Other Applications in General Integer Programming

In this chapter, we briefly review an alternative application of dual-feasible functions in general integer programming. We explore these functions in particular to derive valid inequalities for integer programs. Since the notion of superadditivity is essential for this purpose, we start by reviewing superadditivity in the scope of valid inequalities. Different examples are provided with alternative families of dualfeasible functions. We discuss also the difference between the valid inequalities derived by dual-feasible functions and the well-known Chvatal-Gomory cuts.

[1]  Sándor P. Fekete,et al.  A General Framework for Bounds for Higher-Dimensional Orthogonal Packing Problems , 2004, Math. Methods Oper. Res..

[2]  George S. Lueker,et al.  Bin packing with items uniformly distributed over intervals [a,b] , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[3]  Marco A. Boschetti,et al.  The two-dimensional finite bin packing problem. Part I: New lower bounds for the oriented case , 2003, 4OR.

[4]  Andrea Lodi,et al.  Strengthening Chvátal-Gomory cuts and Gomory fractional cuts , 2002, Oper. Res. Lett..

[5]  Jacques Carlier,et al.  Computing redundant resources for the resource constrained project scheduling problem , 2007, Eur. J. Oper. Res..

[6]  Cláudio Alves,et al.  Theoretical investigations on maximal dual feasible functions , 2010, Oper. Res. Lett..

[7]  Frits C. R. Spieksma,et al.  A branch-and-bound algorithm for the two-dimensional vector packing problem , 1994, Comput. Oper. Res..

[8]  Ralph E. Gomory,et al.  An algorithm for integer solutions to linear programs , 1958 .

[9]  George B. Dantzig,et al.  Decomposition Principle for Linear Programs , 1960 .

[10]  Pamela H. Vance,et al.  Branch-and-Price Algorithms for the One-Dimensional Cutting Stock Problem , 1998, Comput. Optim. Appl..

[11]  François Clautiaux.  New collaborative approaches for bin-packing problems , 2010 .

[12]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[13]  Cláudio Alves,et al.  On the Properties of General Dual-Feasible Functions , 2014, ICCSA.

[14]  José M. Valério de Carvalho,et al.  Exact solution of bin-packing problems using column generation and branch-and-bound , 1999, Ann. Oper. Res..

[15]  Antoine Jouglet,et al.  A new lower bound for the non-oriented two-dimensional bin-packing problem , 2007, Oper. Res. Lett..

[16]  Allan Borodin,et al.  On the Number of Additions to Compute Specific Polynomials , 1976, SIAM J. Comput..

[17]  Cláudio Alves,et al.  Multidimensional dual-feasible functions and fast lower bounds for the vector packing problem , 2014, Eur. J. Oper. Res..

[18]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[19]  R. Gomory,et al.  A Linear Programming Approach to the Cutting-Stock Problem , 1961 .

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

[21]  Hanif D. Sherali,et al.  Linear Programming and Network Flows , 1977 .

[22]  Cláudio Alves,et al.  A survey of dual-feasible and superadditive functions , 2010, Ann. Oper. Res..

[23]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[24]  El-Ghazali Talbi,et al.  New lower bounds for bin packing problems with conflicts , 2010, Eur. J. Oper. Res..

[25]  Cláudio Alves,et al.  Constructing general dual-feasible functions , 2015, Oper. Res. Lett..

[26]  Alberto Caprara,et al.  Bidimensional Packing by Bilinear Programming , 2005, IPCO.

[27]  Cláudio Alves,et al.  Cutting and packing : problems, models and exact algorithms , 2005 .

[28]  Cláudio Alves,et al.  Computing Valid Inequalities for General Integer Programs using an Extension of Maximal Dual Feasible Functions to Negative Arguments , 2012, ICORES.

[29]  Paolo Toth,et al.  Knapsack Problems: Algorithms and Computer Implementations , 1990 .

[30]  François Vanderbeck,et al.  Exact Algorithm for Minimising the Number of Setups in the One-Dimensional Cutting Stock Problem , 2000, Oper. Res..

[31]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[32]  J. M. Valério de Carvalho,et al.  On the extremality of maximal dual feasible functions , 2012, Oper. Res. Lett..

[33]  Jacques Carlier,et al.  New reduction procedures and lower bounds for the two-dimensional bin packing problem with fixed orientation , 2007, Comput. Oper. Res..

[34]  Sándor P. Fekete,et al.  New classes of fast lower bounds for bin packing problems , 2001, Math. Program..

[35]  José M. Valério de Carvalho.  A Note on Branch-and-Price Algorithms for the One-Dimensional Cutting Stock Problems , 2002, Comput. Optim. Appl..

[36]  Volker Kaibel,et al.  Integer Programming and Combinatorial Optimization, 11th International IPCO Conference, Berlin, Germany, June 8-10, 2005, Proceedings , 2005, IPCO.

[37]  Sanjeeb Dash,et al.  Valid inequalities based on simple mixed-integer sets , 2006, Math. Program..

[38]  David S. Johnson,et al.  Near-optimal bin packing algorithms , 1973 .

[39]  Claude-Alain Burdet,et al.  A Subadditive Approach to Solve Linear Integer Programs , 1977 .