Eule , Rica do; Lindne , Niels; Bo ndö e , Ral
A icle
P ice op imal ou ing in public anspo a ion
EURO Jou nal on T anspo a ion and Logis ics (EJTL)
P o ided in Coope a ion wi h:
Associa ion o Eu opean Ope a ional Resea ch Socie ies (EURO), F ibou g
Sugges ed Ci a ion: Eule , Rica do; Lindne , Niels; Bo ndö e , Ral (2024) : P ice op imal ou ing
in public anspo a ion, EURO Jou nal on T anspo a ion and Logis ics (EJTL), ISSN 2192-4384,
Else ie , Ams e dam, Vol. 13, Iss. 1, pp. 1-15,
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EURO Jou nal on T anspo a ion and Logis ics
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P ice op imal ou ing in public anspo a ion
Rica do Eule ∗, Niels Lindne , Ral Bo ndö e
Zuse Ins i u e Be lin, Takus aße 7, Be lin, 14195, Be lin, Ge many
ARTICLE INFO
Keywo ds:
Mul i-objec i e sho es pa h
Fa e s uc u e
Public anspo a ion
Monoid
Condi ional a e ne wo k
Ticke g aph
RAPTOR
ABSTRACT
We conside he p ice-op imal ea lies a i al p oblem in public ansi (POEAP) in which we aim o calcula e he
Pa e o-se o jou neys wi h espec o icke p ice and a i al ime in a public anspo a ion ne wo k. Public
ansi a e s uc u es a e o en a combina ion o a ious a e s a egies such as, e.g., dis ance-based a es, zone-
based a es o la a es. The ules ha de e mine he ac ual icke p ice a e o en e y complex. Acco dingly,
a e s uc u es a e no o iously di icul o model, as i is in gene al no su icien o simply assign cos s o
a cs in a ou ing g aph. Resea ch in o POEAP is sca ce and usually ei he elies on heu is ics o only conside s
es ic i e a e models ha a e oo limi ed o co e he ull scope o mos eal-wo ld applica ions. We he e o e
in oduce condi ional a e ne wo ks (CFNs), he i s amewo k o ep esen ing a la ge numbe o eal-wo ld
a e s uc u es. We show ha by elaxing label domina ion c i e ia, CFNs can be used as a building block in
label-se ing mul i-objec i e sho es pa h algo i hms. By he na u e o hei ex ensi e modeling capabili ies,
op imizing o e CFNs is NP-ha d. Howe e , we demons a e ha adap ing he mul i-c i e ia RAPTOR (McRAP)
algo i hm o CFNs yields an algo i hm capable o sol ing POEAP o op imali y in less han 400 ms on a e age
on a eal-wo ld da ase . By es ic ing he size o he Pa e o-se , unning imes a e u he educed o below
10 ms.
1. In oduc ion
The desi ed shi o mo e sus ainable means o anspo a ion neces-
si a es an inc ease in he modal sha e o public anspo a ion sys ems.
Recen s udies show ha discoun ed o e en ee a es signi ican ly in-
c ease such a sys em’s adap a ion (B ough e al.,2022;Bull e al.,2021;
Chen e al.,2020), indica ing ha , om a a ele ’s pe spec i e, he
design o a es and icke p ices a e key pa ame e s o os e i s a ac-
i eness. This e ec , howe e , a ies be ween socioeconomic g oups:
While many ide s may alue as connec ions wi h ew ans e s, low-
income ide s a e mo e likely o choose a mo e a o dable mode o
public anspo a ion o o e en abs ain om using anspo a ion a all
i hey deem a es o be oo expensi e (Blumenbe g and Ag awal,2014;
Rosenblum,2020). This unde lines he need o ou ing algo i hms
ha enable passenge s o selec jou neys op imal wi h espec o hei
indi idual needs. Since hese needs a e a ely known, we p esen an
app oach ha calcula es Pa e o se s o op imal jou neys wi h espec
o he ea lies a i al ime, he numbe o ans e s and cos . I is hen
on he use o choose om he op ions p esen ed he one mos sui ed
o hei pe sonal needs.
Ticke p ices a e de e mined by he public ansi p o ide s’ a e
s uc u e. Following Fleishman e al. (1996) a a e s uc u e is ‘‘ he
combina ion o one o mo e a e s a egies wi h speci ic icke s’’ while
he e m a e s a egy e e s o a ‘‘gene al a e collec ion and paymen
∗Co esponding au ho .
E-mail add ess: [email p o ec ed] (R. Eule ).
s uc u e app oach’’. This can be, e.g, a dis ance-based o zone-based
a e, a sho -dis ance discoun , a la a e o a su cha ge. Un il now, a
uni ying amewo k o he algo i hmic ea men o public ansi a e
s uc u es has been lacking. Some a e s a egies, o example dis ance-
based a es, can be add essed easily using label-se ing sho es pa h
algo i hms. Finding a jou ney ha c osses he leas amoun o a e
zones, howe e , is NP-ha d (Blanco e al.,2016) and canno be modeled
using eal- alued a c weigh s. In gene al, he subpa h op imali y p in-
ciple does no hold o a e s uc u es and hus label-se ing algo i hms
canno be di ec ly applied o POEAP. Conside he ollowing example:
A a ele akes a de ou ha passes h ough an addi ional a e zone
o a oid paying he su cha ge o a special connec ion (e.g., a e y).
I is no unlikely ha , a a la e poin , he su cha ge has o be paid
ega dless (e.g., because he a ge s op can only be eached ia a
e y). In ha case, aking he de ou was a subop imal decision and
using he co esponding label o p une o he pa ial jou neys b eaks
he op imali y gua an ee o sho es pa h algo i hms.
1.1. Ou con ibu ion
We de ise algo i hms ha sol e he p ice-op imal ea lies a i al
p oblem (POEAP) e icien ly in p ac ice. To his end, we build upon
h ps://doi.o g/10.1016/j.ej l.2024.100128
Recei ed 21 Ma ch 2023; Recei ed in e ised o m 20 Decembe 2023; Accep ed 8 Feb ua y 2024
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100128
A ailable online 13 Feb ua y 2024
2192-4376/© 2024 The Au ho s. Published by Else ie B.V. on behal o Associa ion o Eu opean Ope a ional Resea ch Socie ies (EURO). This is an open access
a icle unde he CC BY-NC-ND license ( h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/ ).
R. Eule e al.
he s a e-o - he-a public anspo ou ing algo i hms McRAP (Delling
e al.,2015) and Tigh -BMRAP (Delling e al.,2019), which combine
an in elligen enume a ion scheme wi h dominance checks. Ou domi-
na ion ules a e based on condi ional a e ne wo ks (CFN), a no el and
lexible amewo k o modeling a e s uc u es o public anspo a-
ion p o ide s capable o aking mos unde lying a e s a egies in o
accoun . A CFN models a e s uc u es as a icke g aph ep esen ing
ela ions be ween icke s. T ansi ions be ween di e en icke s a e mod-
eled as di ec ed a cs and usually depend on a numbe o addi ional
pa ame e s such as, e.g., a e zones o he a eled dis ance. These
a e modeled ia pa ially o de ed monoids and e en s. Fa e s a e-
gies ha can be exp essed ia CFNs include (bu a e no limi ed o):
zone-based a es, dis ance-based a es, su cha ges o special ehicles
o nigh line s, discoun ed sho -dis ance a es, ans e a es and all
combina ions he eo . We de elop domina ion ules o CFNs based on
pa h ela ions in he icke g aph ins ead o he p ice alone. This allows
us o e ain subpa h op imali y and p o e ha using hese ules in label-
se ing MOSP algo i hm does in ac yield lowes -p ice jou neys. By
u he es ic ing he size o he Pa e o-se , we sol e POEAP in less
han en milliseconds o e he in ica e a e s uc u e o a mid-sized
public ansi p o ide om Ge many.
1.2. Rela ed li e a u e
The e is ample esea ch on ou ing p oblems in public ansi ne -
wo ks. Fo an o e iew, see Bas e al. (2016). Delling e al. (2015)
in oduced he RAPTOR algo i hm o e y as public ansi ou ing.
We e e occasionally o ypical label-se ing mul i-objec i e sho es
pa h algo i hms, by which we mean, e.g., McRAP (Delling e al.,2015),
Ma ins’ algo i hm (Ma ins,1984), and ecen ly Mul i-Objec i e Di-
jks a (Ma is any de las Casas e al.,2021). Delling e al. (2019)
in oduce a e sion o McRAP, Tigh -BMRAP, o compu ing es ic ed
Pa e o-se s.
The concep o elaxed subpa h op imali y is discussed in Be ge and
Mülle -Hannemann (2009). Las ly, we e e o Disse e al. (2008) on
how o model public ansi sys ems wi h ans e s in ime-dependen
g aphs.
