STUDIES IN INTEGER PROGRAMMING Managing Editor Peter L HAMMER, University of Waterloo, Ont., Canada Advisory Editors C BERGE, UniversitC de Paris, France M.A HARRISON, University of California, Berkeley, CA, U.S.A V KLEE, University of Washington, Seattle, WA, U.S.A J.H VAN LINT, California Institute of Technology, Pasadena, CA, U.S.A G.-C ROTA, Massachusetts Institute of Technology, Cambridge, MA, U.S.A Based on material presented at theworkshop on Integer Programming, Bonn, 8-12 September 1975, organised by the Institute of Operations Research (Sonderforschungsbereich21), University of Bonn Sponsored by IBM Germany NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK* OXFORD ANNALS OF DISCRETE MATHEMATICS STUDIES IN INTEGER PROGRAMMING Edited by P.L HAMMER, University of Waterloo, Ont., Canada E.L JOHNSON, 1BM Research, Yorktown Heights, NY, U.S.A B.H KORTE, University of Bonn, Federal Republic of Germany G.L NEMHAUSER, Cornell University, Ithaca, NY, U.S.A 1977 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD I @ NORTH-HOLLAND PUBLISHING COMPANY - 1977 All rights reserved N o part of this publication niay he reproduced, stored in a retrieval systen? or transmitted, in any f o r m or by any means, electronic, mechanical, photocop.ving, recording or otherwise, without the prior permission of the copyright owner Reprinted from the journal 4nnals of Discrete Mathematics Volume I North-Holland ISBN for this Volume: 7204 0765 Published by: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OX1:ORD Sole distributors for the U.S.A a n d Canada: k.LSEVIER NORTH-HOLLAND, INC VANDERBILT AVENUE NEW YORK, NY 0 Printed in T h e Netherlands PREFACE This volume constitutes the proceedings of the Workshop on Integer Programming that was held in Bonn, September 8-12, 1975 The Workshop was organized by the Institute of Operations Research (Sonderforschungsbereich 21), University of Bonn and was generously sponsored by IBM Germany In all, 71 participants frnm 13 different countries took part in the Workshop Integer programming is one of the most fascinating and difficult areas of mathematical optimization There are a great many real-world problems of large dimension that urgently need to be solved, but there is a large gap between the practical requirements and the theoretical development Since combinatorial problems in general are among the most difficult in mathematics, a great deal of theoretical research is necessary before substantial advances in the practical solution of problems can be expected Nevertheless the rapid progress of research in this field has produced mathematical results significant in their own right and has also borne substantial fruit for practical applications We believe that this will be adequately demonstrated by the papers in this volume The 37 papers appearing in this volume cover a wide spectrum of topics in integer programming The volume includes works on the theoretical foundations of integer programming, on algorithmic aspects of discrete optimization, on specific types of integer programming problems, as well as on some related questions on polytopes and on graphs and networks All the papers have been carefully referred We express our sincere thanks to all authors for their cooperation, to the referees for their useful support, to numerous participants for stimulating discussions, and to the editors of the Annals of Discrete Mathematics for their willingness to include this volume in their new series The Program Committee Bonn, 1976 P Schweitzer IBM Germany P.L Hammer E.L Johnson B.H Korte G.L Nemhauser V CONTENTS Preface Con tents V vi A BACHEM, Reduction and decomposition of integer programs over cones E BALAS,Some valid inequalities for t h e set partitioning problem M BALLand R.M V A N SLYKE, Backtracking algorithms for network reliability analysis Coloring the edges of a hypergraph and linear C BERGEand E.L JOHNSON, programming techniques BILDEand J KRARUP, Sharp lower bounds and efficient algorithms for the simple plant location problem V.J BOWMAN, JR and J.H STARR, Partial orderings in implicit enumeration A subadditive approach to solve linear C.