beautiful unities: within the significant classes considered, the problems that are not known to be polynomial time computable are all reducible to each other by a small number of elegant techniques. The treatment is original, comprehensive and thought provoking.'

Leslie Valiant - Harvard University, Massachusetts

‘This book provides a thorough study of the complexity of counting. These basic problems arise in statistical physics, optimization, algebraic combinatorics and computational complexity. The past fifteen years of research have led to a (surprisingly clean) complete characterization of their complexity in the form of a ‘dichotomy theorem', whose proof is the main goal of this volume. Along the way, the authors provide detailed explanations of basic methods for studying such problems, including holographic algorithms and reductions. The book is very well written and organized, and should be useful to researchers and graduate students in the fields above.'

Avi Wigderson - Institute for Advanced Study, Princeton, New Jersey

'This book would make an excellent introduction to the field of counting complexity for a mathematically literate reader with little or no background in computational complexity … Many of the results included in the book are recent, and have not appeared in book form before. As such, this thorough, self-contained treatment of the subject makes a very valuable contribution to the literature.'

Kitty Meeks Source: MathSciNet

‘There’s no doubt in my mind that anyone interested in computational complexity would benefit greatly by reading it. It is a remarkable and indispensable book about a deep and important topic.’

Frederic Green Source: SIGACT News

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Contents

Contents
References
Bibliography
[AB09] Sanjeev, Arora and Boaz, Barak. Computational complexity: A modern approach. Cambridge University Press, 2009.
[ALM+98] Sanjeev, Arora, Carsten, Lund, Rajeev, Motwani, Madhu, Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM 45 (1998), no. 3, 501–555.
[Bar83] Francisco, Barahona Balancing signed toroidal graphs in polynomial time. Preprint, University of Chile, 1983.
[Bar04] Regis, Barbanchon On unique graph 3-colorability and parsimonious reductions in the plane. Theoretical Computer Science 319 (2004), no. 1–3, 455–482.
[Bax82] Rodney J., Baxter Exactly solved models in statistical mechanics. Academic press London, 1982.
[BCSS98] Lenore, Blum, Felipe, Cucker, Michael, Shub, and Steve, Smale. Complexity and real computation, Springer, 1998.
[BD07] Andrei A., Bulatov and Victor, Dalmau. Towards a dichotomy theorem for the counting constraint satisfaction problem. Information and Computation 205 (2007), no. 5, 651–678.
[BG05] Andrei A., Bulatov and Martin, Grohe. The complexity of partition functions. Theoretical Computer Science 348 (2005), no. 2, 148–186.
[BOKR86] Michael, Ben-Or, Dexter, Kozen, and John, Reif. The complexity of elementary algebra and geometry. Journal of Computer and Systems Sciences 32 (1986), no. 2, 251–264.
[Bol98] Bela, Bollobas Modern graph theory. Springer-Verlag, 1998.
[BPR06] Saugata, Basu, Richard, Pollack, and Marie-Francoise, Roy. Algorithms in real algebraic geometry, Springer-Verlag, 2006.
[BS94] Robert, Burton and Jeffrey E., Steif Non-uniqueness of measures of maximal entropy for subshifts of finite type. Ergodic Theory and Dynamical Systems 14 (1994), no. 2, 213–235.
[BS95] New results on measures of maximal entropy. Israel Journal of Mathematics 89 (1995), 275–300.
[BSS89] Lenore, Blum, Mike, Shub, and Steve, Smale. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society 21 (1989), 1–46.
[Bul13] Andrei A., Bulatov The complexity of the counting constraint satisfaction problem. Journal of the ACM 60 (2013), no. 5, 34:1–34:41.
[BW05] Graham R., Brightwell and Peter, Winkler. Counting Eulerian circuits is #Pcomplete, in Proceedings of the 2nd Workshop on Analytic Algorithmics and Combinatorics, 2005.