In con as o he gene al ac i i y o he ield, li e a u e on p ice-
op imal ou ing is gene ally a he sca ce. This is ce ainly due o
he usually in ica e na u e o public ansi a e s uc u es. Mos ap-
p oaches deal wi h a es on a heu is ic basis o only conside a e y
na ow se o a e s a egies. Mos no ably, Mülle -Hannemann and
Schnee (2005) s udy a e s uc u es ha en ail dis ance- and ela ion-
based p ices, i.e., s uc u es ha a e usually associa ed wi h long-
dis ance public anspo a ion. They app oxima e a es by assigning
a ixed p ice o e e y a c. This app oach, howe e , does no accoun
o a e s a egies such as, e.g., a e zones and sho -dis ance discoun
icke s. Bo h a e usually mo e p ominen in local public anspo a ion.
Reinha d and Pisinge (2011) conside op imizing he numbe o
a e zones as a special case in hei s udy o non-addi i e objec i e
unc ions in (mul i-c i e ia) sho es pa h p oblems. Howe e , hei
app oach elies on a ge p uning as he sole domina ion echnique,
so ha pa ial pa hs canno be p uned un il a 𝑠,𝑡-pa h is known. In
con as , ou app oach applies o mo e gene al a e s uc u es, includes
a ge p uning, bu also allows o p uning a ea lie s ages.
Schöbel and U ban (2021) iden i ied condi ions unde which p ice-
op imized ou ing is ac able o zone- and dis ance-based a e s a e-
gies. Addi ionally, hey iden i y he no-elonga ion and no-s opo e
p ope ies as desi able p ope ies o a e s uc u es. This ollows a
line o esea ch conce ned wi h he design o a e s uc u es. Fo a
e iew o ecen wo k, see Schöbel and U ban (2021). Blanco e al.
(2016) showed zone-based a es o esul in NP-ha d ou ing p oblems
i een e ing a zone does no en ail addi ional cos s. The p oo elies on
a educ ion o he p oblem o inding a pa h wi h a minimum numbe
o colo ed edges (B oe sma e al.,2005). I was gi en in he con ex o
ligh ajec o y op imiza ion wi h o e ligh cos s (Blanco e al.,2017).
A e sion adap ed o public anspo is gi en by Schöbel and U ban
(2021). Delling e al. (2015) used RAPTOR o compu e jou neys ha
ouch he smalles numbe o a e zones. Recen ly, Gündling (2020)
conside ed p ice-op imized ou ing in in e modal anspo a ion. They,
howe e , only conside ed mileage-based and la a es.
Ou app oach combines ideas om au oma a heo y and op imiza-
ion o e monoids. Fo an o e iew o au oma a heo y, see Hopc o
and Ullman (1979). The icke g aph concep is inspi ed by he ap-
plica ion o ini e au oma a o he language-cons ained sho es pa h
p oblem (Ba e e al.,2000). I is di e en , howe e , in ha i se es
o e alua e pa hs ins ead o es ic ing he se o easible pa hs. Fu he -
mo e, ou app oach also co e s a es based on nume ical pa ame e s
ha a e no exp essed as pa o a o mal language. Finally, i has been
known o a while ha sho es pa h algo i hms can be gene alized o
o de ed monoids (Zimme mann,1981) and semi ings (Moh i,2002)
in a s aigh o wa d ashion. Recen ly, monoids we e also p oposed
as a gene al cons ain model o (single-c i e ia) esou ce cons ained
sho es pa h p oblems (Pa men ie ,2019).
This pape is an ex ended and imp o ed e sion o wo k p esen ed
a he ATMOS’19 con e ence (Eule and Bo ndö e ,2019). Apa om
s eamlining he p esen a ion and p oo s, he ollowing addi ions we e
made: We now be e mo i a e he in e play be ween he monoid and
he icke g aph. We p opose a supe io app oach o dealing wi h
o e lap a eas. We p o ide an adap ion o he ecen Tigh -BMRAP
algo i hm o ou use case. This leads o an imp o emen in algo i h-
mic pe o mance o up o wo o de s o magni ude compa ed o he
p e ious esul s in Eule and Bo ndö e (2019). Finally, we p o ide
a complexi y analysis and in es iga e he ela ion o ou app oach o
au oma a heo y in Appendix A.
Ticke g aphs o a ious Ge man public ansi p o ide s can be
ound in Bo ndö e e al. (2018,2021).
1.3. O e iew
In Sec ion 2, we in oduce he a e s uc u e o MDV, an associa ion
ha is esponsible o he public ansi a es o a ious ope a o s in
he Leipzig–Halle egion o Ge many. This a e s uc u e will se e as
a unning example o he es o he pape . We p esen condi ional
a e ne wo ks in de ail in Sec ion 3and show how hey can be used
o model a ious aspec s o a e s uc u es. The algo i hmic ea men
o a es and domina ion ules is laid ou in Sec ion 4. Sec ion 5
discusses how he mul i-c i e ia RAPTOR and Tigh -BMRAP algo i hms
can be modi ied o use CFNs o p ice-op imal sea ch. An e alua ion
o he amewo k’s pe o mance is conduc ed in Sec ion 6using he
ne wo k and a e s uc u e o MDV. Sec ion 7concludes he pape wi h
some closing ema ks. In Appendix A, we p o ide a supplemen a y
complexi y analysis o POEAP and explo e links o au oma a heo y.
2. Running example: MDV
We in oduce he eade o some in icacies o a e s uc u es in
public ansi using he example o Mi eldeu sche Ve keh s e bund
(MDV) (Mi eldeu sche Ve keh s e bund GmbH,2019a). Th oughou
Sec ions 3and 4, he MDV a e s uc u e will se e as a unning
example o illus a e ou co e concep s. A schema ic depic ion o MDV’s
a e plan is gi en in Fig. 1.
Example 2.1 (The a e sys em o MDV).MDV’s a ea o ope a ions co e s
la ge u al a eas in Eas e n Ge many, as well as he conu ba ion o
Halle and Leipzig. As o 2019, his a ea is di ided in o a se o 56
pai wise disjoin a e zones. In mos cases, he p ice depends on he
numbe o isi ed a e zones: he e a e p ice le els o one o six a e
zones. We deno e he espec i e icke s by 𝑍𝑖wi h 𝑖∈ [6]. Fo example,
a eling om s a ion o s a ion in Fig. 1 equi es icke 𝑍4. Fo all
pa hs co e ing mo e han six a e zones, a icke o MDV’s whole a ea
o ope a ions has o be pu chased, which we deno e by 𝑀. The wo
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100128
2
R. Eule e al.
Fig. 1. A sec ion o MDVs a e plan wi h wo lines and six a e zones. Two o he a e zones, colo ed in ligh g ay, a e he ci ies o Halle and Leipzig. Ve ically ha ched hexagons
ep esen o e lap a eas ha can be coun ed as ei he o he neighbo ing zones. The ho izon ally ha ched ci cle ep esen s he small ci y Me sebu g in which a special discoun ed
a e is applicable. Small black nodes ep esen public ansi s ops. Foo pa hs a e indica ed by do ed lines.
la ge ci ies, Halle and Leipzig, each o m a single a e zone. T a eling
in hese zones equi es special icke s mo e expensi e han 𝑍1. These,
we deno e by 𝐻and 𝐿, espec i ely.
Fo all pa hs ha pass h ough mul iple a e zones, hey, howe e ,
coun as no mal zones, i.e., one o he icke s 𝑍2,…, 𝑍6, 𝑀 is applied.
Hence, he pa hs −and −incu icke s 𝐻and 𝐿, espec i ely,
while he pa h −incu s 𝑍2. Se e al smalle ci ies a e pa o la ge
a e zones, bu allow o discoun ed a es (ci y a es) when a eling
only in ha ci y. Fo each ci y 𝑐in he se o such ci ies 𝐶, we deno e
he icke by 𝐶𝑐. The pa h −in Me sebu g (𝑚), hence, equi es he
icke 𝐶𝑚. When ex ending he pa h o s op , he icke 𝑍1becomes
applicable. As o 2019, he e a e 17 ci ies wi h ci y a es and wo p ice
le els (which we deno e by 𝐶1and 𝐶2). Fo pa hs s a ing in Halle and
Leipzig, he e a e discoun ed icke s o sho ips (𝐷𝐻and 𝐷𝐿), which
can be used o a maximum numbe o ou s ops wi hou ans e s.
Hence, pa hs −and −a e admissible o discoun ed icke s 𝐷𝐻
and 𝐷𝐿, espec i ely, while pa hs −and −a e no . Discoun ed
icke s also exis o o he zones (𝐷). These a e a li le cheape and
depend on he leng h o he jou ney (4 km maximum) ins ead o he
numbe o isi ed s ops. Some imes i is possible o choose be ween ci y
a es and leng h-based discoun s. In his case, he ci y a e is applied
because i is cheape . To no unduly bu den people li ing a he bo de s
o a e zones, MDV uses o e lap a eas. These can be coun ed as pa
o ei he o hei adjacen a e zones, whiche e is mos bene olen o
he a ele . Fo example, when a eling om o , all s ops a e
coun ed as pa o a e zone 162 and hus icke 𝑍1would be applicable.
When a eling om o ,coun s as pa o he a e zone 233 bu
in he pa h −i coun s as pa o Halle. Hence, icke s 𝑍1and 𝐻
a e applicable, espec i ely.
