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