The Mcf-separator: detecting and exploiting multi-commodity flow structures in MIPs

Given a general mixed integer program, we automatically detect block structures in the constraint matrix together with the coupling by capacity constraints arising from multi-commodity flow formulations. We identify the underlying graph and generate cutting planes based on cuts in the detected network. Our implementation adds a separator to the branch-and-cut libraries of Scip and Cplex. We make use of the complemented mixed integer rounding framework but provide a special purpose aggregation heuristic that exploits the network structure. Our separation scheme speeds-up the computation for a large set of mixed integer programs coming from network design problems by a factor two on average. We show that almost 10% of the instances in general testsets contain consistent embedded networks. For these instances the computation time is decreased by 18% on average.

[1]  Geir Dahl,et al.  A Cutting Plane Algorithm for Multicommodity Survivable Network Design Problems , 1998, INFORMS J. Comput..

[2]  Thomas L. Magnanti,et al.  Shortest paths, single origin-destination network design, and associated polyhedra , 1993, Networks.

[3]  Alper Atamtürk,et al.  On the facets of the mixed–integer knapsack polyhedron , 2003, Math. Program..

[4]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[5]  G. Nemhauser,et al.  Integer Programming , 2020 .

[6]  Arie M. C. A. Koster,et al.  On the strength of cut-based inequalities for capacitated network design polyhedra , 2007 .

[7]  Laurence A. Wolsey,et al.  Aggregation and Mixed Integer Rounding to Solve MIPs , 2001, Oper. Res..

[8]  Gerald G. Brown,et al.  Automatic identification of embedded network rows in large-scale optimization models , 1984, Math. Program..

[9]  Laurence A. Wolsey,et al.  Integer Knapsack and Flow Covers with Divisible Coefficients: Polyhedra, optimization and Separation , 1995, Discret. Appl. Math..

[10]  Laurence A. Wolsey,et al.  Faces for a linear inequality in 0–1 variables , 1975, Math. Program..

[11]  Thorsten Koch,et al.  Konrad-zuse-zentrum F ¨ Ur Informationstechnik Berlin Miplib 2003 , 2022 .

[12]  Laurence A. Wolsey,et al.  Valid inequalities for mixed 0-1 programs , 1986, Discret. Appl. Math..

[13]  Ted Ralphs,et al.  COmputational INfrastructure for Operations Research (COIN-OR) , 2004 .

[14]  Oktay Günlük,et al.  Capacitated Network Design - Polyhedral Structure and Computation , 1996, INFORMS J. Comput..

[15]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.

[16]  Thomas L. Magnanti,et al.  Network Design and Transportation Planning: Models and Algorithms , 1984, Transp. Sci..

[17]  Tobias Achterberg,et al.  Constraint integer programming , 2007 .

[18]  Michael R. Bussieck,et al.  Discrete optimization in public rail transport , 1997, Math. Program..

[19]  Oktay Günlük,et al.  A branch-and-cut algorithm for capacitated network design problems , 1999, Math. Program..

[20]  Martin W. P. Savelsbergh,et al.  Bidirected and unidirected capacity installation in telecommunication networks , 2003, Discret. Appl. Math..

[21]  Thomas L. Magnanti,et al.  Modeling and Solving the Two-Facility Capacitated Network Loading Problem , 1995, Oper. Res..

[22]  A. Koster,et al.  Capacitated network design using general flow-cutset inequalities , 2007 .

[23]  Martin W. P. Savelsbergh,et al.  Polyhedral results for the edge capacity polytope , 2002, Math. Program..

[24]  Laurence A. Wolsey,et al.  Valid Linear Inequalities for Fixed Charge Problems , 1985, Oper. Res..

[25]  Itzhak Gilboa,et al.  Source Sink Flows with Capacity Installation in Batches , 1998, Discret. Appl. Math..

[26]  Tobias Achterberg,et al.  SCIP: solving constraint integer programs , 2009, Math. Program. Comput..

[27]  Egon Balas,et al.  Facets of the knapsack polytope , 1975, Math. Program..

[28]  Alper Atamtürk,et al.  Sequence Independent Lifting for Mixed-Integer Programming , 2004, Oper. Res..

[29]  Alper Atamtürk,et al.  On capacitated network design cut–set polyhedra , 2002, Math. Program..

[30]  Arie M. C. A. Koster,et al.  On cut‐based inequalities for capacitated network design polyhedra , 2011, Networks.

[31]  Martin W. P. Savelsbergh,et al.  An Updated Mixed Integer Programming Library: MIPLIB 3.0 , 1998 .

[32]  R. Fourer,et al.  Finding Embedded Network Rows in Linear Programs I. Extraction Heuristics , 1988 .

[33]  Thomas L. Magnanti,et al.  The convex hull of two core capacitated network design problems , 1993, Math. Program..

[34]  Laurence A. Wolsey,et al.  Valid Inequalities and Superadditivity for 0-1 Integer Programs , 1977, Math. Oper. Res..

[35]  Martin W. P. Savelsbergh,et al.  Lifted flow cover inequalities for mixed 0-1 integer programs , 1999, Math. Program..

[36]  Panos M. Pardalos,et al.  Handbook of Optimization in Telecommunications , 2006 .

[37]  Alper Atamtürk,et al.  Flow pack facets of the single node fixed-charge flow polytope , 2001, Oper. Res. Lett..

[38]  Alper Atamtürk.  Cover and Pack Inequalities for (Mixed) Integer Programming , 2005, Ann. Oper. Res..

[39]  M. Stoer,et al.  A polyhedral approach to multicommodity survivable network design , 1994 .

[40]  Alper Atamtürk,et al.  On splittable and unsplittable flow capacitated network design arc–set polyhedra , 2002, Math. Program..

[41]  Laurence A. Wolsey,et al.  A branch-and-cut algorithm for the single-commodity, uncapacitated, fixed-charge network flow problem , 2003, Networks.

[42]  L. Wolsey,et al.  Designing Private Line Networks - Polyhedral Analysis and Computation , 1996 .

[43]  Deep Medhi,et al.  Routing, flow, and capacity design in communication and computer networks , 2004 .

[44]  Martin W. P. Savelsbergh,et al.  Valid inequalities for problems with additive variable upper bounds , 1999, Math. Program..

[45]  Laurence A. Wolsey,et al.  The 0-1 Knapsack problem with a single continuous variable , 1999, Math. Program..

[46]  Michal Pióro,et al.  SNDlib 1.0—Survivable Network Design Library , 2010, Networks.

[47]  Oktay Günlük,et al.  Minimum cost capacity installation for multicommodity network flows , 1998, Math. Program..

[48]  Laurence A. Wolsey,et al.  Lifting, superadditivity, mixed integer rounding and single node flow sets revisited , 2003, 4OR.

[49]  Robert Weismantel,et al.  On the 0/1 knapsack polytope , 1997, Math. Program..

[50]  R. Wessäly.  Dimensioning Survivable Capacitated Networks , 2000 .

[51]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[52]  Robert E. Bixby,et al.  Progress in computational mixed integer programming—A look back from the other side of the tipping point , 2007, Ann. Oper. Res..