[CC06] Jin-Yi, Cai and Vinay, Choudhary. Some results on matchgates and holographic algorithms, in Proceedings of the 33rd International Colloquium on Automata, Languages, and Programming, 2006, pp. 703–714.
[CC07] Valiant's Holant theorem and matchgate tensors. Theoretical Computer Science 384 (2007), no. 1, 22–32.
[CC12] Jin-Yi, Cai and Xi, Chen. Complexity of counting CSP with complex weights, in Proceedings of the 44th Annual ACM Symposium on Theory of Computing, 2012, pp. 909–920.
[CCL09] Jin-Yi, Cai, Vinay, Choudhary, and Pinyan, Lu. On the theory of matchgate computations. Theory of Computing Systems 45 (2009), no. 1, 108–132.
[CCL13] Jin-Yi, Cai, Xi, Chen, and Pinyan, Lu. Graph homomorphisms with complex values: A dichotomy theorem. SIAM Journal on Computing 42 (2013), no. 3, 924–1029.
[CCL16] Nonnegative weighted #CSP: An effective complexity dichotomy. SIAM Journal on Computing 45 (2016), no. 6, 2177–2198.
[CF16] Jin-Yi, Cai and Zhiguo, Fu. Holographic algorithm with matchgates is universal for planar #CSP over boolean domain, CoRR abs/1603.07046 (2016).
[CFGW15] Jin-Yi, Cai, Zhiguo, Fu, Heng, Guo, and Tyson, Williams. A Holant dichotomy: Is the FKT algorithm universal?, in Proceedings of the IEEE 56th Annual Symposium on Foundations of Computer Science, 2015, pp. 1259–1276.
[CG14] Jin-Yi, Cai and Aaron, Gorenstein. Matchgates revisited. Theory of Computing 10 (2014), no. 7, 167–197.
[CGG+16] Jin-Yi, Cai, Andreas, Galanis, Leslie Ann, Goldberg, Heng, Guo, Mark, Jerrum, Daniel, Štefankovic, and Eric, Vigoda. #BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region. Journal of Computer and System Sciences 82 (2016), no. 5, 690–711.
[CGW14] Jin-Yi, Cai, Heng, Guo, and Tyson, Williams. Holographic algorithms beyond matchgates. Proceedings of the 41st International Colloquium on Automata, Languages, and Programming, 2014, pp. 271–282.
[CGW16] Jin-Yi, Cai, Heng, Guo, and Tyson, Williams. A complete dichotomy rises from the capture of vanishing signatures. SIAM Journal on Computing 45 (2016), no. 5, 1671–1728.
[CK12] Jin-Yi, Cai and Michael, Kowalczyk. Spin systems on k-regular graphs with complex edge functions. Theoretical Computer Science 461 (2012), 2–16.
[CK13] Partition functions on k-regular graphs with 0, 1-vertex assignments and real edge functions. Theoretical Computer Science 494 (2013), 63–74.
[CKS01] Nadia, Creignou, Sanjeev, Khanna, and Madhu, Sudan. Complexity classifications of boolean constraint satisfaction problems. Society for Industrial and Applied Mathematics, 2001.
[CLX10] Jin-Yi, Cai, Pinyan, Lu, and Mingji, Xia. Holographic algorithms with matchgates capture precisely tractable planar #CSP. Proceedings of the IEEE 51st Annual Symposium on Foundations of Computer Science, 2010, pp. 427–436.
[CLX11] Jin-Yi, Cai, Pinyan, Lu, and Mingji, Xia. A computational proof of complexity of some restricted counting problems. Theoretical Computer Science 412 (2011), no. 23, 2468–2485.
[CNR89] Hyeong-Ah, Choi, Kazuo, Nakajima, and Chong S., Rim Graph bipartization and via minimization. SIAM Journal on Discrete Mathematics 2 (1989), no. 1, 38–47.
