Studies on Graphs and Discrete Programming

388 Pages·1981·8.446 MB·English

by  P. Hansen (Eds.)

General Editor
Peter L. HAMMER, University of Waterloo, Ont., Canada
Advisory Editors
C. BERCE, 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.
C.-C. ROTA, Massachusetts Institute of Technology, Cambridge, MA, USA
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD
NORTH-HOLLAND
MATHEMATICS STUDIES 59
Annals of Discrete Mathematics (11 )
General Editor: Peter L. Hammer
University of Waterloo, Ont., Canada
Studies on Graphs and
Discrete Programming
Editor:
P. HANSEN
Faculte Universitaire
Catholique de Mons
Mons, Belgium
NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM NEW YORK OXFORD
Q North- Holland Publishing Company, 1981
All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem,
or rransmirted. in any form or by any means, elerrronic. mechanical. phorocopying. recording
or otherwise. without thepriorpermissiono f thecopyrighr owner.
ISBN: 0 444 86216 1
Publishers
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Library of Congress Cataloging in Publication Data
Main entry under title:
Studies on graphs and discrete programming.
(Annals of discrete mathematics ; 11) (North-
Holland mathematics studies ; 59)
Includes proceedings of the Workshop on
Applications of Graph Theory to Management, held
at the European Institute for Advanced Studies in
Management, Brussels, Mar. 20-21, 1979, and also
papers from the loth Mathematical Programming
Symposium, Montreal, Aug. 27-31, 1979.
Bibliography: p.
1. Networks analysis (Planning) --Congresses.
2. Graph theory--Congresses. 3. Integer program-
ming--Congresses. I. Hansen, P. (Pierre) 11. Work-
shop on Applications of Graph Theory to Manage-
ment (1979 : European Institute for Advanced
Studies in Management) 111. International Mathe-
matical -ramming Symposium (1Qth : 1979 :
Montreal, Qu6bec) IV. Series. V. Series: North-
Holland mathematics studies ; 59.
w.a5.~78 519.7'7 81-14027
ISBN 0,444 - 862 16- 1 AACR2
PRINTED IN THE NETHERLANDS
This volume contains the proceedings of the Workshop on Applications of
Graph Theory to Management, held at the European Institute for Advanced
Studies in Management, Brussels, March 20-21, 1979. In addition, contributed
papers are included, many of which were presented at the Xth Mathematical
Programming Symposium, Montreal, August 27-31, 1979.
Research on Graph Theory and on Discrete Programming has been very
active during the last decade; the intersection of both areas constitutes a
particularly fruitful research domain. A variety of studies in the present volume
illustrates the point, both as far as theory and applications are concerned. Some
progress is made towards proving the deep Chvatal-Hammer conjecture on the
aggregation of inequalities in integer programming. The intriguing relationship
between the optimal values of variables in linear 0-1 programs and their
continuous relaxations is further analysed, for matching and for packing
problems. Polynomial algorithms are proposed for new or little studied prob-
lems such as coloring claw-free perfect graphs or optimizing totally dual
integral systems. Improved resolution methods are suggested for classical
problems, e.g. assignment and quadratic assignment. Several new theorems
provide original perspectives on important applications to school timetabling
and scheduling the games of a sports league. Other applications concern basis
restructuring and reinversion in linear programming-for which a surprising
connection with a machine maintenance problem is exhibited-network synth-
esis with non-simultaneous multicommodity flows and testing the reliability of
a tree system. Studies on recent developments of discrete programming without
direct ties with graphs are also presented: solving non-separable non-convex
programs, a noteworthy yet neglected subject is taken up; parametric multi-
criteria integer programming is explored. Complexity theory is used to analyse
the administration of standard length telephone cable reels and to provide
polynomial algorithms for some continuous locational problems.
To all those who have generously given of their time during this volume’s
preparation, allowing all papers to be refereed, sincere thanks are due.
