Table Of ContentProgress
in
Combinatorial
Optimization
Edited by
WILLIAM R. PULLEYBLANK
Department of Combinatorics and Optimization
University of Waterloo
Waterloo, Ontario, Canada
1984
®
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Canadian Cataloguing in Publication Data
Main entry under title:
Progress in combinatorial optimization
Papers presented at a conference held at the
University of Waterloo, summer 1982.
ISBN 0-12-566780-9
1. Mathematical optimization - Congresses.
2. Combinatorial analysis - Congresses.
I. Pulleyblank, W. R.
QA402.5.P76 1984 519.5 C84-098433-2
Library of Congress Cataloging in Publication Data
Main entry under title:
Progress in combinatorial optimization.
1. Combinatorial optimization—Congresses.
I. Pulleyblank, W. R. (William R.)
QA164.P78 1984 519 84-2964
ISBN 0-12-566780-9 (alk. paper)
PRINTED IN THE UNITED STATES OF AMERICA
84 85 86 87 9 8 7 6 5 4 3 2 1
CONTRIBUTORS
Numbers in parentheses indicate the pages on which the authors' contributions begin.
Julian Arâoz (3), Departamento de Matématica y Cs. Computacion, Universidad
Simon Bolivar, Caracas 1081, Venezuela
Earl R. Barnes (13), IBM Thomas J. Watson Research Center, Yorktown Heights,
New York 10598
Louis J. Billera (27), Cornell University, Ithaca, New York 14853
R. Bixby (39), Department of Industrial Engineering and Management Sciences,
Northwestern University, Evanston, Illinois 60201
M. Burlet (69), U.S.M.G.-IMAG-BP 53, 38041 Grenoble Cedex, France
R. Chandrasekaran (101), The University of Texas at Dallas, Richardson, Texas
75075
V. Chvâtal (107), School of Computer Science, McGill University, Montreal,
Quebec H3A 2K6, Canada
Jack Edmonds (3, 117), Department of Combinatorics and Optimization, Univer-
sity of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Peter C. Fishburn (131), Bell Telephone Laboratories, Inc., Murray Hill, New Jer-
sey 07974
J. Fonlupt (69), U.S.M.G.-IMAG-BP 53, 38041 Grenoble Cedex, France
Andrâs Frank (147), Research Institute for Telecommunications, Budapest,
Hungary
F. R. Giles (117), Department of Mathematics, Acadia University, Wolfville, Nova
Scotia BOP 1X0, Canada
V. Griffin (3), #314A, Sector G, Urbanization Caralinda, Naguanagua, Estado
Carabobo, Venezuela
M. Grötschel (167), Lehrstuhl für Angewandte Mathematick II, Universität
Augsburg, Memminger Str. 6, D-8900 Augsburg, Germany
Alan J. Hoffman (13, 185), IBM Thomas J. Watson Research Center, Yorktown
Heights, New York 10598
Masao Iri (197), Department of Mathematical Engineering and Instrumentation
Physics, Faculty of Engineering, University of Tokyo, Bunkyo-ku, Tokyo, Japan
Bernhard Körte (221), Institut für Ökonometrie und Operations Research,
Abteilung OR, Universität Bonn, Nassestrasse 2, 53 Bonn 1, Germany
Vil
Vlll CONTRIBUTORS
J. Labetoulle (245), Centre National d'Etudes des Télécommunications, Issy les
Moulineaux, France
E. L. Lawler (245, 363), Computer Science Division and Electronic Research Lab-
oratory, University of California at Berkeley, Berkeley, California 94720
J. K. Lenstra (245), Centre for Mathematics and Computer Science, 1009 AB
Amsterdam, The Netherlands
L. Lovâsz (167, 221), Mathematical Institute, Eotvos L. University, Budapest
H-1088, Hungary
O. Marcotte (263), Département de Mathématiques et d'Informatique, Université
de Sherbrooke, Sherbrooke, Quebec, Canada
S. Thomas McCormick (185), Department of Operations Research, Stanford Uni-
versity, Stanford, California 94305
Beth Spellman Munson (27), Bell Laboratories, WB 1H-312, Crawfords Corner
Road, Holmdel, New Jersey 07733
James B. Orlin (273), Sloan School of Management, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139
C. H. Papadimitriou (295), Department of Operations Research, Stanford Univer-
sity, Stanford, California 94305
A. Recski (307), Research Institute for Telecommunication, Budapest, H-1026
Gabor A.u.65, Hungary
A. H. G. Rinnooy Kan (245), Erasmus University, Rotterdam, The Netherlands
A. Schrijver (167, 315), Universiteit van Amsterdam, Instituut voor Actuariaat en