While we co e he mos impo an ea u es o he a e s uc u e,
we do igno e some edge cases and explici excep ions. These a e
among o he hings: Sligh ly di e en discoun ules o speci ic ains,
coun ing s a ions ha a e passed wi hou a s op o discoun ed icke s
and excep ions o a speci ic unnel. This is done in pa because
hey a e no p ope ly e lec ed in ou da ase , and in pa o simpli y
p esen a ion.
3. A o mal amewo k o a e s uc u es
Conside a (di ec ed) ou ing g aph 𝐺= (𝑉 , 𝐴), in which a cs
ep esen ei he public anspo connec ions, oo pa hs, o ans e s
be ween lines and/o modes o anspo a ion. Public ansi jou neys
can hen be in e p e ed as pa hs in 𝐺. In he ollowing, we will conside
a ime-dependen o mula ion as p esen ed, o example, by Disse e al.
(2008), i.e., we a e gi en a ime-dependen FIFO a el ime unc ion
𝑐(𝑎) ∶ 𝐼→𝐼on each a c 𝑎∈𝐴, whe e 𝐼is he se o ime poin s.
In 𝐺, e e y pa h 𝑝is associa ed wi h a icke 𝜏∈𝑇 om a icke se
𝑇 ha has o be bough o use 𝑝. Each icke has a co esponding p ice
𝜋(𝜏) ∈ Q+. In he ollowing, we migh also w i e 𝜋(𝑝)ins ead o 𝜋(𝜏)i
𝜏is he icke associa ed wi h 𝑝. The icke o a pa h is de e mined by
he a e s uc u e.
We aim o sol e he p ice-op imal ea lies a i al p oblem (POEAP).
We e e o De ini ion 3.6 o a p ecise de ini ion, bu he essence
is ha , o gi en 𝑠, 𝑡 ∈𝑉, we wan o ind a Pa e o-se o 𝑠, 𝑡-pa hs
𝑃∗
𝑠,𝑡 ⊆ 𝑃𝑠,𝑡 wi h espec o a i al ime and icke p ice in 𝐺. He e, 𝑃𝑠,𝑡
deno es he se o all 𝑠, 𝑡-pa hs in 𝐺.
Ideally, we wan o sol e POEAP by aking ad an age o he exis ing
li e a u e on label-se ing MOSP algo i hms. Ticke p ices, howe e ,
usually canno be modeled ia eal- alued FIFO unc ions on a cs.
Hence, a amewo k o a e s uc u es is needed ha allows us o label
pa hs in a way ha (a) labels can be upda ed quickly when a new a c is
elaxed, (b) dominance ela ionships be ween labels can be es ablished
ha espec he subpa h op imali y p ope y.
3.1. Modeling wi h monoids
No e ha in Example 2.1, he p ice o a pa h depends on se e al pa-
ame e s: he numbe o isi ed s a ions, he o al dis ance a eled, he
se o isi ed a e zones and on indica o s epo ing whe he ans e s
we e made o whe he he pa h c ossed ci y bo de s.
All hese pa ame e s sha e se e al key p ope ies: Fi s , he e is a
na u al pa ial o de on hem, indica ing which con igu a ion equi es
a mo e expensi e icke . Fo example, o a e zones 𝐴, 𝐵, 𝐶 we ha e
{𝐴, 𝐵}⊂{𝐴, 𝐵, 𝐶}; o dis ances and ans e s i is he canonical
o de on N. The pa ame e s can be summed up along a pa h using
an app op ia e no ion o addi ion. Fo dis ances, his is he no mal
addi ion o na u al numbe s; o a e zones, i is he union o se s; we
can use he logical OR (∨) on he se {0,1} o indica o s. Finally, we
can assume he exis ence o a neu al elemen o e e y pa ame e .
The abo e p ope ies sugges ha he s uc u e o a pa ially o -
de ed posi i e monoid is an app op ia e model o a la ge numbe o
a e- ele an pa ame e s.
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100128
3
R. Eule e al.
De ini ion 3.1 (Pa ially o de ed monoid).Amonoid (𝐻, +) is a se 𝐻
oge he wi h an associa i e ope a ion +(called addi ion) and a neu al
elemen 𝟎∈𝐻, i.e., ℎ+𝟎=ℎ∀ℎ∈𝐻. We call (𝐻, +,≤)apa ially
o de ed monoid i ≤is a pa ial o de on 𝐻 ha is ansla ion-in a ian
wi h espec o he monoid ope a ion +, i.e., ℎ1≤ℎ2⇒ℎ1+𝑥≤ℎ2+
𝑥∀ℎ1, ℎ2, 𝑥 ∈𝐻. I addi ionally 𝟎≤ℎ∀ℎ∈𝐻, we call (𝐻, +,≤)a
pa ially o de ed posi i e monoid.
No e ha we can de ine he c oss-p oduc o wo pa ially o de ed
monoids (𝐻1,+1,≤1)and (𝐻2,+2≤2)by (𝐻1×𝐻2,+12,≤1,2), whe e
(ℎ1, ℎ2)+1,2(𝑖1, 𝑖2) ∶= (ℎ1+1𝑖1, ℎ2+2𝑖2)and (ℎ1, ℎ2)≤1,2(𝑖1, 𝑖2)i and only
i ℎ1≤𝑖1and ℎ2≤𝑖2 o ℎ1, 𝑖1∈𝐻1and ℎ2, 𝑖2∈𝐻2. The c oss-p oduc
o wo pa ially o de ed posi i e monoids is again a pa ially o de ed
posi i e monoid. This cons uc ion allows us o ep esen all he a e-
ele an pa ame e s in Example 2.1 abo e as a single pa ially o de ed
posi i e monoid (𝐻, +,≤).
P ice-op imal pa hs can hen be ound in he ollowing way: We
label each a c 𝑎∈𝐴wi h a weigh in 𝐻 ep esen ing he ele an
pa ame e s on his a c. The weigh o a pa h 𝑝, deno ed by w(𝑝), li es in
𝐻as well and can be ob ained by summing up he weigh s o he a cs o
𝑝. The icke 𝜏 o 𝑝and i s p ice can hen be de i ed om 𝑤(𝑝)using he
ules o he a e s uc u e. Finding a p ice-op imal 𝑠, 𝑡-pa h wi h 𝑠, 𝑡 ∈𝑉
can now be achie ed by inding he Pa e o-se o 𝑠, 𝑡-pa hs wi h ega d
o he pa ial o de o (𝐻, +,≤). He e, we can apply a label-se ing
MOSP algo i hm such as, e.g., Ma ins’ algo i hm, by using elemen s
o 𝐻as labels and he pa ial o de o (𝐻, +,≤) o es ablish dominance
be ween labels (Pa men ie ,2019). No e, howe e , ha his se will
likely s ill con ain many domina ed pa hs wi h espec o p ice. This
necessi a es he il e ing ou o supe luous pa hs in a pos -p ocessing
s ep.
Example 3.1 (MDV).Fo MDV, we can cons uc a monoid in he
ollowing way: o each ci y 𝑐∈𝐶we de ine he monoid (𝐻𝑐∶=
{0,1},∨,≤)as an indica o whe he ou pa h s a ed in 𝑐and hen le
he ci y. Hence, all a cs ep esen ing a connec ion lea ing he ci y ca y
he weigh 1 ∈ 𝐻𝑐. Fu he mo e, we ep esen he dis ance a eled
by he monoid (𝐻𝑑𝑖𝑠𝑡 ∶= N,+,≤), he numbe o isi ed s a ions by
(𝐻𝑠𝑡𝑜𝑝 ∶= N,+,≤), he se o a e zones by (𝐻𝑧𝑜𝑛𝑒 ∶= 2𝑍,∪, ⊆)and inally
he ans e s by (𝐻𝑡𝑟𝑎𝑛 ∶= {0,1},∨,≤). P ice-op imal pa hs can hen be
compu ed by inding he Pa e o-se o e he monoid (𝐻𝑑𝑖𝑠𝑡 ×𝐻𝑠𝑡𝑜𝑝 ×
𝐻𝑡𝑟𝑎𝑛 ×𝐻𝑧𝑜𝑛𝑒 ×∏𝑐∈𝐶𝐻𝑐,+,≤)and il e ing ou domina ed pa hs in a
pos -p ocessing s ep. He e, +and ≤a e induced om he componen
monoids.
