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Submodular Functions and Optimization 2nd ed Edition
Satoru Fujishige (Eds.) Digital Instant Download
Author(s): Satoru Fujishige (Eds.)
ISBN(s): 9780444520869, 0444520864
Edition: 2nd ed
File Details: PDF, 15.63 MB
Year: 2005
Language: english

Submodular Functions
and Optimization
Second Edition

ANNALS OF DISCRETE MATHEMATICS 58
Series Editor: Peter L. HAMMER
Rutgers University, Piscataway, NJ, U.S.A
Please refer to this volume as follows:
S. Fujishige: Submodular Functions and Optimization
(Second Edition)
(Annalsof Discrete Mathematics,Vol.58) (2005)

Submodular Functions
and Optimization
Second Edition
Satoru Fujishige
Research Institute for Mathematical Sciences
Kyoto University
Kyoto, Japan
2005
Amsterdam - Boston - Heidelberg - London - New York - Oxford
Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo

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Preface
Submodular functions frequently appear in the analysis of combinatorial
systems such as graphs, networks, and algebraic systems, and reveal com-
binatorially nice and deep structures of the systems. The importance of
submodular functions has widely been recognized in recent years in com-
binatorial optimization and other fields of combinatorial analysis. The
present book provides the readers with an exposition of the theory of sub-
modular functions from an elementary technical level to an advanced one.
The theory of submodular functions was developed in the earliest stage
till 1950's by H. Whitney and W. T. Tutte for matroids, by G. Choquet
for the capacity theory and by 0. Ore for graphs. A nourishing stage of
the theory came with J. Edmonds' work on matroids and polymatroids
in 1960's. Related studies were also made in the theory of characteristic
function games by L. S. Shapley and others. Since 1970, applications of
(poly-)matroids to practical engineering problems have been extensively
made by M. Iri, A. Recski and others, and also theoretical developments
in submodular functions by W. H. Cunningham, J. Edmonds, A. Frank,
M. Iri, E. L. Lawler, L. Lovasz, A. Schrijver, E. Tardos, N. Tomizawa,
D. J. A. Welsh, U. Zimmermann and others.
The theory of submodular functions is now becoming mature, but a lot
of fundamental and useful results on submodular functions are scattered
in the literature. The main purpose of the present book is to put these
materials together and to show the author's unifying view of the theory
of submodular functions by means of base polyhedra and duality for sub-
modular and supermodular systems. Special emphasis is placed on the
constructive aspects of the theory, which will lead us to practical efficient
algorithms. No comprehensive survey of submodular functions is aimed at
here. I had to omit important results on submodular functions such as a
strongly polynomial time algorithm for minimizing submodular functions
due to M. Grotschel, L. Lovasz and A. Schrijver. This is mainly because
the precise description and validation of the results would require further
technical developments outside the mainstream of this book.
A sketch of the author's view of submodular functions was given in a
survey paper [Puji84c], which was written while I was visiting Professor
Bernhard Korte's Institute in Bonn as an Alexander von Humboldt fellow
in 1982-83, and laid a basis of the project of writing this book, which I
gratefully acknowledge. I also acknowledge that part of my work, upon
V

VI
which the present book is primarily based, has been supported by grants-
in-aid of the Ministry of Education, Science and Culture of Japan.
I would like to express my deep sincere thanks to Professor Masao Iri of
the University of Tokyo who first drew my attention to the theory of ma-
troids, a promising and enjoyable research field of combinatorics, in 1975
and has since then been keeping giving me invaluable advice and stimulat-
ing discussions on submodular functions and other related discrete systems.
Without his advice most of my work would not have been accomplished.
Thanks are also due to Professor Nobuaki Tomizawa, now at Niigata Uni-
versity, with whom I enjoyed inspiring discussions and joint work. I am also
very much grateful to Professor Peter L. Hammer for his encouragement to
write this book.
I used a preliminary version of this book for a lecture of the Doctoral
Program in Socio-Economic Planning at the University of Tsukuba and I
thank the students who attended the lecture for their useful comments.
I have also benefited from comments and communications received from
Bill Cunningham, Tetsuo Ichimori, Naoki Katoh, Kazuo Murota, Masataka
Nakamura, Hans Rock and Uwe Zimmermann, to name a few, in the course
of my research on submodular functions and writing this book.
July 1990 S.F.