[DFJ02] Martin, Dyer, Alan, Frieze, and Mark, Jerrum. On counting independent sets in sparse graphs. SIAM Journal on Computing 31 (2002), no. 5, 1527–1541.
[DG00] Martin E., Dyer and Catherine, Greenhill. The complexity of counting graph homomorphisms. Random Structures and Algorithms 17 (2000), 260–289.
[DGJ09] Martin, Dyer, Leslie Ann, Goldberg, and Mark, Jerrum. The complexity of weighted boolean CSP. SIAM Journal on Computing 38 (2009), no. 5, 1970–1986.
[DGJ10]An approximation trichotomy for boolean #CSP. Journal of Computer and System Sciences 76 (2010), no. 3, 267–277.
[DGJR12] Martin, Dyer, Leslie, AnnGoldberg,Markus Jalsenius, andDavid Richerby. The complexity of approximating bounded-degree boolean #CSP. Information and Computation 220–221 (2012), 1–14.
[DL92] Paul, Dagum and Michael, Luby. Approximating the permanent of graphs with large factors. Theoretical Computer Science 102 (1992), no. 2, 283–305.
[DP91] Christopher T.J., Dodson and Timothy, Poston. Tensor geometry: The geometric viewpoint and its uses, Graduate Texts in Mathematics 130, Springer-Verlag, 1991.
[DR13] Martin E., Dyer and David Richerby. An effective dichotomy for the counting constraint satisfaction problem. SIAM Journal on Computing 42 (2013), no. 3, 1245–1274.
[Eul36] Leonhard, Euler Solutio problematis ad geometriam situs pertinentis. Commentarii Academiae Scientiarum Petropolitanae 8 (1736), 128–140.
[GG16] Andreas, Galanis and Leslie Ann, Goldberg. The complexity of approximately counting in 2-spin systems on k-uniform bounded-degree hypergraphs. Information and Computation 251 (2016), 36–66.
[GGJ16] Andreas, Galanis, Leslie Ann, Goldberg, and Mark, Jerrum. A complexity trichotomy for approximately counting list H-colourings, in Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming, 2016, pp. 46:1–46:13.
[GGJT10] Leslie Ann, Goldberg, Martin, Grohe, Mark, Jerrum, and Marc, Thurley. A complexity dichotomy for partition functions with mixed signs. SIAM Journal on Computing 39 (2010), no. 7, 3336–3402.
[GGv+11] Andreas, Galanis, Qi, Ge, Daniel, Štefankovič, Eric, Vigoda, and Linji, Yang. Improved inapproximability results for counting independent sets in the hard-core model, in Proceedings of the 15th International Workshop on Randomization and Computation, 2011.
[GGY16] Andreas, Galanis, Leslie Ann Goldberg, and Kuan Yang. Approximating partition functions of bounded-degree boolean counting constraint satisfaction problems. CoRR abs/1610.04055 (2016).
[GJP03] Leslie Ann, Goldberg, Mark, Jerrum, and Mike, Paterson. The computational complexity of two-state spin systems. Random Structures and Algorithms 23 (2003), no. 2, 133–154.
[GL16] Heng, Guo and Pinyan, Lu. Uniqueness, spatial mixing, and approximation for ferromagnetic 2-spin systems, in Proceedings of the 20th International Workshop on Randomization and Computation, 2016, pp. 31:1–31:26.
[GM94] G., Galbiati and F., Maffioli. On the computation of Pfaffians. Discrete Applied Mathematics 51 (1994), no. 3, 269–275.
[Gov15] Artem, Govorov, private communication, 2015.
[GŠ12] Qi, Ge and Daniel, Štefankovič. The complexity of counting Eulerian tours in 4-regular graphs. Algorithmica 63 (2012), no. 3, 588–601.
[GT11] Martin, Grohe and Marc, Thurley. Counting homomorphisms and partition functions. Model Theoretic Methods in Finite Combinatorics. Contemporary Mathematics 558 (Martin Grohe and Johann Makowsky, eds.), American Mathematical Society, 2011.