P. HANSEN
1981
P. Hansen, ed., Studies on Graphs and Discrete Programming
@ North-Holland Publishing Company (1981) 1-13
INTEGER AND FRACTIONAL MATCHINGS*
Egon BALAS
Graduate School of Indusrrial Administration, Carnegie-Mellon Unioersity, Pittsburgh,
PA 15213, USA
We examine the connections between maximum cardinality edge matchings in a graph
and optimal solutions to the associated linear program, which we call maximum f-matchings
(fractional matchings). We say that a maximum matching M separates an odd cycle with
vertex set S, if M has no edge with exactly one end in S. An odd cycle is separable if it is
separated by at least one maximum matching. We show that (1) a graph G has a maximum
f-matching that is integer, if and only if it has no separable odd cycles; (2) the minimum
number q of vertex-disjoint odd cycles for which a maximum f-matching has fractional
components, equals the maximum number s of vertex-disjoint odd cycles, separated by a
maximum matching; (3) the difference between the cardinality of a maximum f-matching
and that of a maximum matching in G is one half times s ; (4) any maximum f-matching with
fractional components for a minimum number s of vertex-disjoint odd cycles defines a
maximum matching obtainable from it in s steps; and (5) if a maximum f-matching has
fractional components for a set Q of odd cycles that it not minimum, there exists another
maximum f-matching with fractional components for a minimum-cardinality set S of odd
cycles, such that S c Q, lQ\Sl is even, and the cycles in Q\S are pairwise connected by
alternating paths. Statement (4) duplicates an earlier result of J.P. Uhry [7].
1. Introduction
Let G = (V, E) be an undirected graph with 1 V(= n and (El = rn, let A be the
vertex-edge incidence matrix of G, and ep the p-vector of 1's. Consider the
problem of finding a maximum-cardinality edge-matching in G
1
EM(G) max{emx Ax G en, x E {0,1)")
and the associated linear program
LEM(G) max{emx I Ax ss en, x 2 0).
We will call EM(G) the matching problem, and LEM(G) the fractional
matching, or shortly the f-matching problem. A feasible solution to EM(G)
will be called a matching, a feasible solution to LEM(G), an f-matching.
Clearly, every matching is an f-matching.
* Research supported by the National Science Foundation through grant MCS76-12026 A02,
and by the U.S. Office of Naval Research through contract N0014-C-0621 NR 047-048.
1
2 E. Balm
We wish to investigate the relationship between maximum matchings and
maximum f-matchings in a graph G, i.e., between optimal solutions to EM(G)
and LEM(G). For a problem P, we denote by v(P)t he value of (an optimal
solution to) P. For vertex sets S, TGV , we denote by (S, T) the set of edges of
G with one end in S and the other in T, and by (S) the subgraph of G induced
s=
by S. Also, V\S.
An alternating path relative to a matching M is a path whose edges alternate
fi.
between M and An M-augmenting path is an alternating path whose end
vertices are distinct and are not incident with M. A matching M in G is of
maximum cardinality if and only if G contains no M-augmenting path (Berge
PI).
If G has no odd cycles (i.e., is bipartite), A is totally unimodular and every
basic solution to LEM(G) is integer, i.e.,
v(LEM(G))= u(EM(G)). (1)
If G is not bipartite, LEM(G) has fractional basic solutions, and the convex
hull of feasible integer solutions, i.e., the matching polytope, is defined (to-
gether with the constraints of LEM(G)) by the inequalities
1
xes$(\Sl-l), SGV, (S(33a nd odd. (2)
ee(S,S )
(Edmonds [3]). Any basic feasible solution to LEM(G) has components equal to
0,l or $ (Balinski [l]).F urthermore, if x, =$,t hen e belongs to some odd cycle C,
and xf = i, Vf E C.
Let M be a maximum matching in G. If G is bipartite, M is a maximum
f-matching; but the converse if false, as illustrated by the graph in Fig. 1, where
the maximum matching shown in heavy lines is also a maximum f-matching, in
spite of the presence of a 5-cycle.
It would seem that whether a maximum matching M is also a maximum
f-matching, depends on whether M covers all vertices of every odd cycle; but
this is not so. The matching in the graph of Fig. 2 leaves a vertex of an odd
cycle exposed, yet it is a maximum f-matching. On the other hand, the
maximum matching M in the graph of Fig. 3(a) covers all vertices of the only
odd cycle of the graph, yet it is not a maximum f-matching, as shown by the
f-matching M' of Fig. 3(b): lM(=3, while the value of M' is 34.