While he abo e modeling app oach co e s a easonable se o eal-
wo ld applica ions, i is no exp essi e enough o be o much use o
complex a e s uc u es. Fi s , no e ha i equi es an o de -p ese ing
ela ionship be ween he monoid and he icke p ices. This assump ion
does no hold in gene al: Fo example, some public ansi associa ions
(e.g., in he ci y o B emen, Ge many, be o e 2020 (Ve keh s e bund
B emen/Niede sachsen GmbH,2019)) apply nigh su cha ges on se-
lec ed lines. This means ha , e.g., in he ea ly mo ning, i can be
bene icial o s a a jou ney la e because i becomes cheape . In pa -
icula , i is necessa y o cap u e ime in he monoid, bu he mapping
be ween ime and p ice does no p ese e o de . Second, monoids
o en canno exp ess logical p icing condi ions: Fo example, a p ice
depending on he o de on which s ops a e isi ed canno be modeled
using he monoid-based app oach. Thi d, sho es pa h sea ch o e he
monoid is agnos ic o he a e s uc u e and he e o e ends o conside
unnecessa ily la ge sea ch ees. In he MDV case, i we al eady know
an 𝑠, 𝑡-pa h o which, e.g., he icke 𝐷𝐿is applicable, all pa ial pa hs
s a ing in 𝑠 ha equi e a mo e expensi e icke can be p uned e en
hough hey migh no be domina ed w. . o ≤. To do his, howe e ,
we mus ind a way o quickly ob ain he co esponding icke o labels
om 𝐻.
Finally, no e ha he labels o a label-se ing MOSP algo i hm li e in
𝐻and migh be qui e la ge. In mos cases, howe e , i is no necessa y
Fig. 2. Ticke g aph associa ed wi h he MDV public ansi ne wo k. To simpli y he
p esen a ion, all icke s o ci y a es a e collapsed o 𝐶1and 𝐶2 ep esen ing he wo
p ice le els o ci y a es. Possible s a ing icke s a e highligh ed in ligh g ay.
o ca y he whole label along. Conside again he MDV case. I a pa h
has le a ci y 𝑐, all ci y a es become una ailable and labels o all
monoids 𝐻𝑐, 𝑐 ∈𝐶, need no longe be conside ed. This in o ma ion
emains una ailable o an algo i hm using he monoid-based model.
Compu a ional esul s in Sec ion 6 e eal ha pu ely monoid-based
modeling quickly becomes in ac able, e en when only conside ing he
a e zone monoid (𝐻𝑧𝑜𝑛𝑒,∪, ⊆).
3.2. Ticke g aphs and a e e en s
To o e come he challenges laid ou in he p e ious sec ion, we
ex end ou modeling in wo di ec ions: Fi s , we wan o ake he
pa h’s icke in o accoun . To do so, we de elop a model o ep esen
icke s and hei ela ionships. Second, we in oduce e en s ha model
logical a e s a egies. They also se e o educe he size o he monoid.
Add essing he i s poin , we no ice ha he e is a na u al p og ession
o he applicable icke along a pa h 𝑝. I 𝑝is sho , a sho -dis ance
icke migh su ice. When 𝑝is ex ended by adding ano he s op 𝑣a
i s end, his icke migh no longe be applicable and now, e.g., a zone
icke migh apply. When a eling e en u he , a icke co e ing wo
zones migh be needed. Hence, we can ela e icke s o each o he ia
hei abili y o ansi ion in o one ano he along pa hs in he ou ing
g aph. We o malize his obse a ion by in oducing a icke g aph =
(𝑇 , 𝐸) ha con ains an a c 𝑒= (𝜏1, 𝜏2) ∈ 𝐸i icke 𝜏1can ansi ion
in o icke 𝜏2. T ansi ions depend on he weigh 𝑤(𝑝)and do no occu
a e e y s op. Hence, we in oduce a icke ansi ion unc ion ha
checks 𝑤(𝑝)and selec s he app op ia e icke om he neighbo hood o
he icke o 𝑝in . The icke g aph p o ides c ucial ad an ages o e
he pu ely monoid-based app oach: Fi s , when we wan o use p ice-
based a ge -p uning (c . Sec ion 5.4), we mus compu e he cu en
icke p ice. The icke g aph allows doing his by upda ing icke s
along a pa h, he eby only checking a ew ansi ion condi ions on he
ou going edges. Wi hou he icke g aph, all ules o he a e s uc u e
would need o be checked whene e a e ex is elaxed. Second, he
icke g aph also ca ies in o ma ion abou he possible u he icke
p ice de elopmen o a pa ial pa h: This knowledge is use ul o design
dominance ules.
Example 3.2 (Running example: Ticke g aph o MDV).Conside he
icke g aph in Fig. 2. All possible icke s in oduced in Example 2.1
a e ep esen ed as nodes. Whene e , a icke can ansi ion in o ano he
one, we in oduce an a c. No e ha , e.g., a discoun ed icke o Halle
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𝐷𝐻can ne e ansi ion in o a Leipzig icke 𝐿. Now, conside again he
pa h −in Fig. 1. This pa h equi es a icke co e ing one a e zone.
Appending he s a ion will equi e a icke co e ing wo a e zones.
This is modeled by pe o ming a icke ansi ion in he icke g aph
along he edge (𝑍1, 𝑍2)induced by he icke ansi ion unc ion o 𝑍1.
When calcula ing he icke p ice o a pa h ha con ains a subpa h wi h
icke 𝑍3now only a condi ion on he numbe o a e zones needs o
be checked o de e mine whe he 𝑍3o 𝑍4is applicable. Wi hou using
he icke g aph, e.g., he applicabili y o all discoun s would need o
be ecalcula ed. Fu he mo e, when compa ing wo pa hs ha ing, say,
icke s 𝑍2and 𝑍3, espec i ely, a dominance check can sa ely ma k he
pa h wi h icke 𝑍3as domina ed e en i i is sho e in dis ance. In he
pu ely monoid-based app oach, bo h pa hs would be nondomina ed.
To educe he dimension o he monoid, no e ha he pa ame e s
de e mining he icke o a pa h can be b oadly ca ego ized as ( a e)
s a es, e.g., numbe o s ops, and ( a e) e en s, e.g., a ans e o he
boa ding o a ain ha equi es a su cha ge. Up o now, e en s we e
included in he monoid ia indica o s. Howe e , his is no s ic ly
necessa y: A ans e a c could cause a icke ansi ion in he icke
g aph. The ea e , ans e s migh be ine ec ual and hence i is unnec-
essa y o eco d hem in he monoid. To do so, we need o anno a e
a cs in he ou ing g aph no only wi h elemen s o a monoid bu also
wi h e en s. The dis inc ion be ween s a es and e en s in oduces some
lexibili y o he modeling, as some imes aspec s o a a e s uc u e can
be modeled as bo h. Howe e , we na u ally aim o keep he monoid as
low-dimensional as possible. Using e en s, p ices may now also depend
on logical condi ions, e.g., he o de o e en s, which is no possible in
he monoid-based app oach.
Example 3.3 (Running example: Fa e e en s o MDV).The monoid in Ex-
ample 3.1 was in oduced as (𝐻𝑑𝑖𝑠𝑡 ×𝐻𝑠𝑡𝑜𝑝×𝐻𝑡𝑟𝑎𝑛×𝐻𝑧𝑜𝑛𝑒×∏𝑐∈𝐶𝐻𝑐,+,≤).
We can see any ans e as an e en 𝑡𝑟𝑎 occu ing on a ans e a c o
he ou ing g aph. In he same way, lea ing any ci y 𝑐∈𝐶can be seen
as an e en 𝑐𝑖𝑡𝑦 occu ing on a cs c ossing he ci y’s bo de s. Hence,
we can sh ink he monoid o (𝐻𝑑𝑖𝑠𝑡 ×𝐻𝑠𝑡𝑜𝑝 ×𝐻𝑧𝑜𝑛𝑒,+,≤), educing
he size o a label in a MOSP algo i hm by 17 en ies o ci ies and
by one en y o ans e s. Finally, we in oduce e en s ℎ𝑎𝑙 and 𝑙𝑒𝑖 o
public ansi a cs ending in he special a e zones o Halle and Leipzig,
espec i ely. This is pu ely a design decision, since he in o ma ion
could also be ead om he s a e o 𝐻𝑧𝑜𝑛𝑒. We ob ain a se o e en s
𝑆= {𝑐𝑖𝑡𝑦, 𝑡𝑟𝑎, ℎ𝑎𝑙, 𝑙𝑒𝑖, 𝑠0}whe e 𝑠0is a dummy e en wi h no e ec , see
also Example 3.4.
3.3. Condi ional a e ne wo ks
We now combine he h ee co e ideas o a icke g aph, a e e en s
and modeling wi h monoids o a o mal model o public ansi a e
s uc u es.
Again, le 𝐺= (𝑉 , 𝐴)be a ou ing g aph. Addi ionally, le (𝐻, +,≤)
be a posi i e, pa ially o de ed monoid, = (𝑇 , 𝐸)a icke g aph and
𝑆a se o a e e en s. We label each a c 𝑎∈𝐴wi h a weigh 𝑤(𝑎) ∈ 𝐻
and a a e e en 𝑒(𝑎) ∈ 𝑆. By collec ing hese weigh s and e en s along
a pa h 𝑝in 𝐺, we build i s a e s a e 𝑓(𝑝).
De ini ion 3.2 (Fa e s a e).A a e s a e 𝑓∈𝑇×𝐻is a pai o a icke
𝜏(𝑓)and a weigh 𝑤(𝑓). We w i e 𝐹∶= 𝑇×𝐻 o he space o all a e
s a es.