[GW95] Michel X., Goemans and David P., Williamson Improved approximation algorithms formaximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42 (1995), no. 6, 1115–1145.
[GW13] Heng, Guo and Tyson, Williams. The complexity of planar boolean #CSP with complex weights, in Proceedings of the 40th International Colloquium on Automata, Languages, and Programming, 2013, pp. 516–527.
[Had75] F., Hadlock Finding a maximum cut of a planar graph in polynomial time, SIAM Journal on Computing 4 (1975), no. 3, 221–225.
[Hal66] John H., Halton A combinatorial proof of Cayley's theorem on Pfaffians. Journal of Combinatorial Theory 1 (1966), no. 2, 224–232.
[Has01] Johan, Hastad Some optimal inapproximability results. Journal of the ACM 48 (2001), no. 4, 798–859.
[Hie73] Carl, Hierholzer Ueber die moglichkeit, einen linienzug ohne wiederholung und ohne unterbrechung zu umfahren. Mathematische Annalen 6 (1873), no. 1, 30–32.
[HL16] SangxiaHuangPinyan, Lu. Adichotomy for realweighted holant problems. Computational Complexity 25 (2016), no. 1, 255–304.
[HT74] John, Hopcroft and Robert E., Tarjan Efficient planarity testing. Journal of the Association for Computing Machinery 21 (1974), no. 4, 549–568.
[IMRS98] Harry B., Hunt III, Madhav V., Marathe, Venkatesh, Radhakrishnan, and Richard E., Stearns The complexity of planar counting problems. SIAM Journal on Computing 27 (1998), no. 4, 1142–1167.
[Isi25] Ernst Ising. Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik 31 (1925), no. 1, 253–258.
[Jac85a] Nathan, Jacobson Basic algebra , volume 1, W.H. Freeman, 1985.
[Jac85b] Basic algebra , volume 2, W.H. Freeman, 1985.
[Jer87] Mark, Jerrum Two-dimensional monomer-dimer systems are computationally intractable. Journal of Statistical Physics 48 (1987), no. 1, 121–134.
[Jer15] Mark, Jerrum, private communication, 2015.
[JS93] Mark, Jerrum and Alistair, Sinclair. Polynomial-time approximation algorithms for the Ising model. SIAM Journal on Computing 22 (1993), no. 5, 1087–1116.
[JVW90] F., Jaeger, D.L., Vertigan, and D.J.A., Welsh. On the computational complexity of the Jones and Tutte polynomials. Mathematical Proceedings of the Cambridge Philosophical Society 108 (1990), no. 1, 35–53.
[KC10] Michael, Kowalczyk and Jin-Yi, Cai. Holant problems for regular graphs with complex edge functions, in Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science, 2010, pp. 525–536.
[KD79] M. S., Kirshnamoorthy and Narsingh Deo. Node-deletion NP-complete problems. SIAM Journal on Computing 8 (1979), no. 4, 619–625.
[Kir47] G., Kirchhoff Ueber die auflosung der gleichungen, auf welcheman bei der Untersuchung der linearen Verteilung galvanischer strome gefuhrt wird. Annalen der Physik und Chemie 12 (1847), 497–508.
[Kow10] Michael, Kowalczyk Dichotomy theorems for Holant problems, Ph.D. thesis, University of Wisconsin–Madison, 2010.
[Lic82] David, Lichtenstein Planar formulae and their uses. SIAM Journal on Computing 11 (1982), no. 2, 329–343.
[Lie67a] Elliott H., Lieb Exact solution of the F model of an antiferroelectric. Physical Review Letters 18 (1967), no. 24, 1046–1048.
[Lie67b] Exact solution of the problem of the entropy of two-dimensional ice. Physical Review Letters 18 (1967), no. 17, 692–694.