Econometrie, Jodenbreestraat 23, 1011 NH, Amsterdam, The Netherlands
Po Tong (363), Computer Science Division and Electronic Research Laboratory,
University of California at Berkeley, Berkeley, California 94720
L. E. Trotter, Jr. (263), School of Operations Research & Industrial Engineering,
Cornell University, Ithaca, New York 14853
V. V. Vazirani (363), Computer Science Division and Electronic Research Labora-
tory, University of California at Berkeley, Berkeley, California 94720
PREFACE
The University of Waterloo marked its twenty-fifth anniversary in 1982. As a part
of the silver jubilee celebrations, the Department of Combinatorics and
Optimization organized a Summer School and Conference on Combinatorics. This
was the fourth of a series of international meetings on combinatorics held by the
Department: the first in 1966, the second in 1968, and the third in 1977 to celebrate
the sixtieth birthday of Professor W. T. Tutte.
Each week of the conference focussed on a specific field of combinatorial theory.
The first was on combinatorial optimization, the second on graph theory, and the
third on enumeration and design. Each week consisted of two series of instructional
lectures, a number of invited talks by specialists in the field, and a number of con-
tributed papers.
The instructional series on combinatorial optimization were given by J. Edmonds
and F. R. Giles (Total integrality of linear inequality systems) and by L. Lovâsz
(Geometric methods in combinatorial optimization). This volume contains articles
summarizing the contents of these series of lectures, as well as most of the other
presentations given at the meeting.
Combinatorial optimization and, more generally, the study of algorithmic prob-
lems is one of the newest and fastest growing areas in the field of combinatorial
mathematics. In 1963, Herbert Ryser wrote in the introduction to his monograph
Combinatorial Mathematics [The Mathematical Association of America, distrib-
uted by John Wiley and Sons, Inc. (1963)]: "Two general types of problems appear
throughout the literature. In the first, the existence of the prescribed configuration is
in doubt, and the study attempts to settle this issue. In the second, the existence of
the configuration is known, and the study attempts to determine the number of con-
figurations. . . ." In 1971 at a plenary session of the Second Louisiana Conference
on Combinatorics, Graph Theory, and Computing, Professor Ryser lectured on
"Recent Activity in Combinatorics." He described a third type of combinatorial
problem: How to determine efficiently whether or not a particular structure exists
and how to find efficiently a structure with certain properties.
During the 1970s and 1980s, this third type of problem has proved to be the most
popular in the entire area of combinatorics. Evidence of this is given by the large
number of meetings, journals, and articles dealing with combinatorial optimization
and graphical algorithms. The papers contained in this volume provide an excellent
cross section of the field up to the summer of 1982.
IX
ACKNOWLEDGMENTS
This volume was prepared at the University of Waterloo using TROFF on the
Watmath UNIX system. Thanks are due to Helen Warren, Archana Kundra, Elinor
Hawn, and Christy Folliott for the fine job they did in typesetting the manuscripts.
Technical advice was provided by R. Goebel, J. C. Beatty, and especially by I.
Telford. Their contributions are gratefully acknowledged.
Special thanks are also due to the staff of the Department of Combinatorics and
Optimization, Joan Selwood, Kathy Knight, and Christy Folliott, for the large
amount of support they provided in the preparation of this volume. Moreover, the
staff of Academic Press provided considerable assistance in the preparation of the
final copy.
Major financial support was provided by the Natural Sciences and Engineering
Research Council of Canada and the Faculty of Mathematics of the University of
Waterloo, and is gratefully acknowledged.
