An Application of Graph Theory in the Process of Mutual Debt Compensation

Transcription

1 Acta Poltechca Hugarca ol. 12 No A Applcato of Graph Theor the Process of Mutual Debt Compesato ladmír Gazda Des Horváth Marcel Rešovský Techcal Uverst of Košce Facult of Ecoomcs; Němcove Košce Slovaka vladmr.gazda@tuke.sk Pavol Jozef Šafárk Uverst Košce Facult of Scece; Park Agelum Košce Slovaka des.horvath@ups.sk Techcal Uverst of Košce Facult of Ecoomcs; Němcove Košce Slovaka marcel.resovsk@tuke.sk Abstract: I our paper the class of graph tractable problems of mutual debt compesato MDC amog frms s troduced. We demostrate that the debt compesato s related to the optmzato problems mamzg the crculato of the cosecutve compesato the dgraph of debts. Each optmzato problem s frstl formulated b lear programmg methods. It s subsequetl redeveloped order to appl the more effcet Kle s ccle-cacellg algorthm. The class of the formulated compesato problems cossts of the followg models: the model mamzg the returs of the MDC orgazer; the model of proftable subsdes ad the model mmzg the clams eposto of the MDC orgazer. Kewords: mutual debt compesato; dgraph of debts; Kle s ccle cacellg 1 Itroducto The creasg complet of the ecoomc evromet ad the large umber of heterogeeous agets partcpatg the market ecesstate a ew approach modelg credtor debtor relatoshps. The semal papers of Damod ad Dbvg [6] ad Alle ad Gale [2] [3] offer the graph theor as a effcet tool for modelg ths feld. The subsequet boostg research performed mal durg the last decade was prmarl motvated b the estg facal crses the developmet of the complet theor ad the lmtatos of the tradtoal ecoometrc techques. The research results demostrate a sgfcat mpact of the facal etwork structure o the facal rsks gve b the potetal etet of the solvec cotago spreadg through the et. Here Alle ad Gale [3] comparg ccle ad complete graph structures coclude that the latter s more 7

2 . Gazda et al. A Applcato of Graph Theor the Process of Mutual Debt Compesato reslet to lqudt shocks. O the other had the applcato of more realstc etwork structures usg dfferet methodologes leads to ambguous results or eve cotrovers. For stace Ner et al. [14] troduce the dea that the effect of the degree of coectvt s o-mootoc.e. tall a small crease coectvt creases the cotago effect; but after a certa threshold value coectvt mproves the ablt of a bakg sstem to absorb shocks. O the other had accordg to ver-lrmot [19] facal fraglt clearl depeds o several etwork characterstcs. I partcular the hgher the etwork coectvt the larger the umber of baks volved the cotago process ad the qucker the cotago pheomeo. Despte the above-gve cotroverses we base our approach o the tuto that the creasg umber of debt relatos has a postve mpact o the dager of facal solvec. That s also a reaso wh the reducto of the debt relatos becomes the prmar goal of our artcle. However determg a wa to fd possbltes for debt reducto facal etworks s ot a ew topc. Actvtes moblzg ecoomc agets to settle debts ad cut losses from the upad debts have bee dscussed facal research for a log tme. Here despte havg ma commo features the approaches dffer some detals. Rotemberg [15] deals wth the problem of debt repamet betwee ultmate leders ultmate borrowers ad termedares. He cosders rather cosecutve repamet of the debts that s restrcted b the umber of trasactos per tme ut tha debt compesato the structure. The presece of a ccle a debt structure represeted b a drected graph demostrates the ecesst for a outer source of lqudt. Settlg all the debts the structure depeds o the order of the debt repamets made b partcular subects. Rotemberg [15] s cocered wth the amout of addtoal lqudt provded b some subects from outsde the sstem to settle all the oblgatos sde the structure. erhoeff [18] uses graph represetato to solve the problem of the settlg of multple debts betwee agets. The goal s to fd the optmal soluto wth a effcet method that mmzes the umber of lks trasfers betwee credtors ad debtors ad the total amout of moe trasferred. A mportat assumpto about the zero total balace of trasfers has bee made. The effcec of the trasfer process ma be epressed b the lmtato that o more tha N 1 trasfers are eeded the graph wth N vertces N represets the umber of debted agets. Moreover f the ccle sub-graph ests the structure of debts the mmum amout of owed moe ca be subtracted from each graph lk. Boerer ad Hatfeld [4] aalze the clearg process of debts betwee agets the balaced the debts vs. clams posto s zero for each aget ad the ubalaced facal posto. I the balaced case the ccle removal mechasm that clears all debts s used whle the case of ubalaced postos of agets cosecutve removal of debts the cha s appled. Our model s bult o the aforemetoed cocepts; however t dffers ma mportat aspects ad motvatos. The approach we use s devoted to the 8