E e y e ex 𝑣∈𝑉is labeled wi h an ini ial a e s a e 𝜇(𝑣) ∈ 𝐹. Fa e
s a es will se e as pa h labels o sho es -pa h algo i hms. They con-
ain all he in o ma ion necessa y o decide domina ion be ween pa hs.
In con as o common (mul i-objec i e) sho es -pa h applica ions, he
a c labels 𝐻×𝑆li e no in he same space as he pa h labels 𝐹.
We now wan o enable he acking o a e s a es along pa hs in
𝐺. To do so, we o malize he no ion o he icke ansi ion unc ion on
icke s 𝜏∈𝑇in he icke g aph = (𝑇 , 𝐸). A icke ansi ion unc ion
Table 1
MDV a e monoid.
Name Rep esen s G ound se Ope a o Pa ial o de Neu al elemen
𝐻𝑑𝑖𝑠𝑡 Dis ance N+≤0
𝐻𝑠𝑡𝑜𝑝 S ops N+≤0
𝐻𝑧𝑜𝑛𝑒 Fa e zones 2𝑍∪⊆∅
e u ns he icke 𝜏2∈𝑇a icke 𝜏1∈𝑇 ansi ions in o gi en an
accumula ed weigh ℎ∈𝐻and a a e e en 𝑠∈𝑆. Possible candida es
a e he neighbo hood o 𝜏1in as well as 𝜏1i sel .
De ini ion 3.3 (Ticke ansi ion unc ion).The icke ansi ion unc ion
𝛤∶𝑇×𝐻×𝑆→𝑇o is a unc ion ha , gi en a weigh ℎ∈𝐻and
e en 𝑠∈𝑆, maps each icke 𝜏∈𝑇in o i s closed ou -neighbo hood
𝛿+(𝜏)∪{𝜏}.
The de ini ion is in en ionally kep as gene al as possible o cap u e
a la ge numbe o possible ansi ion condi ions. We use he no ion o
icke ansi ion unc ions o de ine he upda e o a a e s a e when
elaxing an a c o he ou ing g aph.
De ini ion 3.4 (Fa e upda e unc ion).Le 𝑓∈𝐹and 𝑎∈𝐴. Then, he
a e upda e unc ion Up ∶ 𝐹×𝐴→𝐹is gi en by 𝑔∶= Up(𝑓 , 𝑎)wi h
𝑤(𝑔) ∶= 𝑤(𝑓) + 𝑤(𝑎)
𝜏(𝑔) ∶= 𝛤(𝜏(𝑓), 𝑤(𝑔), 𝑒(𝑎)).
The a e s a e o a pa h 𝑝= (𝑣1,…, 𝑣𝑛)can now be acked by le ing
𝑓1∶= 𝜇(𝑣1)and 𝑓𝑖∶= Up(𝑓𝑖−1,(𝑣𝑖−1, 𝑣𝑖)) ∀ 𝑖= 2,…, 𝑛. In pa icula ,
when calcula ing 𝑓𝑖we need only conside he ou -neighbo hood o
𝜏(𝑓𝑛)ins ead o ee alua ing all a e s a egies.
Combining all he abo e de ini ions, we a i e a he no ion o
condi ional a e ne wo ks which can p ecisely desc ibe a a e s uc u e.
De ini ion 3.5 (Condi ional a e ne wo k).Le 𝐺= (𝑉 , 𝐴)be a ou ing
g aph and le he ollowing be gi en:
1. a di ec ed acyclic icke g aph = (𝑇 , 𝐸)wi h ansi ion unc-
ion 𝛤,
2. a c weigh s 𝑤∶𝐴→𝐻 om a pa ially o de ed, posi i e
monoid (𝐻, +,≤),
3. a c e en s 𝑒∶𝐴→𝑆,
4. ini ial a e s a es 𝜇∶𝑉→𝐹and
5. a p ice unc ion 𝜋∶𝑇→Q+ ha is mono onously non-
dec easing along di ec ed pa hs in 𝑇, i.e., i he e is a di ec ed
𝜏1−𝜏2-pa h in o 𝜏1, 𝜏2∈𝑇, hen 𝜋(𝜏1)≤𝜋(𝜏2). We w i e 𝜋(𝑝)
ins ead o 𝜋(𝜏(𝑓(𝑝))) o a pa h 𝑝∈𝑃.
We call he six- uple (, 𝛤 , 𝑤, 𝑒, 𝜇, 𝜋)acondi ional a e ne wo k o 𝐺.
No e ha cycle- eeness in , he mono onici y condi ion on 𝜋and
he posi i i y o 𝐻ensu e ha no p ice-dec easing cycles exis in 𝐺.
We conside hose assump ions na u al enough ha any easonable a e
s uc u es should sa is y hem.
Example 3.4 (Running example: Condi ional a e ne wo k o MDV).We
can now gi e he comple e condi ional a e ne wo k o he a e
s uc u e o MDV. We ha e al eady in oduced he icke g aph (Exam-
ple 3.2). As in Example 3.3, we de ine he monoid (𝐻, +,≤)as (𝐻𝑑𝑖𝑠𝑡 ×
𝐻𝑠𝑡𝑜𝑝 ×𝐻𝑧𝑜𝑛𝑒,+,≤). The componen s o (𝐻, +,≤)a e summa ized in
Table 1.
The se o a e e en s is 𝑆= {𝑐𝑖𝑡𝑦, 𝑡𝑟𝑎, ℎ𝑎𝑙, 𝑙𝑒𝑖, 𝑠0}. The meaning o
hese e en s is summa ized in Table 2. Ve ices 𝑣∈𝑉in he ou ing
g aph can now be anno a ed wi h an ini ial a e s a e om 𝐹=𝑇×𝐻.
Fo example, o s a ion in Fig. 1 we ha e 𝜇(𝑣)=(𝐷, (0,0,{156})),
i.e., we s a wi h he sho -dis ance icke , and he weigh is composed
o 0m o dis ance, 0 isi ed s ops and he a e zone 156. A cs 𝑎∈𝐴a e
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𝛤(𝑀, ℎ, 𝑠) = 𝑀
𝛤(𝑍𝑖, ℎ, 𝑠) = {𝑍𝑖+1 |ℎ𝑧𝑜𝑛𝑒|=𝑖+ 1
𝑍𝑖o he wise 𝛤(𝐷, ℎ, 𝑠) = ⎧
⎪
⎪
⎨
⎪
⎪
⎩
𝑍1|ℎ𝑧𝑜𝑛𝑒|= 1 ∧ (ℎ𝑑𝑖𝑠𝑡 >4 ∨ 𝑠=𝑡𝑟𝑎)
𝑍2|ℎ𝑧𝑜𝑛𝑒|= 2 ∧ (ℎ𝑑𝑖𝑠𝑡 >4 ∨ 𝑠=𝑡𝑟𝑎)
𝑍3|ℎ𝑧𝑜𝑛𝑒|= 3 ∧ (ℎ𝑑𝑖𝑠𝑡 >4 ∨ 𝑠=𝑡𝑟𝑎)
𝐷o he wise
𝛤(𝐿, ℎ, 𝑠) = {𝐿 𝑠 =𝑙𝑒𝑖 ∨𝑠=𝑡𝑟𝑎
𝑍2o he wise 𝛤(𝐷𝐿, ℎ, 𝑠) = ⎧
⎪
⎨
⎪
⎩
𝑍2𝑠≠𝑙𝑒𝑖 ∧ℎ𝑠𝑡𝑜𝑝 >4
𝐿 𝑠 =𝑡𝑟𝑎 ∨ (𝑠=𝑙𝑒𝑖 ∧ℎ𝑠𝑡𝑜𝑝 >4)
𝐷𝐿o he wise
𝛤(𝐻, ℎ, 𝑠) = {𝐻 𝑠 =ℎ𝑎𝑙 ∨𝑠=𝑡𝑟𝑎
𝑍2o he wise 𝛤(𝐷𝐻, ℎ, 𝑠) = ⎧
⎪
⎨
⎪
⎩
𝑍2𝑠≠ℎ𝑎𝑙 ∧ℎ𝑠𝑡𝑜𝑝 >4
𝐻 𝑠 =𝑡𝑟𝑎 ∨ (𝑠=ℎ𝑎𝑙 ∧ℎ𝑠𝑡𝑜𝑝 >4)
𝐷𝐻o he wise
𝛤(𝐶1, ℎ, 𝑠) = ⎧
⎪
⎨
⎪
⎩
𝑍1𝑠=𝑐𝑖𝑡𝑦 ∧ℎ𝑑𝑖𝑠𝑡 >4
𝐷 𝑠 =𝑐𝑖𝑡𝑦 ∧ℎ𝑑𝑖𝑠𝑡 ≤4
𝐶1o he wise
𝛤(𝐶2, ℎ, 𝑠) = ⎧
⎪
⎨
⎪
⎩
𝑍1𝑠=𝑐𝑖𝑡𝑦 ∧ℎ𝑑𝑖𝑠𝑡 >4
𝐷 𝑠 =𝑐𝑖𝑡𝑦 ∧ℎ𝑑𝑖𝑠𝑡 ≤4
𝐶2o he wise
Box I.
