Table Of ContentEURO Advanced Tutorials on Operational Research
Series Editors: M. Grazia Speranza · Jose Fernando Oliviera
Giuseppe Lancia
Paolo Serafini
Compact
Extended Linear
Programming
Models
EURO Advanced Tutorials on Operational
Research
Series editors
M. Grazia Speranza, Brescia, Italy
Jose Fernando Oliviera, Porto, Portugal
More information about this series at http://www.springer.com/series/13840
fi
Giuseppe Lancia Paolo Sera ni
(cid:129)
Compact Extended Linear
Programming Models
123
Giuseppe Lancia PaoloSerafini
Department ofMathematics, Computer Department ofMathematics, Computer
Science, andPhysics Science, andPhysics
University of Udine University of Udine
Udine Udine
Italy Italy
ISSN 2364-687X ISSN 2364-6888 (electronic)
EURO AdvancedTutorials onOperational Research
ISBN978-3-319-63975-8 ISBN978-3-319-63976-5 (eBook)
DOI 10.1007/978-3-319-63976-5
LibraryofCongressControlNumber:2017948212
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Contents
1 Introduction... .... .... ..... .... .... .... .... .... ..... .... 1
2 Polyhedra. .... .... .... ..... .... .... .... .... .... ..... .... 7
2.1 Basic Definitions... ..... .... .... .... .... .... ..... .... 7
2.2 Convex Hulls of Infinitely Many Points.. .... .... ..... .... 11
2.3 The Slack Matrix .. ..... .... .... .... .... .... ..... .... 12
2.4 Projections of Polyhedra.. .... .... .... .... .... ..... .... 13
2.5 Union of Polyhedra Defined by Inequalities... .... ..... .... 24
2.6 Union of Polyhedra Defined by Vertices and Extreme Rays.... 28
3 Linear Programming.... ..... .... .... .... .... .... ..... .... 33
3.1 Basic Definitions... ..... .... .... .... .... .... ..... .... 33
3.2 Polyhedral Characterization ... .... .... .... .... ..... .... 35
3.3 Duality .. .... .... ..... .... .... .... .... .... ..... .... 36
3.4 Algorithms ... .... ..... .... .... .... .... .... ..... .... 39
4 Integer Linear Programming.. .... .... .... .... .... ..... .... 43
4.1 Basic Definitions... ..... .... .... .... .... .... ..... .... 43
4.2 Modeling .... .... ..... .... .... .... .... .... ..... .... 45
4.3 Formulations with Integral LP-Relaxation. .... .... ..... .... 48
4.4 The Branch-and-Bound Approach... .... .... .... ..... .... 51
4.5 The Cutting-Plane Approach... .... .... .... .... ..... .... 58
4.6 General-Purpose MILP Solvers. .... .... .... .... ..... .... 65
5 Large-Scale Linear Programming .. .... .... .... .... ..... .... 67
5.1 LP with Exponentially Many Columns... .... .... ..... .... 68
5.2 Column Generation and Branch-and-Bound... .... ..... .... 70
5.3 LP with Exponentially Many Rows . .... .... .... ..... .... 71
5.4 LP with Exponentially Many Columns and Rows .. ..... .... 72
v
vi Contents
6 General Techniques for Compact Formulations... .... ..... .... 75
6.1 Primal Compact Extended Formulations.. .... .... ..... .... 75
6.2 Two Examples of Compact Extended Formulations. ..... .... 78
6.3 Compact Equivalent Formulations .. .... .... .... ..... .... 81
6.4 A First Example of Compact Equivalent Formulation .... .... 84
6.5 A Second Example of Compact Equivalent Formulation .. .... 86
6.6 A Common Feature for Path Problems... .... .... ..... .... 90
6.7 The Dantzig–Wolfe Decomposition Technique. .... ..... .... 92
6.8 Projections and Slack Matrix Factorization.... .... ..... .... 94
6.9 Union of Polyhedra ..... .... .... .... .... .... ..... .... 99
7 The Permutahedron .... ..... .... .... .... .... .... ..... .... 103
7.1 Basic Definitions... ..... .... .... .... .... .... ..... .... 103
7.2 A Compact Extended Formulation by LP Techniques .... .... 106
7.3 A Direct Compact Extended Formulation. .... .... ..... .... 108
7.4 A Minimal Compact Extended Formulation... .... ..... .... 109
8 The Parity Polytope. .... ..... .... .... .... .... .... ..... .... 113
8.1 External Representation of the Parity Polytope. .... ..... .... 114
8.2 A Compact Extended Formulation by Union of Polyhedra. .... 117
8.3 A Compact Extended Formulation by LP Techniques .... .... 118
9 Trees .... .... .... .... ..... .... .... .... .... .... ..... .... 123
9.1 Steiner and Spanning Trees ... .... .... .... .... ..... .... 123
9.2 Spanning Trees.... ..... .... .... .... .... .... ..... .... 125
9.3 Bounded-Degree Spanning Trees ... .... .... .... ..... .... 131
9.4 Minimum Routing Cost Trees . .... .... .... .... ..... .... 133
10 Cuts and Induced Bipartite Subgraphs.. .... .... .... ..... .... 137
10.1 Basic Definitions... ..... .... .... .... .... .... ..... .... 137
10.2 Max-Cut and Edge-Induced Bipartite Subgraphs ... ..... .... 138
10.3 Compact Versions.. ..... .... .... .... .... .... ..... .... 141
10.4 Model Comparison. ..... .... .... .... .... .... ..... .... 144
10.5 Node-Induced Bipartite Subgraphs.. .... .... .... ..... .... 146
11 Stable Sets .... .... .... ..... .... .... .... .... .... ..... .... 149
11.1 Basic Definitions... ..... .... .... .... .... .... ..... .... 149
11.2 Compact Models... ..... .... .... .... .... .... ..... .... 150
11.3 Comparability Graphs.... .... .... .... .... .... ..... .... 152