Fig. 1. Fig. 2.
Integer and fractional matchings 3
2. Separable odd cycles
The key concept for understanding the connection between maximum
matchings and maximum f-matchings is the following.
Let M be a maximum matching and C an odd cycle in G, and let S be the
vertex set of C. We will say that M separates C if
Mn(s, 3)=@ (3)
An odd cycle which is separated by at least one maximum matching in G will
be called separable. If C is a separable odd cycle in G, with vertex set S, then G
has a maximum matching of the form M = MI UM,, where M1 and M2 are
maximum matchings in (S) and in (3) respectively. If C is a nonseparable odd
+
cycle with vertex set S of cardinality 2k 1, then every maximum matching
+
contains at least k 1 edges incident with S.
Theorem 1. G has a maximum f-matching that is integer if and only if G has
no separable odd cycle.
Proof. Necessity. Let M be a maximum matching in G which separates an odd
cycle C with vertex set S, and (S(=2k+1.S ince M is maximal and
M n (S, 3) = 8, M contains k edges of (S, S). Let X be the incidence vector of
M, and define f by
i
4 egC,
ie=0 eE(S,S)\C,
e E E\(S, S).
X,
Clearly, f is a feasible solution to LEM(G), and
u(LEM(G))ae,f = (M(-k+t(k+l)>u(EM(G)),
hence G has no maximum f-matching that is integer.
Sufficiency. Let u(LEM(G))>u(EM(G)), and let f be a basic maximum
f-matching in G having fractional components for a minimum number of odd
cycles, say C1,.. . , C,. For i = 1,. . . , q, let Si be the vertex set of Ci, with
4 E. Balas
lSil = 2ki + 1. Clearly, Si flS i = 8, Vi,j E (1, . . . , q}. Further, let S = UfZlS i. The
components fe of f such that ie=delfi ne a matching M' in (3) which is
clearly maximum, or else f itself could not be a maximum f-matching.
For i = 1,.. . , q, let Mi be a set of ki mutually nonadjacent edges of Ci, i.e., a
maximum matching in (Si). Then the edge set
M = M'UM I u - - u M, (4)
*
is a matching in G such that M fl (Si, $1 = 8, i = 1,. . . , q.
Next we prove by contradiction that M is of maximum cardinality, which
implies that the odd cycles C1,.. . , C, are all separable and thus completes the
proof the theorem.
Suppose M is not maximum. Then there exists an M-augmenting path P in
G. Moreover, we claim that the maximum matchings Mi in (S,), i = 1,.. . , q,
which are not unique, can always be chosen such that P has no edge in
common with any of the cycles Ci, i = 1, . . . , q. For suppose P r lC i # 8 for some
i E { 1, . . . , q}. Since Ci has only one vertex exposed with respect to M, at least
one of the end vertices of the path P, say u, is not contained in Ci. Let w be the
first vertex of Ci encountered when traversing P starting from u. Then the
maximum matching Mi in (Si) can be replaced by fii, containing those ki
mutually nonadjacent edges of Ci that leave exposed w. This replaces the
matching M in G by
A2 = (M\Mi) U fii,
such that \fil=\Ml, and the portion of P between w and u is then an
6f-augmenting path p, with pn Ci = $3. If p still has some edge in common
with some Ci, j ~ ( 1., . . , q}-{i}, the above procedure can be repeated. After at
most q applications of this procedure, we obtain a matching I\;r in G such that
IMl= \MI,
fi, u u fiq
= M'U * *
p,
(where each is a maximum matching in (Si)), and an 6f-augmenting path
not containing any edge of any Ci, i = 1, . . . , q. This proves that w.l.o.g., the
path P can be assumed to have no edge in common with any Ci, i = 1,. . . , q.
If we now reverse the assignment of edges in P with respect to M, we obtain
a matching M" in G, such that IM*( = (MIS1 . But from M" one can construct
an f-matching iin G such that e i5 eP and ih as fractional components for fewer
odd cycles than 2, contrary to our assumptions on 2.