Vll
Preface to the Second Edition
When I finished my monograph (the first edition) in 1990, there was a
polynomial-time algorithm forminimizing submodular functions bymeans
of the ellipsoid method, due toGrotschel, Lovasz, and Schrijver, and devis-
ing a combinatorial polynomial-time algorithm forminimizing submodular
functions wasstill an open problem. Submodular function minimization
is sofundamental in thetheory ofsubmodular functions andoptimization
that the monograph could not be completed without treating algorithmsfor
submodular function minimization, butI hesitated to include theellipsoid
method for submodular function minimization dueto Grotschel, Lovasz,
and Schrijver because of its non-combinatorial feature. Hence in the first
edition submodular function minimization was treated algorithmically in a
very unsatisfactory way. However, in 1999 the long-standing open problem
of submodular function minimization was resolved independently by Satoru
Iwata, Lisa Fleischer andmyself, andbyLex Schrijver, in different ways
though both algorithms were based onthe framework ofBill Cunningham.
I amvery happy to addChapter VIthereby including the combinatorial
strongly polynomial algorithms forsubmodular function minimization.
Moreover, among other related developments after 1990 one ofthe most
important results is the discrete convex analysis dueto Kazuo Murota,
which has been described in another new chapter, Chapter VII.
Chapters VIand VII form Part IIofthe second edition. Part I includes
Chapters I~V, which formed theoriginal edition of themonograph. In
Chapters I~Vof Part I typos and minor errors have been corrected. I
appreciate comments received from readers andmy friends andcolleagues
on the first edition of this monograph. I have also added remarksand
references related to thedevelopments after 1990, which areput between
brackets [and ].
I would like to mention here another important recent development
related to submodular functions that I could not include in Part II.It
is concerned with theconnectivity augmentation problem for graphsand
its generalization, which waspioneered by T. Watanabe and A. Naka-
mura [Watanabe+Nakamura87] and further developed by A. Frank, T.
Jordan, H.Nagamochi, T.Ibaraki, and others. Readers should be referred
to [Frank92, 94b, 05], [Frank+Jordan95], [Frank+KiralyO2], [Jordan95],
[Jackson+Jordan05], [Nagamochi+Ibaraki02], [NagamochiOO, 04]and the
references therein for later andrecent developments. I also would like to

Vlll
mention the source location problem, which is closely related to the connec-
tivity augmentation problem. The source location problem was first con-
sidered in [Tamura+Sengoku+Shinoda+Abe92] and [Tamura+Sugawara+
Sengoku+Shinoda98], and was further investigated in [Ito+Uehara+Yoko-
yamaOO], [Ito+IINUY02] and [Arata+Iwata+Makino+Fuji02] for undirected
networks, and in [Nagamochi+Ishii+ItoOl], [Ito+MAHIF03] and [Barasz+
Becker+FrankO5] for directed networks (also see the references therein).
I am very grateful to Hiroshi Hirai, Satoru Iwata, Tom McCormick,
Kazuo Murota, Takeshi Naitoh, Akiyoshi Shioura, Akihisa Tamura, and
Zaifu Yang for their useful comments on an earlier version of Chapters
VI and VII in Part II, which helped rectifying errors and improved the
presentation of Part II. Special thanks are also due to Kazuo Murota, who
gave me valuable detailed comments on Chapter VII. I also thank Andras
Frank and Tibor Jordan for useful information about recent developments
in connectivity augmentation.
Kyoto, January 2005 S.F.

IX
Contents
Preface v
Preface to the Second Edition vii
PART I 1
Chapter I. Introduction 3
1. Introduction 3
1.1. Introduction 3
1.2. Mathematical Preliminaries 4
(a) Sets 4
(b) Algebraic structures 5
(c) Graphs 9
(d) Network flows 13
(e) Elements of convex analysis and linear inequalities 15
Chapter II. Submodular Systems and Base Polyhedra 21
2. From Matroids to Submodular Systems 21
2.1. Matroids 21
2.2. Polymatroids 25
2.3. Submodular Systems 33
3. Submodular Systems 45
3.1. Fundamental Operations on Submodular Systems 45
(a) Reductions and contractions by sets 45
(b) Reductions and contractions by vectors 46
(c) Translations and sums 51
(d) Other operations 53

3.2. Greedy Algorithm 55
(a) Distributive lattices and posets 55
(b) Greedy algorithm 58
3.3. Structures of Base Polyhedra 66
(a) Extreme points and rays 66
(b) Elementary transformations of bases 70
(c) Tangent cones 72
(d) Faces, dimensions and connected components 75
3.4. Intersecting- and Crossing-Submodular Functions 86
(a) Tree representations of cross-free families 87
(b) Crossing-submodular functions 91
(c) Intersecting-submodular functions 101
&
3.5. Related Polyhedra 102
(a) Generalized polymatroids 102
(b) Polypseudomatroids 106
(c) Ternary semimodular polyhedra 112
3.6. Submodular Systems of Network Type 122
Chapter III. Neoflows 127
4. The Intersection Problem 127
4.1. The Intersection Theorem 127
(a) Preliminaries 128
(b) An algorithm and the intersection theorem 131
(c) A refinement of the algorithm 136
4.2. The Discrete Separation Theorem 140
4.3. The Common Base Problem 142
X

XI
5. Neoflows 145
5.1. Neoflows 145
(a) Submodular flows 145
(b) Independent flows 146
(c) Polymatroidal flows 147
5.2. The Equivalence of the Neoflow Problems 148
(a) From submodular flows to independent flows 148
(b) From independent flows to polymatroidal flows 149
(c) From polymatroidal flows to submodular flows 150
5.3. Feasibility for Submodular Flows 153
5.4. Optimality for Submodular Flows 155
5.5. Algorithms for Neoflows 167
(a) Maximum independent flows 167
(b) Maximum submodular flows 172
(c) Minimum-cost submodular flows 175
5.6. Matroid Optimization 188
(a) Maximum independent matchings 188
(b) Optimal independent assignments 194
Chapter IV. Submodular Analysis 199
6. Submodular Functions and Convexity 199
6.1. Conjugate Functions and a Fenchel-Type Min-Max Theorem
for Submodular and Supermodular Functions 199
(a) Conjugate functions 199
(b) A Fenchel-type min-max theorem 201
6.2. Subgradients of Submodular Functions 203
(a) Subgradients and subdifferentials 203
(b) Structures of subdifferentials 209

