Cornuejols G. Combinatorial Optimization. Packing and Covering 2001

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The integer programming models known as set packing and set covering have a wide range of applications, such as pattern recognition, plant location, and airline crew scheduling. Sometimes, because of the special structure of the constraint matrix, the natural linear programming relaxation yields an optimal solution that is integral, thus solving the problem. Sometimes, both the linear programming relaxation and its dual have integral optimal solutions. Under which conditions do such integrality properties hold? This question is of both theoretical and practical interest. Min-max theorems, polyhedral combinatorics, and graph theory all come together in this rich area of discrete mathematics. In addition to min-max and polyhedral results, some of the deepest results in this area come in two flavors: "excluded minor" results and "decomposition" results. In this book, we present several of these beautiful results. Three chapters cover min-max and polyhedral results. The next four chapters cover excluded minor results. In the last three chapters, we present decomposition results. We hope that this book will encourage research on the many intriguing open questions that still remain. In particular, we state 18 conjectures. For each of these conjectures, we offer $5000 for the first paper solving or refuting it. The paper must be accepted by a quality refereed journal (such as Journal of Combinatorial Theory B, Combinatorica, SIAM Journal on Discrete Mathematics, or others to be determined by Gerard Cornuejols) before 2020 . Claims must be sent to Gerard Cornuejols at Carnegie Mellon University during his lifetime.
Clutters
T-Cuts and 7-Joins
Perfect Graphs and Matrices
Ideal Matrices
Odd Cycles in Graphs
0,±1 Matrices and Integral Polyhedra
Signing 0,1 Matrices to Be Totally Unimodular or Balanced
Decomposition by A-Sum
Decomposition of Balanced Matrices
Decomposition of Perfect Graphs