I also wish to express my thanks to the referees who did a very prompt and com-
petent job of reviewing the papers submitted for inclusion in this volume.
Lifting the Facets of Polyhedra
/. Araoz
Dto. do Mat. y Cs. Computatcidn,
Univ. Simonon Bolivar, Caracas,
1081, Venezuela
and Institut für Operations Research,
Nassestr.2,
5300 Bonn, West Germany
J. Edmonds
Dept. of Combinatorics and Optimization,
Univ. of Waterloo,
Waterloo, Ontario,
Canada
V. Griffin
Area de Postgrado,
Univ. de Carabobo,
Valencia, Veneuela.
This paper presents polarity results about lifting the facets of polyhedra based on
die polyhedral polarity work of the authors. It also shows how known theorems related
to integral polyhedra are special cases of these results.
1. Introduction.
We present a polar characterization of facets of polyhedra
obtained by lifting facets of lower dimensional polyhedra. We also
show how known theorems related to lifting the facets of integral
polyhedra are special cases of these results. To be more specific, let H
and L denote a partition of the index set N.
PROGRESS IN COMBINATORIAL OPTIMIZATION 3 Copyright © 1984 by Academic Press Canada
All rights of reproduction in any form reserved.
ISBN 0-12-566780-9
4 ARAOZETAL
For any polyhedron PQIfUL and r ί JR* such that
1.1 PL = {xL € ff* : (Γ,Χ^) ί P} is full dimensional, let
1.2 x u ^a be a facet of P .
L L L
Any facet of P of the form
1.3 xay+XLUL^OL+ry
is called a fact extension or a lifting of (1.2).
Pollatschck [12] and Padberg [9] independently considered exten-
sion of facets when P was a special (0, l)-polyhedron and r was a vec-
tor of zeros. Padberg introduced "sequential lifting** i.e. a sequence of
extensions with \H\ =1, and other authors including Balas [4], Ham-
mer, Johnson and Peled [7], Wolsey [15], Nemhauser and Trotter [8]
and Trotter [14] discussed properties of these sequential liftings. Wol-
sey [16] applied sequential lifting to bounded integer programs where r
was not necessarily 0. Zemel [17] introduced "simultaneous lifting" for
general (0, l)-programs with r=0. For knapsack problems, simultane-
ous lifting is also used in Balas and Zemel [5].
Peled [10,11] used also simultaneous lifting for general (0,1)-
programs but allows r to be a (0, l)-vector, and related the lifting of
(1.2) to the vertices of
1.4 P°"{y:(1.3) is a valid inequality for P}.
It was this idea of Peled that stimulated this paper.
In section 2 we outline the polyhedral polarity theory developed
in Arâoz, Edmonds, Griffin [1,2], Griffin [6] and use it in section 3 to
show for any full dimensional polyhedra P and P as defined in (1.1),
L
that (1.3) is a facet extension of (1.2) if and only if y is a vertex of PQ
and that xjiy^ry is a facet of P if and only if y is an extreme ray of the
recessional cone of P°.
An early version of this paper is Aräoz, Edmonds, Griffin [3].
2. Polarity of Polyhedra.
The polarity of polyhedra given by general bilinear relations have
been described in Aräoz, Edmonds and Griffin [2] and Griffin [6].
There we show that there are six different polarity relations, four of
which have been extensively studied: The Cone Polarity, The Min-
kowski Polarity, The Reverse Minkowski Polarity and the Polarity of
Covex Sets.
In this paper we will look at the polarity of polyhedra given by
(1.3) with relation to extensions of facets.
Let xily if and only if
LIFTING THE FACETS OF POLYHEDRA 5
(2.1) xWy+xu+vy*a where W€*"xir, u€J^, vOR*, at*.
For any set PZ5t" the Ω-polar of P is
(2.2) Ρΰ={?Οΐ":χΩ? for all xtP)
={yEJ?":(2.1) is valid forP}.
For ßOR*, β' is defined anlogously.
P is Ω-dosed when P=PQQ.