XI1
6.3. The Lovasz Extensions of Submodular Functions 211
7. Submodular Programs 216
7.1. Submodular Programs — Unconstrained Optimization .... 216
(a) Minimizing submodular functions 217
(b) Minimizing modular functions 223
7.2. Submodular Programs — Constrained Optimization 228
(a) Lagrangian functions and optimality conditions 229
(b) Related problems 234
(b.l) The principal partition 234
(b.2) The principal structures of submodular systems .. 245
(b.3) The minimum-ratio problem 248
Chapter V. Nonlinear Optimization
with Submodular Constraints 253
8. Separable Convex Optimization 253
8.1. Optimality Conditions 253
8.2. A Decomposition Algorithm 257
8.3. Discrete Optimization 260
9. The Lexicographically Optimal Base Problem 261
9.1. Nonlinear Weight Functions 262
9.2. Linear Weight Functions 264
10. The Weighted Max-Min and Min-Max Problems 269
10.1. Continuous Variables 269
10.2. Discrete Variables 272
11. The Fair Resource Allocation Problem 273
11.1. Continuous Variables 273
11.2. Discrete Variables 274

Xlll
12. The Neoflow Problem with a Separable Convex Cost
Function 280
PART II 285
Chapter VI. Submodular Function Minimization 287
13. Symmetric Submodular Function Minimization: Queyranne's
Algorithm 287
14. Submodular Function Minimization 290
14.1. The Iwata-Fleischer-Fujishige Algorithm 293
(a) A weakly polynomial algorithm 293
(b) A strongly polynomial algorithm 300
(c) Modification with multiple exchanges 303
(d) Submodular functions on distributive lattices 305
14.2. Schrijver's Algorithm 308
14.3. Further Progress in Submodular Function
Minimization 313
Chapter VII. Discrete Convex Analysis 315
15. Locally Polyhedral Convex Functions and Conjugacy 315
16. L- and L''-convex Functions 319
16.1. L- and L''-convex Sets 319
16.2. L- and L''-convex Functions 322
16.3. Domain-integral L- and L^-convex Functions 326
17. M- and M^-convex Functions 331
18. Conjugacy between L'/L^-convex Functions and M'/M^-convex
Functions 338
19. The Discrete Fenchel-Duality Theorem 341

XIV
20. Algorithmic and Structural Properties of Discrete Convex
Function 344
20.1. L- and iAconvex Functions 344
20.2. M- and M^-convex Functions 345
20.3. Proximity Theorems 351
21. Other Related Topics 356
21.1. The M-convex Submodular Flow Problem 356
21.2. A Two-sided Discrete-Concave Market Model 357
22. Historical Notes 360
References 365
Index 389

Exploring the Variety of Random
Documents with Different Content

A
CAP. XI.
The waye to Babylon whereas the Soudan dwelleth.
ND whoso wyll go through the lande of Babylon where the
Soudan1 dwelleth, to have leave to go more sykerly2 throughe
the Churches & countreys, and to go to mount Sinay before he come
to Hierusalem, and then turne agayne by Hierusalem; he shall goe
from Gaza to the castell Dayre. And after a man commeth out of
Surry, and goeth in the wildernesse, where the waye is full sandy,
and the wyldernesse lasteth eyght Journeys,3 where men findeth all
that them nedeth of vytayles and men call that wyldernesse
Archelleke,4
and whan a man commeth out of this deserte, hee
entreth into Egypte, and they call Egypte, Canopat,5 and in another

language men call it Mersyne,6 and the fyrste goode towne that men
fynde is called Beleth, and it is at the ende of the Kingdome of
Alape,7
and from thence men come to Babylon and to Kayre,8
and in
Babylon is a fayre churche of our lady, where she dwelled vii yeare
when she was oute of the lande of Jewes, for dreade of Kynge
Herode. And there lyeth the bodye of Saynte Barbara vyrgyn, and
there dwelled Joseph whan he was solde of his brethrene, and there
made Nabugodonosor put the children in (the) fire, for they were of
right9 trouth, the which chyldren men call Anania, Azaria, and Misael
(as ye
psalme of Benedicite saith) but Nabugodonosor called them
thus, Sydrac, Mysac, Abdenago, that is to say, God glorious, God
victorious, God over all Kingedomes, and that was for myracle that
he made Goddes sonne, as he sayd, go wyth those chyldren
throughe the fyre. There dwelleth the Soudan, for there is a faire
citie and a stronge castell and it standeth upon a rocke. In that
Castell is always dwellyng to kepe the castell and to serve the
soudan, above viii10
thousand persons or folk that take all theyr
necessaries at the Soudans courte. I should well knowe it, for I
dwelled with him soudiour11 in his warres a great while agayne the
Bedions,12 and he wold haue wedded me to a great princes
daughter ryght richly, if I would haue forsaken my faith.
1: Sultan.
2: Certainly, surely.
3: Day's march.
4: Athylec, Abylech, Alhylet, Alhelet, Abylet.
5: Query Canopus, a city 12 miles from Alexandria, named after the
pilot of Menelaus' vessel, who was buried here.
6: Mersur, Morsyn.
7: Aleppo.