Table 2
MDV a e e en s.
Fa e e en Rep esen s
𝑐𝑖𝑡𝑦 Lea ing ci y wi h ci y icke 𝐶1o 𝐶2
𝑡𝑟𝑎 T ans e
ℎ𝑎𝑙 Head o A c is in Halle
𝑙𝑒𝑖 Head o A c is in Leipzig
𝑠0No hing
now anno a ed wi h weigh s om 𝐻and e en s om 𝑆. Fo example,
i 𝑎1= (,)had a leng h o 231 m, i would be anno a ed wi h
𝑤(𝑎1) = (231,1,{233}) and 𝑒(𝑎1) = 𝑐𝑖𝑡𝑦. The a c 𝑎2= (,) ep esen s a
oo pa h and is anno a ed wi h 𝑤(𝑎2) = (0,0,∅) and 𝑒(𝑎2) = 𝑡𝑟𝑎.
Hence, o cons uc a condi ional a e ne wo k o MDV, we now
only need o gi e he ansi ion unc ions o . Le 𝑠∈𝑆and ℎ=
(ℎ𝑑𝑖𝑠𝑡, ℎ𝑠𝑡𝑜𝑝, ℎ𝑧𝑜𝑛𝑒) ∈ 𝐻. Then, he icke ansi ion unc ion 𝛤o is
de ined by he equa ions gi en in Box I. Fo example, he condi ions
o 𝛤(𝐷𝐿, ℎ, 𝑠)mean ha we need o ansi ion o he wo zones icke
𝑍2whene e we lea e Leipzig and a el o mo e han ou s ops, and
ha we ansi ion o he s anda d Leipzig icke 𝐿whene e we ans e
o su pass he ou s ops limi wi hin Leipzig. In all o he cases, we can
s ick wi h he discoun ed Leipzig icke 𝐷𝐿. No e ha we did no ye
co e MDV’s o e lap a eas. We discuss in Sec ion 3.4 why i is bes o
add ess hese in a p ep ocessing s ep.
We can now inally o malize he p ice-op imal ea lies a i al
p oblem.
De ini ion 3.6 (P ice-op imal ea lies a i al p oblem (POEAP)).Le a
public anspo a ion ne wo k be gi en as a di ec ed g aph 𝐺= (𝑉 , 𝐴)
oge he wi h a condi ional a e ne wo k (, 𝛤 , 𝑤, 𝑒, 𝜇, 𝜋)and a ime-
dependen FIFO a el ime unc ion 𝑐(𝑎) ∶ 𝐼→𝐼∀𝑎∈𝐴. Then, he
p ice-op imal ea lies a i al p oblem (POEAP) asks o ind a Pa e o-se
(w. . . p ice and a i al ime) o 𝑠, 𝑡-pa hs 𝑃∗
𝑠,𝑡 ⊆ 𝑃𝑠,𝑡 in 𝐺, i.e.,
∀𝑝∗∈𝑃∗
𝑠,𝑡∄𝑝∈𝑃𝑠,𝑡 ∶𝜋(𝑝)≤𝜋(𝑝∗) ∧ 𝑐(𝑝)≤𝑐(𝑝∗)∧(𝜋(𝑝)< 𝜋(𝑝∗) ∨ 𝑐(𝑝)< 𝑐(𝑝∗))
(1)
∀𝑝∈𝑃𝑠,𝑡 ∃𝑝∗∈𝑃∗
𝑠,𝑡 ∶𝜋(𝑝∗)≤𝜋(𝑝) ∧ 𝑐(𝑝∗)≤𝑐(𝑝).(2)
3.4. Some hin s on modeling wi h CFNs
In he ollowing, we elabo a e on some common ea u es o a e
s uc u es and how hey can be modeled using condi ional a e ne -
wo ks.
T ans e penal ies, oo pa hs, and su cha ges. When oo pa hs ha e no
in luence on he icke , hey can be modeled as a cs wi h a weigh o
𝟎∈𝐻and an e en 𝑠0 ha canno ac i a e a icke ansi ion. This
way, a oo pa h does no change he cu en a e s a e. The ansi ion
om a oo pa h o a public anspo a ion ehicle equi es some ca e.
Assume we walk om s op 𝑣0 o 𝑣1along a c 𝑎0= (𝑣0, 𝑣1) o ake a
ehicle along 𝑎1= (𝑣1, 𝑣2) o each 𝑣2. Some a e s uc u es use he
numbe o s ops a pa h ouches o calcula e p ices. He e, his numbe
would be wo. Coun ing a s op when elaxing 𝑎0is a mis ake i he
op imal pa h would be o con inue on oo . Coun ing bo h 𝑣1and 𝑣2
when elaxing 𝑎1is also w ong, since his would o e coun he numbe
o s ops o e e y jou ney ha eaches 𝑣1 ia a ehicle. Hence, he
g aph model needs o be ex ended by spli ing up s ops in o e ices o
e e y ou e and a e ex ha is connec ed o oo pa hs. These e ices
a e hen connec ed ia ans e a cs and boa ding a cs. Placing weigh s
and e en s di e en om 𝟎and 𝑠0on ans e a c allows us o make he
applicable icke dependen on he numbe o ans e s, while e en s
on a cs ep esen ing boa ding can be used o model su cha ges o he
boa ded ou e. Fo mo e de ails on how o build hese expanded g aphs,
we e e o Disse e al. (2008).
O e lap a eas. Some a e s uc u es ha a e based on a e zones con-
ain o e lap a eas. S a ions in an o e lap a ea can be coun ed as pa o
ei he o i s neighbo ing zones, whiche e is cheapes o he cus ome .
This is mean o mi iga e sha p p ice inc eases o sho jou neys
a a e zone bo de s. MDV uses hem as well as se e al o he Ge -
man ailway companies (e.g., Ve keh s e bund B emen/Niede sachsen
GmbH (Ve keh s e bund B emen/Niede sachsen GmbH,2023)).
A a i s glance, one migh be emp ed o ep esen o e lap a eas as
icke s in he icke g aph. A label p opaga ed along a pa h s a ing in
an o e lap a ea hen keeps his icke un il a egula a e zone is picked
up along he pa h and ansi ions in he zone icke o his a e zone.
This app oach, howe e , becomes cumbe some when se e al o e lap
a eas bo de each o he . In his case, a icke o each combina ion o
o e lap a eas needs o be in oduced.
Al e na i ely, o e lap a es can be inco po a ed by label duplica ion:
Assume an o e lap a ea neighbo s 𝑛 a e zones. We associa e each a c
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Fig. 3. Example o a ou ing g aph (a) wi h wo possible condi ional a e ne wo ks (b) and (c). Fo bo h ne wo ks, he unde lying pa ially o de ed monoid is (R,+,≤), he a e
e en s a e 𝑆= {𝑠0, 𝑠1, 𝑠2, 𝑠3}and he ini ial a e s a e o all e ices 𝑣𝑖wi h 𝑖= 1,…,5is 𝜇(𝑣𝑖) = (𝐴, 0). We se p ices o he icke s as 𝜋(𝐴) = 0,𝜋(𝐵) = 2,𝜋(𝐶) = 3,𝜋(𝐷) = 1 and
𝜋(𝐸)=5. The alue o he ansi ion unc ion 𝛤 o a gi en weigh ℎand e en 𝑠is gi en ia indica o unc ions on he a e a cs. Using he icke g aph (b), he uppe 𝑣1, 𝑣5-pa h
yields icke 𝐶, while he lowe pa h yields icke 𝐸. Using icke g aph (c), he uppe pa h yields icke 𝐵, he lowe pa h yields icke 𝐶.
𝑎whose ℎ𝑒𝑎𝑑(𝑎) ep esen s a s op in he o e lap a ea wi h 𝑛di e en
weigh s, one o each a e zone i could possibly be pa o . When
se ling he e ex in a sho es pa h sea ch, he cu en a e s a e is
upda ed once o each weigh , he eby c ea ing 𝑛new labels. This,
howe e , leads o an inc eased need o dynamic memo y alloca ion,
which should be a oided.
Hence, we p opose simply ou e duplica ion as he mos con enien
model o o e lap a eas. Whene e a ou e o he ime able con ains a
s op 𝑣in an o e lap a ea neighbo ing 𝑛 a e zones, we simply in oduce
𝑛 ou es each wi h a single a e zone a 𝑣. In he ou ing g aph, his
co esponds o in oducing pa allel a cs wi h each s o ing a di e en
a e zone in i s weigh . To a oid c ea ing unnecessa y duplica es, his
is done block-wise, i.e., only o each consecu i e sequence o s ops
along a ou e ha a e in he same o e lap a ea. Hence, o e lap a eas
a e aken ca e o in a p ep ocessing s ep and a e no ep esen ed in he
condi ional a e ne wo k.
4. CFNs in ou ing algo i hms
Label-se ing MOSP algo i hms ely on dynamic p og amming and
he subpa h op imali y condi ion (Be ge and Mülle -Hannemann,2009).