[Lie67c] Exact solution of the two-dimensional Slater KDP model of a ferroelectric. Physical Review Letters 19 (1967), no. 3, 108–110.
[Lie67d] Residual entropy of square ice. Physical Review 162 (1967), no. 1, 162–172.
[Lin86] Nathan, Linial Hard enumeration problems in geometry and combinatorics. SIAM Journal on Algebraic and Discrete Methods 7 (1986), no. 2, 331–335.
[LL15] Jingcheng, Liu and Pinyan, Lu. FPTAS for counting monotone CNF, in Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, 2015, pp. 1531–1548.
[LLY12] Liang, Li, Pinyan, Lu, and Yitong, Yin. Approximate counting via correlation decay in spin systems, in Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms, 2012.
[LLZ14] Jingcheng, Liu, Pinyan, Lu, and Chihao Zhang. The complexity of ferromagnetic two-spin systems with external fields, in Proceedings of the 18th International Workshop on Randomization and Computation, 2014, pp. 843–856.
[LOT03] Maciej, Li'skiewicz, MitsunoriOgihara, , and Seinosuke, Toda. The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes. Theoretical Computer Science 304 (2003), no. 1, 129–156.
[LW16] Jiabao, Lin and Hanpin, Wang. The complexity of Holant problems over boolean domain with non-negative weights. CoRR abs/1611.00975 (2016).
[LY80] John M., Lewis and Mihalis, Yannakakis. The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences 20 (1980), no. 2, 219–230.
[MSV04] Meena, Mahajan, P.R., Subramanya, and V., Vinay. The combinatorial approach yields an NC algorithm for computing Pfaffians. Discrete Applied Mathematics 143 (2004), no. 1–3, 1–16.
[Mur00] Kazuo, Murota Matrices and matroids for systems analysis. Springer, Berlin, 2000.
[MW96] Milena, Mihail and Peter, Winkler. On the number of eulerian orientations of a graph. Algorithmica 16 (1996), no. 4–5, 402–414.
[MWW09] Elchanan, Mossel, DrorWeitz, , and NicholasWormald, . On the hardness of sampling independent sets beyond the tree threshold. Probability Theory and Related Fields 143 (2009), no. 3, 401–439.
[Nes96a] Yu V., Nesterenko Modular functions and transcendence problems. Comptes Rendus de l'Académie des Sciences –Series I –Mathematics 322 (1996), 909–914.
[Nes96b]Modular functions and transcendence questions. Matematicheskii Sbornik 187 (1996), no. 9, 65–96.
[Oht92] Yasuhiro, Ohta Bilinear theory of soliton, Ph.D. thesis, University of Tokyo, 1992.
[Pap94] Christos H., Papadimitriou Computational complexity, Addison-Wesley, 1994.
[Pau35] Linus, Pauling The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. Journal of the American Chemical Society 57 (1935), no. 12, 2680–2684.
[PB83] J., Scott Provan and Michael O., Ball The complexity of counting cuts and of computing the probability that a graph is connected. SIAM Journal on Computing 12 (1983), no. 4, 777–788.
[Pot52] Renfrey B., Potts Some generalized order-disorder transformations. Mathematical Proceedings of the Cambridge Philosophical Society 48 (1952), no. 1, 106–109.
[Rom12] Antonio, Romano Classical mechanics with mathematica. Birkhauser, 2012.
[Sip96] Michael, Sipser Introduction to the theory of computation. PWS Publishing Company, 1996.
[Sly10] Allan, Sly Computational transition at the uniqueness threshold, in Proceedings of the 51st IEEE Annual Symposium on Foundations of Computer Science, 2010, pp. 287–296.
[SS12] Allan, . Sly and Nike Sun. The computational hardness of counting in twospin models on d-regular graphs, in Proceedings of the IEEE 53rd Annual Symposium on Foundations of Computer Science, 2012, pp. 361–369.
[SST12] Alistair, Sinclair, Piyush, Srivastava, and Marc Thurley. Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs, in Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms, 2012.