8: Cairo.
9: True faith.
10: Other editions say 6,000.
11: Soldier.
12: Bedouins.

A
CAP. XII.
YET here followeth of the Soudan and of his
Kingdomes that he hath conquered, which he
holdeth strongly with force.
ND ye shall understand that the Soudan is lorde of v Kingdomes:
the which he hath conquered and gotten to him by strength, and
these be they—the Kingdome of Canopate (that is) the Kingdome of
Egipte, the Kingdome of Hierusalem: whereof David and Salomon
were Kings, the Kingdome of Surry, of the which the citie of Damas1
was the chiefe, the Kingdome of Alape in the lande of Dameth, and
the Kingdome of Arabya: which was one of the three Kinges that
made offeryng to our Lorde when he was borne, and many other
landes he holdeth in his hande, and also he holdeth Calaphes2
that
is a great thing to the Soudan, that is to say, among them Roys3 yle
and this vale is colde.

And then men go uppon the mount of Sainct Katherina and that is
much higher than the mount Moyses. And there as saint Katheryn
was graven4
is no church ne castell, ne other dwelling place, but

there is an hyll of stones gathered togither, about the place there
she was graven of Aungels, there was wont to be a chapell, but it is
all cast downe & yet lyeth there a great parte of the stones.
But under the foote of mount Sinay is a monasterie of Monkes, and
there is the church of Sainct Katherine wherein be many lamps
brenning, and they have oyle onlye enough to eate and to brenne,
and that they haue by myracle of God, there come certaine of all
maner of byrdes euery yeare once, lyke pylgrymes and eche of them
bringeth a braunch of olyve in token of offering, whereof they make
much oyle.
1: Damascus.
2: Khalifs.
3: Who are accounted there as kings.
4: Buried.

N
CAP. XIII.
For to returne fro Sinay to Hierusalem.
OW sythen a man hath visited this holy place of Sainct Katheryn
and he will torne to Hierusalem, he shall fyrst take leave of the
Monkes, and recommend him specially to their prayers, then those
Monks will freely giue to Pilgrims victuals to pass through the
Wildernesse to Surry & that lasteth well xiii Journeys. And in that
wyldernesse dwell many Arabyns that men call Bedoins and
Ascoperdes,1 these are folk that are full of all maner of yll
condycyons, and they have no houses, but tentes, the wyche they
make of beastes skinnes, as of camelles and other beastes the
whyche they eate, and thereunder they lye, and they dwell in places
where they maye fynde water, as on the rede sea, for in that
wildernesse is greate defaute of water, and it faileth ofte where a
man findeth water one time, he fyndeth it not another tyme, and
therefore make they no houses in those countreys. These men that I
speake of tyll not the land, for they eate no breade, but2
yf it be
anye that dwelleth neare a goode towne. And they rost al theyre
fishes and flesh upon the hote stones agaynst the sonne, and they
are stronge men and well fyghtynge, and they do nothinge but
chace wyld beastes for theyr sustenaunce, and they sette3 not by
theyr lyves, therfore they dreade not the Soudan nor no prince of all
the worlde. And they haue greate warre wythe the Soudan, and the
same tyme that I was dwelling with him they bare but a shelde and
a speare for to defende them with, and they holde4 none other
armour, but they wynde theyr heades and neckes in a great lynnen
clothe,5
and they are men of full yll kynde.

1: Or Giants from the Arabic askhaf, a tall, big-boned man. It will be
remembered that Sir Bevis of Southampton brought home a Giant
Ascapart—who probably was one of them.
2: Unless.
3: They value not.
4: Have.
5: A turban.

A
CAP. XIIII.
As men are passed this wyldernesse againe comming to
Hierusalem.
ND when men are passed this wyldernesse towarde Hierusalem
they come to Barsabe1 that was sometime a fayre and a lykyng
towne of Christen men, and yet is some of their churches, and in
that towne dwelled Abraham the Patryarke. This towne of Barsabe
founded Uryas wife, of whom David engendred Salomon the wyse
that was Kyng of Hierusalem, and of the xii kindes2
of Israell, and he
raigned xl yeare. And from thence go men to the vale of Ebron, that
is from thence nere xii myle and some call it the vale of Mambre,3
and also it is called the vale of Teeres, for as much as Adam in that
vale he wept a hundred yeare the death of his sonne Abel that
Cayne slew. And Ebron was sometime4 the principall Citie of the
Philistines & there dwelled giaunts & there it was so free, that all
that had done evill in other places were there saved. In Ebron Josue
and5
Calope and theyr felowship came fyrst to espy how they might
wynne the lande of promyssion. In Ebron David raigned fyrst vii
yeare and a halfe & in Hierusalem he raigned xxxii6 yeare and a
halfe, and there be the graves of the Patryarkes—Adam, Abraham,
Jacob and theyr wyves, Eve, Sare, Rebecca7 and they are in the
hanging8
in the hyll. And under them is a right fayre Churche
Kirnelde9 after the facion and maner as it were a Castell, the which
the Sarasins keepe right well, and they haue that place in greate
worship for the holy Patryarkes that lieth there, and they suffer no
Christen men ne Jewes to come therein but they have speciall grace
of the Soudan, for they holde Christen men and Jewes but as
houndes that should come in no holy place, and they call the place
Spelunke10 or double cave or double grave; for one lyeth on another,