12 Traveling Salesman Problems.. .... .... .... .... .... ..... .... 155
12.1 Separation of Subtour Inequalities .. .... .... .... ..... .... 156
12.2 A Column Generation Model for the ATSP... .... ..... .... 157
12.3 ATSP - A Compact Reformulation.. .... .... .... ..... .... 160
12.4 Time-Window ATSP .... .... .... .... .... .... ..... .... 161
Contents vii
13 Packing .. .... .... .... ..... .... .... .... .... .... ..... .... 165
13.1 Bin Packing .. .... ..... .... .... .... .... .... ..... .... 165
13.2 Robust Knapsack .. ..... .... .... .... .... .... ..... .... 169
13.3 Cycle Packing. .... ..... .... .... .... .... .... ..... .... 170
14 Scheduling .... .... .... ..... .... .... .... .... .... ..... .... 173
14.1 The Job-Shop Problem... .... .... .... .... .... ..... .... 173
14.2 The One Machine Problem.... .... .... .... .... ..... .... 178
15 Computational Biology Problems... .... .... .... .... ..... .... 183
15.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 183
15.2 Sorting by Reversals and Alternating Cycles .. .... ..... .... 184
15.3 Alignments and Longest Common Subsequences... ..... .... 190
15.4 Protein Fold Comparison . .... .... .... .... .... ..... .... 193
References.... .... .... .... ..... .... .... .... .... .... ..... .... 197
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 205
Notation
sets The set f1;2;...;ng is denoted by ½n(cid:2). Sets are usually named with
upper-case letters S, T, etc. We write S(cid:3)T when S is a subset of T,
including the case S¼T. A proper subset of T is denoted by
ðS(cid:3)T;S6¼TÞ or by S(T.
vectors Vectors are denoted by lower-case letters, e.g., a, b, with possible
superscripts, e.g., a0, b0. Their entries are denoted by the same letters
withasubscriptreferringtothecomponent,e.g.,ai,bi,a0i,b0i.Vectors
can be either row vectors or column vectors. The context will always
make clear the meaning. For instance, if we say ‘the linear form cx’
and both c and x have been defined as vectors, it is clear that c is row
vector and x is a column vector. Similarly, if we say ‘the quadratic
formaHb,’whereaandbaredefinedasvectorsandHisamatrix,itis
clearthataisrowvectorandbisacolumnvector.Sometimes,avector
thatisinitiallydefinedasacolumn(row)vectorneedstobetransposed
as a row (column) vector. Again, if the meaning is clear we avoid to
use the transposition sign, while otherwise we indicate the transposed
vector as aT.
The vectors of the canonical basis of Rn are denoted by
e1 ¼ð1;0;...;0Þ, …, en ¼ð0;...;0;1Þ.
matrices Matrices are denoted by upper-case letters, e.g., A, B, with possible
superscripts, e.g., A0, B0. Their entries are denoted by the corre-
sponding lower-case letters with subindices referring to the rows and
columns,e.g.,aij,bij,a0ij,b0ij.Differentlyfromvectors,ifamatrixhasto
be transposed, this will be denoted by the transposition sign.
equations A convention we use in this book regards the equation numbering. If
we reference an integer linear program as (x), we reference the
integrality relaxation of (x) as (x).
ix
Chapter 1
Introduction
Linear Programming (LP) and Integer Linear Programming (ILP) are two of the
mostpowerfultoolsevercreatedinmathematics.Theirusefulnesscomesfromthe
manyareaswheretheycanprovidesatisfactorymodelingandsolvingtechniquesto
real-life problems. Their appeal comes from the rich combinatorial and geometric
theorytheyarebasedupon.
Solving an LP problem consists in minimizing a linear functional over a poly-
hedron, which, inturn,amounts todetecting avertex of thepolyhedron where the
linearfunctionalachievestheminimum(ifitexists).Themanytheoreticalaspects
ofthestudyoflinearprogramsarenotonlyinterestingperse,butareinstrumental
to the most important goal, i.e., the actual possibility of computing the minimum.
Therefore,thebasicquestionishowtheinputdatathatdescribethepolyhedronand
consistinanexplicitlistofthepolyhedronfacetscanbeturnedintothesolution.
Also solving an ILP problem consists in minimizing a linear functional over a
polyhedron, only that this time the input data do not describe the polyhedron of
interest, but rather a larger, ‘weaker’, polyhedron. This makes the problem much
moredifficulttosolve,becausewehavealsoto‘discover’thepolyhedroninside.
Inbothcasesthecomputationalperformanceofthesolvingalgorithmsdependson
theinputsize,i.e.,onthenumberoffacetsoftheinputpolyhedron.Thevastmajority
oftheLP/ILPinstancesofinterestaremadeofpolyhedraandlinearcostfunctions
derived,indeed,fromthemodelingofcombinatorialoptimizationproblems,whose
instancesarecombinatorialobjectssuchasgraphs,sets,sequences,etc.Therefore,
whenwediscussthesizeofanLP/ILPinstanceitisimportanttodoitinrelationto
thesizeoftheinstanceoftheunderlyingcombinatorialoptimizationproblem.
Itturnsoutthatthemosteffectiveformulationsforsomecombinatorialoptimiza-
tion problems (almost invariably NP-hard problems) are polyhedra with an expo-
nentialnumberoffacets.Thesetypeoflarge-sizemodelswerefirstpursuedbythe
©SpringerInternationalPublishingAG2018 1
G.LanciaandP.Serafini,CompactExtendedLinearProgrammingModels,
EUROAdvancedTutorialsonOperationalResearch,
DOI10.1007/978-3-319-63976-5_1