[Tar51] Alfred, Tarski A decision method for elementary algebra and geometry. Berkeley and Los Angeles, University of California Press, 1951.
[Thu09] Marc, Thurley The complexity of partition functions, Ph.D. thesis, Humboldt-Universitat zu Berlin, 2009.
[Tod91] S., Toda PP is as hard as the polynomial hierarchy. SIAM Journal on Computing 20 (1991), no. 5, 865–877.
[TS41] W. T., Tutte and C.A.B., Smith. On unicursal paths in a network of degree 4. American Mathematical Monthly 48 (1941), 233–237.
[Tut48] W. T., Tutte The disection of equilateral triangles into equilateral triangles. Mathematical Proceedings of the Cambridge Philosophical Society 44 (1948), 463–482.
[Vad01] Salil P., Vadhan The complexity of counting in sparse, regular and planar graphs. SIAM Journal on Computing 8 (2001), no. 1, 398–427.
[vAEdB51] T., van Aardenne-Ehrenfest and N.G., de Bruijn. Circuits and trees in oriented linear graphs. Simon Stevin 28 (1951), 203–217.
[Val79a] Leslie G., Valiant The complexity of computing the permanent. Theoretical Computer Science 8 (1979), no. 2, 189–201.
[Val79b] The complexity of enumeration and reliability problems. SIAM Journal on Computing 8 (1979), no. 3, 410–421.
[Val02a] Expressiveness of matchgates. Theoretical Computer Science 289 (2002), no. 1, 457–471.
[Val02b] Quantum circuits that can be simulated classically in polynomial time. SIAM Journal on Computing 31 (2002), no. 4, 1229–1254.
[Val05] Leslie G., Valiant Completeness for parity problems, in Proceedings of the 11th International Computing and Combinatorics Conference, 2005, pp. 1–8.
[Val06] Leslie G., Valiant Accidental algorithms, in Proceedings of the IEEE 54th Annual Symposium on Foundations of Computer Science, 2006, pp. 509–517.
[Val08] Holographic algorithms. SIAM Journal on Computing 37 (2008), no. 5, 1565–1594.
[Val10] Leslie G., Valiant Some observations on holographic algorithms, in Proceedings of the 9th Latin American Theoretical Informatics Symposium, 2010, pp. 577–590.
[Ver88] Michel Las, Vergnas. On the evaluation at (3, 3) of the Tutte polynomial of a graph. Journal of Combinatorial Theory, Series B 45 (1988), no. 3, 367–372.
[Ver05] Dirk, Vertigan The computational complexity of Tutte invariants for planar graphs. SIAM Journal on Computing 35 (2005), no. 3, 690–712.
[VV86] Leslie G., Valiant and Vijay V., Vazirani NP is as easy as detecting unique solutions. Theoretical Computer Science 47 (1986), 85–93.
[Wei06] Dror, Weitz Counting independent sets up to the tree threshold, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 140–149.
[Wel93] Dominic, Welsh Complexity: Knots, colourings and countings . London Mathematical Society Lecture Note Series. Cambridge University Press, 1993.
[WR70] B., Widom and J.S. Rowlinson. New model for the study of liquid-vapor phase transitions. Journal of Chemical Physics 52 (1970), no. 4.
[XZZ07] Mingji, Xia, Peng, Zhang, and Wenbo Zhao. Computational complexity of counting problems on 3-regular planar graphs. Theoretical Computer Science 384 (2007), no. 1, 111–125.
[Yan81a] Mihalis, Yannakakis Edge-deletion problems. SIAM Journal on Computing 10 (1981), no. 2, 297–309.
[Yan81b] Node-deletion problems on bipartite graphs. SIAM Journal on Computing 10 (1981), no. 2, 310–327.
[Zyk74] A. A., Zykov Hypergraphs. Uspekhi Matematicheskikh Nauk 6 (1974), no. 180, 89–154.

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