and the Sarasins call it in theyr language Caryatharba, that is to say
the place of Patryarkes, and the Jewes call it Arboth. And in that
same place was Abrahams house, and that was the same Abraham
which sat in his dore, and saw three persons and worshipped but
one, as holy wryt witnesseth saying, Tres videt et unum adoravit.
That is to saye, he saw three and worshipped but one, and him took
Abraham into his house.
1: Beersheba.
2: Tribes.
3: Mamre.
4: Formerly.
5: Jehoshua and Caleb (see Numbers, cap. 13).
6: Pynson and others say 33 years and a half.
7: All other editions have "and of Lya," or Leah, who is evidently
here forgotten.
8: Caves cut in the side of the rock.
9: Crenelated or battlemented.
10: Lat. Spelunca, a cave.

A
CAP. XV.
Here foloweth a lyttle of Adam & Eve and other things.
ND right nere to that place is a cave in a Roche where Adam and
Eve dwelled whan they were dryven out of Paradyse, and there
got they theyr chyldren. And in that place was Adam made as some
men saye, for men called sometime that place the felde of
Damasse,1 for it was in the worshippe2 of Damasse; and fro thence
he was translated into Paradyse as they saye, and afterwarde he
was driven out of Paradyse, and put there agayne, for the same
daye that he was put into Paradyse, the same day he was driven
out, for so soone he synned. And there begynneth the yle3 of Ebron
that lasteth nere to Hierusalem, and the Aungell bad Adam that he

should dwell wyth his wyfe, and there they engendred Seth, of the
which kyndred4 Jesu Christ was borne. And in that vale is the felde
where men draw out of the earth a thinge the which men in that
countrey call Chambell and they eate that thinge in the stede of
spyce & they beare it to sell, and men may not grave5
there so
deepe ne so wyde, but it is at the yeares ende full againe up to the
sydes through the grace of God. And two myle from Ebron is the
grave of Loth6 that was Abraham's brother.
1: Damascus.
2: Pynson and others say lordship.
3: Vale.
4: Kindred or tribe.
5: Dig.
6: Lot.

W
CAP. XVI.
Of the dry tree.
HEN a lyttle from Ebron is the mounte of Mambre, of the
which mount the vale toke his name, and there is the tree of
oke that the Sarasins call dypre,1 that is of Abraham's time, that
men call the dry tree. And they say that it hath ben from the
beginning of the worlde, and was sometime grene and bare leaves,
unto the tyme that our Lorde dyed, and so did all the trees in the
worlde, or else they fayled in their heartes, or else they faded, and
yet is there many of those in the worlde. And some prophesies say,
that a lorde or prince of the weste syde of the worlde shall winne
the lande of promission, that is the holy lande, with the helpe of

Christen men, and he shall do singe2 a masse under that tree, and
the tree shall waxe grene and beare fruite and leaves, and through
that miracle many Sarasins and Jewes shal be turned to the Christen
fayth, and therefore they do great worship therto, and kepe it right3
basely. And yet though it be dry, it beareth a great vertue, for
certainly he that hath a lyttle thereof about him, it healeth a
sicknesse called the falling evill, and hath many other vertues also,
and therefore it is holden right precious.
1: Pynson and others read Dyrpe or Dirpe.
2: Cause a mass to be sung.
3: To keep it carefully.

F
CAP. XVII.
Fro Bethlehem.
ROM Ebron men go to Bethlehem in halfe a daye, for it is but five
myle, and it is a fayre waye & thorow1 woddes full pleasaunt.
Bethlem is but a little citie long and narowe, and well walled, and
enclosed with a great diche and it was wont to be called Effrata as
holy wryte sayth Ecce audivimus eum in Effrata &c., That is to saye,
Lo we herde him in Effrata. And toward the ende of the citie toward
the East, is a ryght fayre churche and a gracious and it hath many
toures, pinacles and kirnelles2
full strongly made & within that
Church is xliiii great pyllers of marble & betwene this church the
field3 florished, as ye shall here.
1: Through woods.
2: Battlements.
3: The flowered field.

T
CAP. XVIII.
Of a fayre mayden that should be put to death wrongfully.
HE cause is, for as much as a fayre maiden yt
was blamed wyth
wrong that she hadde done fornication, for the which cause she
was demed1 to dye and to bee brente2 in that place to the which she
was ledde. And as the woode began to brenne about hir, she made
hir prayer to our Lorde as she was not gyltie of that thing, that he
would helpe hir that it might be knowne to all men. And whan she
had thus sayde, she entred the fyre and anone the fyre went out,
and those braunches that were brenninge became red Roses and
those braunches that were not kindled became white Rosiers3
full of
white roses, and those were the fyrst roses and rosyers that any

