Integrality, complexity and colourings in polyhedral combinatorics

Doctoral School: Mathematics Director: Miklos Laczkovich, Member of the Hungarian Academy of Sciences Doctoral Program: Applied Mathematics Director: Gyorgy Michaletzky, Doctor of Sciences Advisor: Andras Frank, Doctor of Sciences, Professor Co-advisor: Tamas Kiraly, Ph.D. Affiliation: Department of Operations Research, Eotvos Lorand University and Egervary Research Group on Combinatorial Optimization, Budapest, Hungary

[1]  A. Schrijver Polyhedral combinatorics , 1996 .

[2]  Paul D. Seymour,et al.  The matroids with the max-flow min-cut property , 1977, J. Comb. Theory B.

[3]  T. Fleiner Stable and crossing structures , 2000 .

[4]  A. Schrijver,et al.  Proving total dual integrality with cross-free families—A general framework , 1984, Math. Program..

[5]  László Lovász,et al.  Normal hypergraphs and the perfect graph conjecture , 1972, Discret. Math..

[6]  Vladimir Gurvich,et al.  Perfect graphs, kernels, and cores of cooperative games , 2006, Discret. Math..

[7]  Alexander Schrijver,et al.  On total dual integrality , 1981 .

[8]  L. S. Shapley,et al.  College Admissions and the Stability of Marriage , 2013, Am. Math. Mon..

[9]  Tamás Király,et al.  Kernels, Stable Matchings, and Scarf's Lemma , 2010 .

[10]  Li Feng,et al.  The complexity of recognizing linear systems with certain integrality properties , 2008, Math. Program..

[11]  H. Scarf The Core of an N Person Game , 1967 .

[12]  Jack Edmonds,et al.  Total Dual Integrality of Linear Inequality Systems , 1984 .

[13]  Grigor Gasparyan,et al.  Bipartite Designs , 1998, IPCO.

[14]  Christos H. Papadimitriou,et al.  On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence , 1994, J. Comput. Syst. Sci..

[15]  Tamás Fleiner,et al.  Fractional Solutions for NTU-Games , 2010 .

[16]  Shiva Kintali Scarf is Ppad-Complete , 2008, ArXiv.

[17]  András Frank,et al.  Generalized polymatroids and submodular flows , 1988, Math. Program..

[18]  András Sebö,et al.  Hilbert Bases, Caratheodory's Theorem and Combinatorial Optimization , 1990, IPCO.

[19]  Claude Berge,et al.  Recent problems and results about kernels in directed graphs , 1991, Discret. Math..

[20]  J. Edmonds,et al.  A Min-Max Relation for Submodular Functions on Graphs , 1977 .

[21]  László Lovász,et al.  Finite and infinite sets , 1984 .

[22]  D. R. Fulkerson,et al.  Anti-blocking polyhedra , 1972 .

[23]  Mihalis Yannakakis,et al.  On recognizing integer polyhedra , 1990, Comb..

[24]  Jimmy J. M. Tan A Necessary and Sufficient Condition for the Existence of a Complete Stable Matching , 1991, J. Algorithms.

[25]  András Frank,et al.  Characterizing and recognizing generalized polymatroids , 2013, Mathematical Programming.

[26]  V. Gurvich,et al.  Stable families of coalitions and normal hypergraphs , 1997 .

[27]  L. Lovász Normal Hypergraphs and the Weak Perfect Graph Conjecture , 1984 .

[28]  Michael Joswig,et al.  polymake: a Framework for Analyzing Convex Polytopes , 2000 .

[29]  G. Cornuéjols,et al.  Combinatorial optimization : packing and covering , 2001 .

[30]  Michel Balinski,et al.  Integer Programming: Methods, Uses, Computations , 1965 .

[31]  W. Pulleyblank,et al.  Total Dual Integrality and Integer Polyhedra , 1979 .

[32]  Robert G. Jeroslow,et al.  Some Basis Theorems for Integral Monoids , 1978, Math. Oper. Res..

[33]  A. Schrijver Total Dual Integrality from Directed Graphs, Crossing Families, and Sub- and Supermodular Functions , 1984 .

[34]  Bert Gerards,et al.  Matrices with the edmonds—Johnson property , 1986, Comb..