Tha is, e e y subpa h o an op imal 𝑠, 𝑡-pa h is in i sel an op imal pa h.
Fo POEAP, when compa ing pa hs in 𝐺nai ely by he p ice unc ion
𝜋, he subpa h op imali y condi ion is usually iola ed. Conside aking
a local de ou o a oid a a e zone: La e on, a ele s may be o ced o
c oss he zone due o he in as uc u e, u ning he locally dominan
de ou in o a subop imal choice. On he o he hand, a locally domina ed
subpa h migh s ill lead o an op imal 𝑠, 𝑡-pa h. This ype o p oblem
pe sis s in CFNs: he ansi ion be ween icke s depends on he weigh s
and e en s al eady collec ed, bu also on he s uc u e o he eachable
icke g aph. Example 4.1 highligh s ha p oblems can al eady a ise
e en in simple cases.
Example 4.1 (Label dominance in Fig. 3).Conside he ou ing g aph
(a) oge he wi h he condi ional a e ne wo k (b). Examining he pa hs
𝑝1= (𝑣1, 𝑣2, 𝑣4)and 𝑝2= (𝑣1, 𝑣3, 𝑣4), we ind hei espec i e a e s a es
a e 𝑓(𝑝1) = (𝐵, 1) and 𝑓(𝑝2) = (𝐷, 2). Ex ending hem by 𝑣5 o 𝑝′
1and
𝑝′
2yields 𝑓(𝑝′
1)=(𝐶, 3) and 𝑓(𝑝′
2)=(𝐸, 4). Compa ing a e s a es by
p ice would indica e ha 𝑝1could be p uned a 𝑣4since 𝜋(𝐵)> 𝜋(𝐷).
This is a subop imal choice as 𝑝′
1domina es 𝑝′
2since 𝜋(𝐶)< 𝜋(𝐸).
Hence, p ice canno be used as dominance c i e ion o a e s a es. A
na u al al e na i e would be o use he pa ial o de de ined by pa hs
in he icke g aph, ins ead. A icke 𝜏1 hen domina es a icke 𝜏2i
he e is a 𝜏1, 𝜏2-pa h. This would ende he icke s 𝐵and 𝐷and he
icke s 𝐶and 𝐸mu ually incompa able. The idea, howe e , comes
wi h p oblems o i s own. To see his, conside he condi ional a e
ne wo k (c). A 𝑣4, we ha e 𝑓(𝑝1)=(𝐴, 1) and 𝑓(𝑝2)=(𝐴, 2) and
hence bo h pa hs a e equi alen and i would be sensible o keep only
one o hem based on he ela ion be ween 𝑤(𝑓(𝑝1)) and 𝑤(𝑓(𝑝2)).
By elaxing (𝑣4, 𝑣5), we ob ain 𝑓(𝑝′
1)=(𝐵, 3) and 𝑓(𝑝′
2)=(𝐶, 4),
which a e incompa able, i.e., he a e s a es o 𝑝′
1and 𝑝′
2di e ged
om compa able o incompa able. Consequen ly, any dominance ule
p uning ei he 𝑝1o 𝑝2would be de ec i e.
To mi iga e hese and simila p oblems, we migh assume a gene al
incompa abili y o a e s a es. This comes down o enume a ing all 𝑠, 𝑡-
pa hs and simply so ing hem by p ice. Howe e , in a sensibly designed
a e s uc u e, i is usually clea which icke is be e , and aking a
cheape subpa h should usually no u n ou mo e expensi e o e all.
In he emainde o his sec ion, we p opose a mo e ailo ed app oach.
I bases domina ion ules on pa h ela ionships bu adds excep ions o
co e cases in which i is no sa e o do so.
4.1. Dominance o a e s a es
We wan o de ine a pa ial o de o a e s a es ha es o es subpa h
op imali y while no elaxing dominance oo gene ously.
To do so, we pa i ion he icke se 𝑇in o h ee disjoin compa abil-
i y g oups:𝐶𝐹( ull compa abili y), 𝐶𝑃(pa ial compa abili y), 𝐶𝑁(no
compa abili y). Based on he pa i ion 𝐶= (𝐶𝐹, 𝐶𝑃, 𝐶𝑁), we de ine he
pa ial o de .
De ini ion 4.1 (Compa abili y o a e s a es).Le 𝑓1= (𝜏1, ℎ1),𝑓2=
(𝜏2, ℎ2)be a e s a es. We say 𝑓1≤𝐶𝑓2i and only i 𝜏1∉𝐶𝑁,ℎ1≤ℎ2
and
𝜏1=𝜏2i 𝜏1∈𝐶𝑃(3)
∃𝜏1, 𝜏2-pa h in i 𝜏1∈𝐶𝐹.(4)
I 𝑓1≤𝐶𝑓2and ei he ℎ1< ℎ2o 𝜏1≠𝜏2, we say ha 𝑓1is s ic ly less
han 𝑓2, i.e., 𝑓1<𝐶𝑓2.
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We deno e by 𝑃𝑓
𝑠,𝑡 he se o all Pa e o-op imal pa hs wi h espec o
≤𝐶, i.e.,
𝑝∗∈𝑃𝑓
𝑠,𝑡 ⇒∄𝑠, 𝑡-pa h 𝑝∶𝑓(𝑝)<𝐶𝑓(𝑝∗).(5)
We call pa hs in 𝑃𝑓
𝑠,𝑡 s a e-op imal. They o m a supe se o he
se o p ice-op imal pa hs. MOSP algo i hms on g aphs wi h weigh s
om pa ially o de ed monoids ely on he monoid ope a ion being
ansla ion-in a ian wi h espec o he pa ial o de . Simila ly, o
CFN’s, we need he upda e unc ion o be mono one along all a cs
𝑎∈𝐴, i.e.,
∀𝑓1, 𝑓2∈𝐹∶𝑓1≤𝐶𝑓2⟹∀𝑎∈𝐴∶ Up(𝑓1, 𝑎)≤𝐶Up(𝑓2, 𝑎).(6)
This condi ion is enough o ensu e ha a weake o m o subpa h
op imali y holds.
P oposi ion 4.1 (Weak subpa h op imali y).Le 𝐺= (𝑉 , 𝐴)be a ou ing
ne wo k and = (, 𝛤 , 𝑤, 𝑒, 𝜇, 𝜋)be i s condi ional a e ne wo k. Le 𝑝∗∈
𝑃𝑓
𝑠,𝑡 be a s a e-op imal 𝑠,𝑡-pa h in 𝐺 o some 𝑠, 𝑡 ∈𝑉. Then, he e is a pa h
𝑝′= (𝑠=𝑣0, 𝑣1,…, 𝑣𝑛−1, 𝑣𝑛=𝑡) ∈ 𝑃𝑓
𝑠,𝑡 wi h 𝑓(𝑝∗) = 𝑓(𝑝′), such ha e e y
subpa h 𝑝′′ = (𝑣0,…, 𝑣𝑙),𝑙 < 𝑛, o 𝑝′is a s a e-op imal 𝑣0, 𝑣𝑙-pa h.
P oo . Le 𝑝∗= (𝑠=𝑣0, 𝑣1,…, 𝑣𝑛−1, 𝑣𝑛=𝑡) ∈ 𝑃𝑓
𝑠,𝑡 be a s a e-
op imal 𝑠, 𝑡-pa h wi h a e s a es (𝑓0,…, 𝑓𝑛). Assume he e is a 𝑠, 𝑡-pa h
𝑝 = (𝑠=𝑢0, 𝑢1,…, 𝑢𝑙−1 =𝑣𝑛−1, 𝑢𝑙=𝑡)wi h a e s a es (𝑔0, 𝑔1,…, 𝑔𝑙)and
𝑔0=𝑓0.By(6), we can choose 𝑝 o be a simple pa h. Le 𝑘be he
la ges in ege such ha 𝑣𝑛−𝑘=𝑢𝑙−𝑘, i.e., he pa hs (𝑣𝑛−𝑘,…, 𝑣𝑛)and
(𝑢𝑙−𝑘,…𝑢𝑙)a e equal. Now assume
𝑓𝑙−𝑘−1 <𝐶𝑓𝑛−𝑘−1. By de ini ion,
𝑓𝑛−𝑘= Up(𝑓𝑛−𝑘−1,(𝑣𝑛−𝑘−1, 𝑣𝑛−𝑘)) and 𝑔𝑙−𝑘= Up(𝑔𝑙−𝑘−1,(𝑢𝑙−𝑘−1, 𝑢𝑙−𝑘)).
We apply (6) o ob ain
𝑓𝑙−𝑘≤𝐶𝑓𝑛−𝑘. By epea ing he p ocess o
𝑖∈ {𝑘−1,…,0}, we ind 𝑔𝑙≤𝐶𝑓𝑛. Since 𝑝∗was s a e-op imal, i ollows
ha 𝑔𝑙=𝑓𝑛, and consequen ly, 𝑝is also s a e-op imal. Since he numbe
o pa hs in 𝐺is ini e, we can epea his p ocedu e o ind he pa h
𝑝′.□
P oposi ion 4.1 does no imply ha e e y subpa h o a s a e-op imal
pa h is s a e-op imal. We can, howe e , disca d all s a e-op imal pa hs
wi hou his p ope y since a pa h wi h an equal a e s a e s ill emains
in 𝑃𝑓
𝑠,𝑡. Hence, label-se ing MOSP algo i hms can s ill be applied.