man sawe, and so was the mayden saved through the grace of God,
and therefore is that felde called the feeld of God florished, for it
was full of Roses. Also besyde the quire of that Church aforesayd at
the right side as men come downwarde xii4 grees5 is the place
where our Lorde was borne that is now full well dyght6
of Marble &
full rychely depaynted of golde, sylver and asure and other colours.
And a lyttle thens by three paces is the crybe7 of the Oxe and the
Asse, and besyde yt
is the place where the sterre8 fell that lede the
three Kinges Jasper, Melchior and Balthasar, but men of Grece call
the Kinges thus, Galgalath, Saraphy, Malgalath. These three Kinges
offered to our Lorde, Encence, Gold & Mirre and they came together
through myracle of God, for they mette togither in a citie that men
call Chasak, that is liii journeys from Bethleem, and there they were
at Bethleem the fourth9
daye after they hadde seene the sterre. And
under the cloyster of this church xviii grees10 at the righte syde is a
great pytte where the bones of the Innocentes lie, and before that
place where Chryst was borne is the tombe of Sainct Jerom that was
a priest and a Cardinal that translated the Byble and the Sauter11
from Hebrew into Latyn, and beside that church is a Church of
Saynte Nycolas, where our Lady rested hir whan she was delivered
of chyld, and for as much as she had so much mylke in hir pappes
that it greved hir, she mylked it out uppon the redde stones of
Marble, so that yet may the traces bee seene whyte uppon the
stones. And ye shall understande that all that dwell in Bethleem are
Chrysten men, and there are fayre vynes all aboute the citie and
great plentie of wine, for their booke that Mahomet betoke12
them,
the which they call Alkaron and some call it Massap and some call it
Harme, forbiddeth them to drinke any wyne, for in that booke
Machomet curseth all those that drynke of that wyne and all that sell
it, for some men saye that he onse slewe a good hermite in his
dronkennesse which13 he loved much, and therefore he cursed the
wyne, and them that drynke wyne, but his malyce is torned to
hymselfe, as holye writ sayth "Et in verticem ipsius iniquitus ejus
descendit," That is to say in Englyshe, His wickednesse shall
descende on his owne head. And also the Sarasins bringeth forthe

no geise,14 ne they eate no swines fleshe, for they say it is brother
to manne and that it was forbidden in the olde lawe. Also in the
lande of Palestine ne in the lande of Egypte they eate but lyttle veale
and beefe but it be so olde that it may no more travayll15
ne werke,
not that it is forbidden but they kepe them to tylling of their lande.
In this castell of Bethleem was Kyng David borne and he had Lx
wives and ccc lemmans. From Bethleem to Hierusalem is two myle,
and in the way of Hierusalem halfe a myle from Bethleem is a
Church where the aungell sayd to the shepherdes of the bearing of
Christ. In that waye is the tombe of Rachel that was Josephs mother
the Patryarke and she dyed as soone as she hadde borne Benjamyn
and there she was buried, and Jacob hir husbande set xii great
stones upon hir in tokening that she had borne xii children. In this
way to Hierusalem are many Christen churches by the which men go
to Hierusalem.
1: Condemned.
2: Burnt.
3: Rose bushes.
4: Other editions say 16.
5: Steps.
6: Adorned.
7: Crib or Manger.
8: Star.
9: Other editions say "thirteenth."
10: Paces.
11: Psalter.
12: Gave.

13: Whom.
14: Breed no pigs.
15: Plough or draw loads.

F
CAP. XIX.
Of the citie of Hierusalem.
OR to speake of Hierusalem, ye shall understande that it
standeth fayre among hylles, and there is neither ryver nor well,
but water commeth by conduit from Ebron, and ye shall wete that
men called it first Jebus and sythen it was called Salem unto the
time of King David, and he set those two names togither and called
it Hierusalem and so it is called yet. And aboute Hierusalem is the
Kingdome of Surry, & thereby is the lande of Palestyne and Askalon,
but Hierusalem is in the lande of Jude, and it is called Judee, for
Judas Maccabeus was King of that lande, and also it marcheth
afterward on the Kingedome of Araby, on the South side on the
lande of Egipt, on the west side on the great sea, on the north syde
on the Kingdome of Surry and the sea of Cipres. About Hierusalem
are these cities. Ebrone at viii1 myle, Jerico at vi myle Barsebe at viii
myle Askalon xviii2
myle, Jaffa at xxv3
Ramatha at iiii4
mile. At
Bethlem towarde the South is a church of saint Markerot,5 that was
abbot there, for whom they made much sorow when he should dy &
it is painted there how they made dole6 when he dyed, and it is a
piteous thing to beholde. This lande of Hierusalem hath ben in
dyvers nations hands, as Jewes, Cananens, Assyrians, Percians,
Macedons, Grekes, Romayns & Chrysten men, Sarasins, Barbaryans,
Turkes & many other nacions. For Chryste wyll not that it be long in
the handes of traytours ne sinners be they Christen or other. And
now hath the mistrowing7
men holden that lande in theyre handes
Lx yeare & more, but they shall not holde it long and if8 God wyll.
1: Other editions say respectively 7, 17, 16.

2: As Footnote 1.
3: As Footnote 1.
4: Other editions say 3 miles.
5: Variously written, Markertot, Karitot, Karscati, and Mercaritot.
6: Grieved, from Lat. Dolor.
7: Unbelieving, or heathen.
8: Unless it is God's pleasure.