-A BURDETand E.L JOHNSON, integer programs V CHVATALand P,L HAMMER,Aggregation of inequalities in integer programming On the uncapacitated G CORNUEJOLS, M FISHERand G.L NEMHAUSER, location problem D DE WERRA,Some coloring techniques and R GILES,A min-max relation for submodular functions on J EDMONDS graphs A.M GEOFFRION, How can specialized discrete and convex optimization methods be married D GRANOTand F GRANOT,On integer and mixed integer fractional programming problems M GROTSCHEL, Graphs with cycles containing given paths Algorithms for exploiting the structure of M GUIGNARD and K SPIELBERG, the simple plant location problem M GUIGNARD and K SPIELBERG, Reduction methods for state enumeration integer programming P HANSEN, Subdegrees and chromatic numbers of hypergraphs R.G JEROSLOW, Cutting-plane theory: disjunctive methods E.L LAWLER, A ‘pseudopolynomial’ algorithm for sequencing jobs to minimize total tardiness J.K LENSTRA, A.H.G RINNOOY KANand P BRUCKER, Complexity of machine scheduling problems L LOVASZ,Certain duality principles in integer programming R.E MARSTEN and T.L MORIN,Parametric integer programming: the righthand-side case vi 13 49 65 79 99 117 145 163 179 185 205 22 233 247 273 287 293 331 343 363 375 Contents J.F MAURRAS, An example of dual polytopes in the unit hypercube P MEVERT and U SUHL,Implicit enumeration with generalized upper bounds I MICHAELI and M.A POLLATSCHEK, On some nonlinear knapsack problems J ORLIN,The minimal integral separator of a threshold graph M.W PADBERG, On the complexity of set packing polyhedra U.N PELED,Properties of facets of binary polytopes D.S RUBIN,Vertex generation methods for problems with logical constraints J.F SHAPIRO, Sensitivity analysis in integer programming T.H.C SMITHand G.L THOMPSON, A lifo implicit enumeration search algorithm for the symmetric traveling salesman problem using Held and Karp’s 1-tree relaxation T.H.C SMITH,V SRINIVASAN and G.L THOMPSON, Computational performance of three subtour elimination algorithms for solving asymmetric traveling salesman problems J TIND,On antiblocking sets and polyhedra L.E TROTTER, On the generality of multi-terminal flow theory L.A WOLSEY,Valid inequalities, covering problems and discrete dynamic programs U ZIMMERMAN, Some partial orders related to boolean optimization and the Greedy-algorithm S ZIONTS,Integer linear programming with multiple objectives uii 391 393 403 415 42 435 457 467 479 495 507 517 527 539 55 This Page Intentionally Left Blank Annals of Discrete Mathematics (1977) 1-11 @ North-Holland Publishing Company REDUCTION AND DECOMPOSITION OF INTEGER PROGRAMS OVER CONES Achim BACHEM Institut fur Okonometrie und Operations Research, Universitat Bonn, Nassestrape 2, 0-53 Bonn, F.R.G Received: August 1975 Revised: November 1975 We consider the problem c'x (t) s.t Nx + B y = b, X E N , yEZ" where N is an ( m , r ) , B an ( m , n ) integer matrix, and b E 2" In Section we characterize all solutions x E 2' of (t) by an explicit formula and give as a corollary a minimal group representation of equality restricted integer programs, where some of the nonnegativity restrictions are relaxed In Section we discuss decomposing integer programs over cones in case the matrix N has special structure Introduction We consider the problem c'x s.t Nx + By = b x E N', y E Z" where N is an ( m , r ) and B an (m,n ) integer matrix As B is an arbitrary (m,n ) integer matrix, the convex hull of the feasible set of (1.1) is a generalized corner polyhedron, that is an equality restricted integer program, where the nonnegativity restriction of some of the variables are relaxed To give a group representation of the problem, we reformulate (1.1) as a congruence problem, c ' x s.t Nx = b modB x E N' ... papers in this volume The 37 papers appearing in this volume cover a wide spectrum of topics in integer programming The volume includes works on the theoretical foundations of integer programming,... sequencing jobs to minimize total tardiness J.K LENSTRA, A.H.G RINNOOY KANand P BRUCKER, Complexity of machine scheduling problems L LOVASZ,Certain duality principles in integer programming R.E MARSTEN... orderings in implicit enumeration A subadditive approach to solve linear C.-A BURDETand E.L JOHNSON, integer programs V CHVATALand P,L HAMMER,Aggregation of inequalities in integer programming