[35]  Michel Balinski,et al.  Integer Programming: Methods, Uses, Computation , 2010, 50 Years of Integer Programming.

[36]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[37]  Jack Edmonds,et al.  Matroids and the greedy algorithm , 1971, Math. Program..

[38]  Kazuo Murota,et al.  Convexity and Steinitz's Exchange Property , 1996, IPCO.

[39]  C. Rioux Scarf's Theorem and Applications in Combinatorics , 2006 .

[40]  George L. Nemhauser,et al.  Polyhedral Characterizations, Perfection of Line Graphs , 1998, Discret. Appl. Math..

[41]  Myriam Preissmann,et al.  Imperfect and Nonideal Clutters: A Common Approach , 2003, Comb..

[42]  Vladimir Gurvich,et al.  Perfect graphs are kernel solvable , 1996, Discret. Math..

[43]  David Pritchard The Hypergraphic Tutte/Nash-Williams Theorem via Integer Decomposition, Total Dual Laminarity, and Power Matroids , 2010 .

[44]  E. Sperner Neuer beweis für die invarianz der dimensionszahl und des gebietes , 1928 .

[45]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[46]  藤重 悟 Submodular functions and optimization , 1991 .

[47]  Ron Holzman,et al.  Fractional Kernels in Digraphs , 1998, J. Comb. Theory, Ser. B.

[48]  Júlia Pap,et al.  Recognizing conic TDI systems is hard , 2011, Math. Program..

[49]  Robert W. Irving An Efficient Algorithm for the "Stable Roommates" Problem , 1985, J. Algorithms.

[50]  S. Thomas McCormick,et al.  Integer Programming and Combinatorial Optimization , 1996, Lecture Notes in Computer Science.

[51]  David Lindley,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 2010 .

[52]  D. Hilbert Über die Theorie der algebraischen Formen , 1890 .

[53]  András Frank,et al.  A Survey on Covering Supermodular Functions , 2009, Bonn Workshop of Combinatorial Optimization.

[54]  L. Lovász,et al.  A POLYNOMIAL-TIME TEST FOR TOTAL DUAL INTEGRALITY IN FIXED DIMENSION , 1984 .

[55]  Ali Ridha Mahjoub,et al.  Compositions of Graphs and Polyhedra III: Graphs with No W4 Minor , 1994, SIAM J. Discret. Math..

[57]  Attila Bernáth,et al.  Covering skew-supermodular functions by hypergraphs of minimum total size , 2009, Oper. Res. Lett..

[58]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[59]  M. Ziegler Volume 152 of Graduate Texts in Mathematics , 1995 .

[60]  Robert Weismantel,et al.  On Hilbert bases of polyhedral cones , 1996 .

[61]  András Sebö,et al.  Characterizing Noninteger Polyhedra with 0-1 Constraints , 1998, IPCO.

[62]  D. Hilbert Ueber die Theorie der algebraischen Formen , 1890 .

[63]  Xiaotie Deng,et al.  On the complexity of 2D discrete fixed point problem , 2009, Theor. Comput. Sci..

[64]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[65]  J. P. Uhry,et al.  A class of h-perfect graphs , 1984, Discret. Math..

[66]  W. R. Pulleyblank,et al.  Polyhedral Combinatorics , 1989, ISMP.

[67]  András Frank,et al.  Augmenting graphs to meet edge-connectivity requirements , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[68]  Alfred Lehman,et al.  The Width-Length Inequality and Degenerate Projective Planes , 1990, Polyhedral Combinatorics.

[69]  Satoru Fujishige A note on Frank's generalized polymatroids , 1984, Discret. Appl. Math..

[70]  A. Frank Connections in Combinatorial Optimization , 2011 .

[71]  F. Bruce Shepherd,et al.  Applying Lehman's theorems to packing problems , 1995, Math. Program..

[72]  Tamás Fleiner On Stable Matchings and Flows , 2014, Algorithms.

[73]  Tamás Király,et al.  PPAD-completeness of polyhedral versions of Sperner's Lemma , 2013, Discret. Math..

[74]  Tamás Király,et al.  A note on kernels and Sperner's Lemma , 2009, Discret. Appl. Math..

[75]  Tamás Fleiner,et al.  On a lemma of Scarf , 2003, J. Comb. Theory, Ser. B.