A
CAP. XX.
Yet of the holy citie of Hierusalem.
ND ye shall understand that whan men fyrst come to
Hierusalem, they go fyrste a pylgrimage to the Church, where
that the holy grave is, the whiche is out of the citie on the North
syde, but it is now closed in with the wall of the towne, and there is
a full fayre church rounde, all open aboue, and well covered with
leede and on the west syde is a fayre toure and a strong for belles.

And in the middes of the church is a Tabernacle made like a little
house, in maner of halfe a compasse, ryght well and richly of gold
and asure and other coloures well dyght & on the ryght syde is the
sepulchre of oure Lorde, and the tabernacle is viii foote long and v
fote wide and xi fote of height. And it is not longe sythen the
sepulchre was all open, yt
men might kysse it and touche it: but for
men that came thether payned1
them to breake the stones in peces
or pouder, therefore the Soudan hath made a wall about the
sepulchre that no man may touch it. On the lefte syde is no
wyndowe, but therein is many lampes light, and there is a lampe
that hangeth before the sepulcre lyght brenning and on the fryday it
goeth oute by itselfe, and lyghteth againe by itselfe at the houre that
our Lorde rose from death to life. And within that church upon that
right side is the mount Calvary, where our Lord was done on the
crosse, and the crosse was sette in a morteys2 in the roche that is
white of coloure, and a lyttle redde medled3 with, and upon that
roche dropped the bloude of the woundes of our Lord whan he was
pained on the crosse & that is called Golgatha and men go up to
that Golgatha upon greces.4
And in that mortays was Adams head
founde after Noyes flood, in token that the sinnes of Adam, shoulde
bee boughte in the same place, and aboue that roche made
Abraham sacryfice to our Lord, and there is an auter,5 and before
that auter lyeth Godfry of Boleyn,6 Bawdewyn7 and other that were
Christen and kinges of Hierusalem. And ther as our Lord was done8
upon the crosse, is thys wrytten in greke, Otheos9 basylon, ysmon
persemas, ergaste sothyas oyos. That is to say in latine, Hic Deus
Rex noster ante secula operatus est salutem in medio terræ, That is
to saye, This God our King before worldes, hath wrought healthe in
the myddes of the earth. And also upon the roche where the crosse
was fyxed is wrytten within the roche, Eros10 guyst basys, thou,
pestes, thoy, thesmoysy. That is to say in latin, Quod vides est
fundamentum totius mundi et hujus fidei. And it is to saye, that thou
seest, is grounde of all the world and of this faith. And ye shall
understande that our Lorde whan he dyed was thyrty & two11
yeare

olde and three monethes and the prophecie of David sayth that he
should haue xl yeares, when he saith thus. Quadraginta annis
proximus fui generatione huic, that is to say, fourtie yeare was I
neighbour to this kinde, and thus it should seme that prophecie were
not sothe,12 but it is. For in olde time men called yeares of ten
monethes of the which Marche was the fyrst & December the last.
But Cayus Cezar13
that was Emperour of Rome dyd sette to these
two moneths Januarie & Februarie and ordeyned the yere of xii
months. That is to say ccc14 dayes without lepe yere the proper
course of the sonne and therefore after the accompting of x
moneths to the yeare, he dyed in xl yeare and three moneths.
Also within mounte Calvary at the ryghte syde is there an aulter
where the piller lyeth that our Lorde was bound to when he was
scourged and thereby are three15 other pyllers that alwaye drop
water, and some saye that those pyllers weepe for our Lords death,

and neare this aulter in a place xlii grees16 depe was founde the
verye crosse by the assent17
of sainct Eleyn18
under a roch where
the Jewes had hydde it and it was assayed, for they founde three
crosses, one of our Lorde and two of theves. And Saint Elene
assayed them on a dead body that rose as sone as the very19 crosse
of our Lorde was laid on him. And thereby, in the vale, is the place
where the foure nayles of our Lord were hyd, for he had two nayles
in his handes and two in his fete, and one of those nailes the
Emperour of Constantinople20 dyde make a bridell for his horse to
beare him in bataile, for by21
the vertue that it had, he overcame his
enimies, and wan22 all the land of Asye, Turky, Damasse the more23
and the lesse, Surrey and Hierusalem, Araby, Percy, and
Mesopotamy, the Kingdome of Alabe,24 Egipt the high and the lowe,
and other kingdomes many full nyghe all unto Ethyope the low, and
also unto Inde the lesse, that then was chrysten. And there was in
that tyme many good men and holy hermits, of whome the booke
of25
the fathers lyves speaketh, and they are now in Paynims &
Sarasins handes, but whan God will righte26 as these landes are lost
through sinne of Christen men, so shall they be won againe by
christen men throygh the helpe of God. And in the myddes of this
Church is a compasse,27 in the which Joseph of Armath28 layd the
body of our Lord whan he had taken him of29
the crosse & upon the
same place dyd he wash the fete of our Lorde, & that compasse
men say is in the myddes of the world.
1: Tried hard.
2: Mortise.
3: Mixed.
4: Steps.
5: Altar.
6: Bouillon.

