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Applications of maxflow minent
edge disjointpaths jt Eyenetuorkvertex disjoint paths 1
c f'tvertex capacities output max flew
bipartite maximum matchingcasualty integers
MaxMatching
Transform input features in solution F paths inTransform output Max flewTime in terms oforiginalinputProve correct
Ex ling N ntr pt total input sizen classes Eft n enrollmentr rooms SCI r seatst timeslots Afa t t p availabilityp proctors ACKD T proctorL
is availableEvery class needs scheduling at EmekESS in any examEI exam per room per time slot
Ea.ch onoctoroverseesE5exam
class E capacityschedule set of 4 tuples i j k l room j g 1one per tindforkedavailable
1 or
do I f sE I t
I
Each class EC jt exam EEthePIIIhas EI exam e inane Iet eachproctor can examsonh natch one
Feasibleexam atanytime
Integer flew 2 paths set of successfullyscheduled exams
IS Imax flow in H1 classesntr ftp.t2 OCN
C En nr Http xp NZTime ONE 406USD
TnpleSelec_tionInp_tifinitesetsX1 Xe Xd
representdiscrete resources
for all x c Xi for all ICCx y for all C Xi yExite for all
Outputtargetset of tuples Cx x xd cXzxXzx xXdSubject to constraints
for each index i each x c Xi appears in Edx tuplesfor each index i
each XE Xi and YE Xi I appear in Echr y tuples
MaxMatchinginput G LUR E X1 LOutpt max matching Xz R
set of pairsCx y ELxR cC y1 if Xy EE0 if xy E
Ix Cly L
of yOFFIT 4
InX1 Xz Xd
largest complete complete complete maximumvalid set of tuples paths feasible flew
DisjointPat_hCoerTpnti gG Cv.ET generalgraphs NP hardOutput min disjointpaths that cover every vertex
b d0 30
08 1 0o oc e with
Intuition We want to assign a successor to as manymatch vertices as possible
a bb e
Iq Path.EE oFIYessorBuild
Reduce to max matching H LUR EL U
R U Ccopyfind max matching.M
I0 0 paths in G
Oe s paths V M
0 VE timeV V
Projectselection Open pit mininginput dag Etv E U projectsC dependenciesprofit v U su means u can onlybedonefor every vertex after u
Output Subset SEVSE for all u u WES E Smax DCS Festa
a IF Tsf D8 cost
IIPartition V S UT S selectreduce to min cat problem T throw out
toz
8 Build H
a Tsf Tgfcomputemaxflowf
return P Ift
f OcuE time
p Emax 63,03 Eo G mpaff.EE Iobhfepend
profit 5 P lls TH claim
Forany XEV cost Cx o tcu cG t
yieldCx Izzo toCu Execsu
profitCx yield X costa IZ u
P yield V yield s yieldCt119TH costs yield TP 115TH yield s cost G profit s U