4.2. The compa abili y pa i ion
In choosing 𝐶𝐹,𝐶𝑃and 𝐶𝑁, he e is some deg ee o eedom. We
wan 𝐶𝐹 o be as big and 𝐶𝑁as small as possible while s ill ul illing (6).
I is clea ha he bes choice does no only depend on he icke g aph
and he ansi ion unc ion 𝛤, bu also on 𝐺and he a c weigh s and
e en s. Such an app oach, howe e , mos ly likely equi es ex ensi e
compu a ions on 𝐺. We p opose a solu ion ha depends only on and
𝛤and needs no ecompu a ion when changes in he ou ing ne wo k
occu .
Fi s , we in oduce some no a ion. I he e is a di ec ed pa h in
be ween 𝜏1, 𝜏2∈𝑇, we w i e 𝜏1→𝜏2. This includes he case 𝜏1=𝜏2.
The each R(𝜏)o a e ex 𝜏∈𝑇is he subg aph induced by all e ices
eachable om 𝜏, i.e., R(𝜏) ∶= [{𝑘∈𝑇∶𝜏→𝑘}].
De ini ion 4.2 (No-o e aking p ope y).Le 𝜏∈𝑇be a icke . We say
i s each R(𝜏)has he no-o e aking p ope y i o all icke s 𝑘, 𝑙 ∈ R(𝜏)
wi h 𝑘→𝑙and (ℎ, 𝑠) ∈ 𝑊i holds ha
∀
ℎ∈𝐻∶ℎ≤
ℎ⟹𝛤(𝑘, ℎ, 𝑠)→𝛤(𝑙,
ℎ, 𝑠).(7)
The no-o e aking p ope y bea s some esemblance o he FIFO
( i s -in, i s -ou ) p ope y: A wo se a e s a e, i.e., a wo se weigh o
icke , canno gi e ise o a be e a e s a e when elaxing he same
a c in he ou ing g aph. No e ha he no-o e aking p ope y has o
be ul illed no only o he neighbo hood o a icke 𝜏bu o he each
R(𝜏). Subg aphs wi h he no-o e aking p ope y allow o he s ic es
domina ion ules. We use hem as compa abili y g oup 𝐶𝐹.
De ini ion 4.3 (Compa abili y pa i ion).Le 𝐺= (𝑉 , 𝐴)be a ou ing
ne wo k wi h a CFN = (, 𝛤 , 𝑤, 𝑒, 𝜇, 𝜋). We de ine
𝐶𝐹∶= {𝜏∈𝑇∶ R(𝜏) aceable and has he no-o e aking p ope y}(8)
𝐶𝑃∶= {𝜏∈𝑇∖𝐶𝐹∶ ∀ 𝑘∈ R(𝜏) ∀ 𝑠∈𝑆∀ℎ1, ℎ2∈𝐻∶𝛤(𝑘, ℎ1, 𝑠) = 𝛤(𝑘, ℎ2, 𝑠)}
(9)
𝐶𝑁∶= {𝜏∈𝑇∖(𝐶𝐹∪𝐶𝑃)}.(10)
I is no enough o ul ill (7) o 𝜏∈𝑇 o be in he se 𝐶𝐹. I s
each R(𝜏)has also o be aceable, i.e., con ain a Hamil onian pa h.
This condi ion is needed o a oid he di e gence seen in Example 4.1.
I a icke has non- aceable each o does no ha e he no-o e aking
p ope y, i is placed in 𝐶𝑃. Fo icke s 𝜏∈𝐶𝑃, he ansi ion unc ions
o icke s 𝑘∈ R(𝜏)mus be independen o (𝐻, +,≤). This, again, is
necessa y o ensu e ha compa able a e s a es do no di e ge in an
incompa able s a e a e an upda e, i.e., all icke s ha can be eached
om a icke in 𝐶𝐹 hemsel es need o be in 𝐶𝐹. All emaining icke s
a e added o 𝐶𝑁. Fa e s a es con aining icke s om 𝐶𝑁can ne e be
domina ed.
Example 4.2 (Dominance o MDV Fa es).In he g aph in Fig. 2,
all nodes ha e aceable each, and i is easy o e i y ha he no-
o e aking p ope y does indeed hold o all icke s. Hence, we can se
he compa abili y pa i ion o 𝐶𝐹=𝑇,𝐶𝑃=𝐶𝑁= ∅.
Example 4.3 (Dominance o Example 4.1).Fo Ticke G aph (b), we
ha e 𝐶𝐹= {𝐵, 𝐶, 𝐷, 𝐸},𝐶𝑃= {𝐴}and 𝐶𝑁= ∅. Fo Ticke G aph (c),
we ha e ha 𝐶𝐹= {𝐵, 𝐶},𝐶𝑃= ∅ and 𝐶𝑁= {𝐴}.
P oposi ion 4.2 (Mono onici y o he cmpa abili y pa i ion).The pa ial
o de ≤𝐶de ined by De ini ions 4.1 and 4.3 ul ills he mono onici y
condi ion (6).
P oo . Le 𝑎∈𝐴and 𝑓1, 𝑓2∈𝐹such ha 𝑓1≤𝐶𝑓2. Fo 𝑖∈
{1,2}, we w i e 𝑔𝑖∶= Up(𝑓𝑖, 𝑎), i.e, 𝑤(𝑔𝑖) = 𝑤(𝑓𝑖) + 𝑤(𝑎)and 𝜏(𝑔𝑖) =
𝛤(𝜏(𝑓𝑖), 𝑤(𝑔𝑖), 𝑒(𝑎)). By posi i i y o he monoid (𝐻, +,≤),𝑤(𝑓1)≤𝑤(𝑓2)
di ec ly implies 𝑤(𝑔1)≤𝑤(𝑔2). I emains o show ha 𝜏(𝑔1)→𝜏(𝑔2).
To do so, we need o dis inguish he cases 𝜏(𝑓1) ∈ 𝐶𝑃and 𝜏(𝑓1) ∈ 𝐶𝐹.
Fi s , assume ha 𝜏(𝑓1) ∈ 𝐶𝑃and hence 𝜏(𝑓1) = 𝜏(𝑓2). By he
de ini ion o 𝐶𝑃, we ob ain
𝜏(𝑔1) = 𝛤(𝜏(𝑓1), 𝑤(𝑔1), 𝑒(𝑎)) = 𝛤(𝜏(𝑓2), 𝑤(𝑔2), 𝑒(𝑎)) = 𝜏(𝑔2).
Thus, 𝜏(𝑔1) = 𝜏(𝑔2). No e ha he de ini ions o 𝐶𝑃and 𝐶𝐹imply ha
𝜏(𝑔1) ∈ 𝐶𝑃∪𝐶𝐹since 𝜏(𝑔1) ∈ R(𝜏(𝑓1)) and hence 𝑔1≤𝐶𝑔2. Now, assume
𝜏(𝑓1) ∈ 𝐶𝐹. No e ha R(𝜏(𝑓1)) ⊂ 𝐶𝐹. This allows us o apply (7) o
ob ain
𝜏(𝑔1) = 𝛤(𝜏(𝑓1), 𝑤(𝑔1), 𝑒(𝑎)) →𝛤(𝜏(𝑓2), 𝑤(𝑔2), 𝑒(𝑎)) = 𝜏(𝑔2),
which concludes he p oo . □
P oposi ions 4.1 and 4.2 allow us o apply label-se ing MOSP algo-
i hms o POEAP using he compa abili y pa i ion om De ini ion 4.3.
Howe e , we ob ain only he se o s a e-op imal pa hs. I emains o
show ha his se con ains he cheapes pa h.
P oposi ion 4.3 (Co ec ness).Le 𝜋∗∶= min𝑃𝑠,𝑡 𝜋(𝑝). Then, he e is a
leas one 𝑠, 𝑡-pa h 𝑝∗wi h 𝜋∗=𝜋(𝑝∗)and 𝑝∗∈𝑃𝑓
𝑠,𝑡.
P oo . Conside a pa h 𝑝∈𝑃𝑠,𝑡 wi h 𝜋(𝑝) = 𝜋∗. I he e is 𝑝′∈𝑃𝑓
𝑠,𝑡 wi h
𝜏(𝑓(𝑝′)) = 𝜏(𝑓(𝑝)), we a e done. I no , all such pa hs mus be domina ed
w. . . o <𝐶and hence he e is a pa h 𝑝′∈𝑃𝑓
𝑠,𝑡 wi h 𝜏(𝑓(𝑝′)) →𝜏(𝑓(𝑝)).
This implies 𝜋(𝜏(𝑓(𝑝′))) ≤𝜋(𝜏(𝑓(𝑝))) and hence a pa h o he same p ice
as 𝑝is p esen in 𝑃𝑓
𝑠,𝑡.□
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