Concerning notation, for any system of linear inequalities
LI-{Ax*zb} let P(U) denote the polyhedron {xt&iAx^b}. For any
sets 5 and T we denote the convex hull of S by CONV(S), the conical
hull of T by CONE{T) and CONV(S)+CONE(T) by C(S,T). dim(J>)
will denote the dimension of P.
Let P be the polyhedron C(S,T). Then Pn is characterized by:
2.3 THEOREM. (Theorem 3.12[2]). J>° equals P(U) where
i for all stS
2.3.1 L/-
for all ίΕΓ·
2.4 COROLLARY The Ω-polar of a polyhedron is a polyhedron.
There are two sets related to Ω which give the different types of
polarities. They are:
2.5 XQ={X:XW+V=0}
2.6 Y ={y:Wy+u=0}.
Q
We will use the following results about XQ and YQ.
2.7 LEMMA. (3.20[2]). If X, >Ό*0, then for all xtX x-u is a
0 Q
constant value denoted by α=χ·ι*.
2.8 LEMMA. (3.21[2]). Χο*0=Γΰ implies for all β there exists
xtXa such that x·«=β.
2.9 COROLLARY. X *0=Y implies dim(X )>0.
a Q n
The different polarity types are:
TYPE 1. Λη, Γα*0, α=α. For example when N=H and Ω is xy=sO
the Ω-closed polyhedra are the polyhedral cones (Cone Polarity).
TYPE 2. Xa, ΥαΦ0, α<α. For example when N=H and Ω is jry<sl
the Ω-closed polyhedra are the polyhedra which contain the origin
(Minkowski Polarity).
TYPE 3. X , Κ *0,α>α. For example when N=H and Ω is xy^l
Q ο
the Ω-closed polyhedra are the polyhedra contained in their reces-
sion cone which do not contain the origin; and R" (Reverse Min-
kowski Polarity).
6 ARAOZETAL
TYPE 4. Xû=0*y . For example when H=NU{0} asndil is xy^yo
o 9
all polyhedra are Ω-dosed (Polarity of convex sets).
TYPE 5. XQ, YQ=0.
TYPE 6. XQ*0=YQ. An example of this case is the Lifting Polarity
when N-H\JL and Ω is given by (1.3) x y+XLU ^a+ry hence
H L
W=|p^J, u=(Oif,i/i)andv=-r.
In [2] we also give the relation between the vertices and extreme
rays, that is, the generators, of a pointed Ω-closed polyhedron P and
the facets of P° for the six cases. This same relation gives us a charac-
terization of the generators of then-polar of a full dimension Ω-closed
polyhedron when P is given by a minimal system of inequalities, that
is, by its facets.
When P is not Ω-closed, the generators of Pa are not in general
related to the facets of P but to the maximal faces of P of a form deter-
mined by the relation Ω. (For more details see [6] which relates all the
faces of Pù to faces of P.) However, even when P is not Ω-closed, for
certain special cases we can relate the generators of PQ to certain facets
of P. Theorem (2.14) which is applied in section 3 to the facet exten-
sion problem, is an example of this.
We need the following lemmas. (See for example [13] for the
first two).
The equality set of a polyhedron P(U) consists of the members
of U which are satisfied as equality by every point in P(JJ).
2.10.1 LEMMA. For any face F of P({xA^b}) the rank of the equal-
ity set of F is equal to \N\ -dim(F).
2.10.2 LEMMA. Let P=P({xA^b}). If x°a'=b for some point x° in
e
the relative interior of P and xag^b is valid for P then xa'-b
g e
belong to the equality set of P.
2.11 LEMMA. Let P=P({xA <;&}), dim(P)>0 and let {xA/=i } be
z
any subset of the equality set of P with rank equal to that of the
equality set of P. Let xu^a in U give a facet F of P, and let x?
be in the relative interior of F. Then
2.11.1 χα^β is valid for P andx*a = ß, if and only if
2.11.2 3e^0, λ such that a=AA + cii, ß=fc/X+ca.
PROOF. (2.11.1) implies (2.11.2). Since xu^a gives a facet F
of P and χα = β is in the equality set of F (by Lemma (2.10.2)), we
obtain by Lemma (2.10.1)