To construct P, first note that since P contains no edge of any Ci and is an
alternating path, P contains no edge of (S), and the only vertices of P
contained in S are one, or possibly both, end vertices of P. The fact that P has
at least one vertex in S follows from the maximality of the matching M'=
(s).
M n(3, 3) in
Integer and fractional matchings 5
Let u1 and uz be the two end vertices of P. We consider two cases.
Case 1. uleSi for some i ~ ( 1 ,.. . , q},u 2$S. Let el,.. . , e2k+lb e the edges
of Ci, and let el and e2k+bl e incident with ul. Define f by
Ife otherwise.
Case 2. U ~SE, a nd u2€ S, for some i, j E (1, . . . , 9). Let e,, r = 1, . . . ,2k + 1,
and d,, s = 1, . . . ,2k, + 1, be the edges of C, and Ci respectively, where el and
ezk,+al re incident with ul, and dl and d2k,+alr e incident with +Then define Z
by
fe1=-ie {eiEeP,
e E {e2,e 4, . . . , ezk,U> (4,d 4, . . . , dZk,},
e €{el, e3,. ezk,+i)U{di,4 ,. - dzk,+i},
9 9
otherwise.
In Case 1, X has fractional components for q - 1 odd cycles, and em%=
emR +i,s ince the reversal of edge assignments in P adds 1 to, while the change
of values for eECi subtracts $ from, the value of en%I n Case 2, X has
fractional components for q - 2 odd cycles, and em%= emf. In both cases the
existence of i contradicts our assumption that i is a maximum f-matching
which has fractional components for a minimum number (9) of odd cycles.
Thus M is a maximum cardinality matching in G, which separates the odd
cycles C1,. . . , C,.
The graphs in Figs. 1 and 2 both have maximum f-matchings that are integer,
since in each case the unique maximum matching shown in heavy lines does not
separate the unique odd cycle of the graph, hence the latter has no separable
odd cycle. The graph of Fig. 3, on the other hand, has a maximum matching
(other than the one shown in Fig. 3(a)) which separates the odd cycle: it
contains two edges of the cycle, and a third edge not incident with any vertex
of the cycle. Therefore there exists no maximum f-matching that is integer.
It is interesting to note that if LEM(G) has an integer optimum, it does not
follow that the linear program dual to LEM(G), namely the fractional vertex
covering problem LVC(G), also has an integer optimum. In other words, the
absence of a separable odd cycle in G does not guarantee that the cardinality
of a maximum edge matching is equal to that of a minimum vertex cover. For
example the graph of Fig. 4 has no separable odd cycle and therefore it has a
6 E. Balas
(4
Fig. 4.
maximum f-matching of value 3 that is integer. On the other hand, the
cardinality of a minimum vertex cover is 4, and the fractional vertex cover that
assigns value 4 to every vertex u is the only one with value 3.
In proving the ‘if’ part of Theorem 1, we have rediscovered what turns out to
be a recent result of J.P. Uhry:
Corollary 1.1 (J.P. Uhry [7]). Let i bc a maximum f-matching in G having
fractional components for a minimum number of odd cycles Cii, = 1, . . . , q. Let
M’ = {e E E 1 ie= l}, and for i = 1, . . . , q, let Mib e a maximum matching in Ci.
Then
M = M’ u MI u * * * u M2
is a maximum matching in G.
This result shows how to obtain a maximum matching in an arbitrary graph
G from a maximum f-matching in G that satisfies an additional requirement
(having fractional components for a minimum number of odd cycles). It turns
out that this procedure can be reversed, to obtain a maximum f-matching from
a maximum matching that, again, satisfies an additional requirement: to
separate a maximum number of disjoint odd cycles. (Two odd cycles are
disjoint if they have no common vertex.) The next Corollary and its proof state
this relationship more precisely.
For a maximum matching M, let a(G,M) be the maximum number of
disjoint odd cycles separated by M. For a maximum f-matching x, let y(G,x )
be the number of odd cycles for which x has fractional components. Finally, let
x
A be the set of maximum matchings, and the set of maximum f-matchings in
G.
Corollary 1.2.
max a(G,M )= min y(G,x ).
ME& XEX
Proof. In the sufficiency part of the proof of Theorem 1, a maximum f-
matching i such that
y(G, i)= q = min y(G, x)
XEX