7: Baldwin.
8: Placed.
9: Should read Ὁ θεὸς βασιλεὺς ἡμῶν πρὸ αἰώνων εἰργάσατο
σωτηρίαν ἐν μέσῳ τῆς γῆς.
10: Should read Ὃ εἲδες, ἐστὶ βάσις τῆς πίστεως ὅλης τοῦ κόσμου
τούτου.
11: Other editions have 33 years and 3 months.
12: Sooth, true.
13: Caius Julius Cæsar.
14: Other editions give the proper number of days, viz., 365.
15: Other editions say four, which is the number represented in the
engraving.
16: Paces.
17: Perception, or sagacity. Lat., sensus.
18: Helena, mother of Constantine.
19: True, veritable.
20: Another is said to be incorporated in the so-called Iron Crown of
Lombardy. Guisto Fontanini, Archbishop of Ancyra, gives a list of
twenty-three places claiming to have a nail—Venice having three. M.
Rohault de Fleury gives six more—whilst, according to tradition,
Helena sent two to her son, and threw one in the sea to still a storm,
thus leaving but one to meet all demands.
21: Through.
22: Won or conquered.
23: Greater.
24: Aleppo.
25: The Vitæ Sanctorum Patrum, many old printed copies of which
exist.

26: When God thinks fit.
27: A linen swathing-band.
28: Arimathæa.
29: Off.

I
CAP. XXI.
Of the church of the holy sepulchre.
N that Churche of the sepulcre on the north syde is the place
where our Lord was done1 (in) prison, and there is a part of the
cheyne with which he was bound, and there he appeared fyrst to
Mary Magdeleyne when he was risen from death and she trowed2
that he had bene a gardeiner. In the Church of the sepulcre was
wont to be3 Chanons of sainct Benet and they had a pryour; but the
Patryarke was theyr soveraigne.
And without the dores of the Churche on the righte syde as men go
up xviii grees,4 our Lorde sayde to his mother5 Ecce filius tuus. That
is to say, Woman beholde thy sonne, De inde dixit discipulo, Ecce
mater tua. That is to say, Then said he to his disciple, Behold thy
mother.6
And these wordes he sayde when he hanged upon the
crosse. And upon these greces went our Lorde when he bare the
crosse uppon his shoulder, and under these greces is a Chappell
where the priestes synge, but not after our lawe, and alway they
make theyr Sacrament of the aulter of bread, say Pater noster &c.,
and other prayers, as with the which thing they say the wordes of
whome the sacrament is made, for they know not of the addicions
that many Popes haue made but they singe in good devocion. And
nere there is the stone wher our Lord rested him when he was wery
for bearing of the crosse. And ye shall understand that before the
Churche of the Sepulcre is the citie most strong7 for the great playne
that is betwene the citie & the church; on the East side without the
walles of the citie is the vale of Josaphat that commeth to the
walles. In that vale of Josaphat without the citie, is the churche of
sainct Stephen where he was stoned to death, and thereby is the

gate gylted that may not be opened. Through this gate our Lord
entred on palme Sonday upon an asse, and the gate opened against
him whan he would go to the Temple, and yet are the steppes of the
asse sene in three places the which stand8 in full harde stones.
Before the churche of the sepulcre two hundred paces, is a great
hospitall of Sainct John, in the which hospytall are liiii pyllers made
of stone.
And to go towarde the East from the hospitall is a righte fayre
churche that men call our lady the greate, and then is there another
church after that, that men call our lady of the latyn,9 and there it
was Mary Cleophe and Magdeleyne drew10
theyr here whan oure
Lord was put to death.
1: Put.
2: Thought or believed.
3: Were formerly Canons of the Order of St. Benedict.

4: Should be greces or steps.
5: The printer has omitted the word "Mulier ecce," &c.
6: Gospel according to St. John, cap. 19, vv. 26, 27.
7: Pynson says, "most wake" or weak, and other editions say,
"feeble."
8: Pynson has this passage: "The wyche are full of harde stones."
9: Pynson says "Nostre dame de Vatyns."
10: Tore.

A
CAP. XXII.
Of the Temple of God.
ND from the churche of the sepulcre towarde the East at xviii1
paces is Templum Domini. That is a fayre house and it is all
rounde and ryghte high & covered with leed,2
and it is well paved
with white marble, but ye
Sarasins wyl suffre no christen men ne
Jewes to come therein, for they say that so3 foule men should not
come into that holye place, but I came therein and in other places
where I woulde, for I had letters of the Soudan, wyth hys great seal,
and, commonly, other men but have of his signet, and men beare
hys letter with his seale before them hanginge on a speare, and men
do great worship thereto, and kneele against4
it as it were against
God's body: for those men that it is sent to, before they take it, they
encline5 thereto and then they take it, and laye it upon their heads,
and afterward they kisse it, and then they reade it, all enclining with
great worship, and then they profer6 them to do all that the bringer
will. And in this Templum Domini were wont to be Chanons regulers,
and they had an Abbot to whome they were obedient, in this Temple
was Charlemaine when the Aungell brought him the prepuis of our
Lorde when he was circumsised, and after King Charles brought it to
Acon7
into our Ladies Chapell.

1: Other editions say 160 paces.
2: Lead.
3: Such unclean.
4: Before.
5: Bow.
6: Proffer or offer.
7: Pynson and others say Paris.

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