3 Acta Poltechca Hugarca ol. 12 No compesato of debts amog the agets. Our methodolog focuses o the applcato of the stadard graph theor algorthm to solve the debt compesato process amog frms although the problem ca be rewrtte as a applcato of the lear programmg method wth the mmzato of debts the gve structure as a obectve fucto. The ovelt ca be see the troducto of the followg models: the model mamzg the returs of the MDC orgazer; the model of proftable subsdes ad the model mmzg the clams eposto of the MDC orgazer. Our paper s orgazed as follows. I Secto 2 the cocept of a dgraph of debts s troduced. The formalsm of the mutual debt compesato MDC process s preseted Secto 3. The stadard graph approach of Kle s ccle cacellg algorthm whch solves MDC s appled Secto 4. Modfcato of ths algorthm leadg to mamal retur of the MDC orgazer s descrbed Secto 5. Sectos 6 ad 7 troduce the cocept of a mutual debt compesato orgazer the form of a subsd cetre ad the Mstr of Face. Fall the coclusos are preseted. 2 Dgraph of Debts Let = {12 } stad for a set of frms volved a process of mutual debt compesato. Let the debt that frm owes to frm be deoted as. We assume that at most oe of two mutuall eclusve stuatos > 0 ad > 0 ma occur. The debtor credtor relatoshps are defed b a set E = 2 {[ ] > 0 as }. 1 Here the fucto : E R assgs a real postve debt amout to each debtor credtor relato. The usg the prevous structures we ma troduce formall a dgraph of debts as follows: = E. G 2 As set deotes the -eghborhood of verte represetg the lst of ts debtors ad set deotes the out-eghborhood of verte the lst of ts credtors the recevables/debts balace of frm ma be epressed as b = for. 3 9

4 . Gazda et al. A Applcato of Graph Theor the Process of Mutual Debt Compesato I ths formula the total recevables of frm represeted b part balaced b ts total debts represeted b are. It s mportat to ote that we assume faress of MDC. Ths meas that the balace betwee the recevables ad the debts epressed b Eq. 3 have to rema costat durg the MDC process. Qute obvousl the debt dgraph propert b = 0 4 =1 states that the total sum of recevables/debts s balaced. 3 The MDC Process as Ccle Elmato Ths secto ma be perceved as a tutve troducto to the methodolog based o the cclc propertes of dgraphs. The presece of a ccle the dgraph meas the estece of a cclcal substructure of mutual debts the world of tercoected frms. Let C G be a ccle G G defed Eq. 2. Aga C ma be wrtte as ordered trple C E C comprsg the set of vertces C together wth a set E C of arcs quatfed b the debt stregth descrbed b the correspodg weghts. B defg the capact of ccle C as C = m ee C e the weghts of the edges of C ma be reduced b subtractg the ccle capact C from the weghts of all the arcs lg the ccle 1. The assgmets := C e E C e e 6 eld debt compesato sce the cause a reducto the total debts the orgal dgraph G. At the same tme the debtor credtor relatos represeted b at least oe arc wth zero weght are removed from ccle C as well as from 5 1 Further we are usg double otato of the arcs. Frst the otato wth subscrpts comprsg the ordered par s used f we wat to stress the role of the respectve verte; secod the geeral otato of the arc e s used to dcate a elemet of E. 10

5 Acta Poltechca Hugarca ol. 12 No dgraph G ad thus the ccle s elmated from G. Eq. 3 rewrtte as b = C [ C] demostrates that the recevables/debts balace of each frm before ad after the compesato remas coserved. The graph becomes acclcal due to cosecutve applcato of the procedure to all the ccles preset G. 3.1 Eample: The Chage from a Cclcal to a Acclcal Graph Cosder a smple structure of fve frms wth the debt structure gve Fgure 1A. The debt structure creates a ccle gve b a sequece of vertces ad arcs 1 [12] 2 [23] 3 [34] 4 [45] 5 [51].e. a ccle wth vertces C ={12345} ad arcs E C ={[12][23][34][45] [51]}. Fgure 1 A The debt structure before compesato; B The debt structure after compesato The mmum debt the ccle s a debt of sze 1 betwee 2 ad 3. The debt structure Fgure 1B s obtaed b the ccle elmato.e. subtractg 1 from all the debts the ccle. The resultg dgraph becomes acclcal. 11

6 . Gazda et al. A Applcato of Graph Theor the Process of Mutual Debt Compesato 4 MDC Based o Mamum Crculato the Dgraph of Debts Cosder a strogl coected dgraph G = E 2. I ths graph we defe a crculato fucto [see [7] p. 146] 0 : E R 0 whch satsfes equalt 7 ad costrat =. 8 Further we wll demostrate the sutablt of the crculato cocept for MDC modelg. The varable s terpreted as part of a debt whch ma be compesated betwee frms ad. Therefore the total amout of debts compesated due to the par-wse effect of crculato ma be wrtte as follows: [ ] E. I the MDC process the recevables/debts balace of a frm must ot be worseed or mproved. The relato Eq. 8 reports the compesato of the recevables ad debts o the level of sgle frm. Therefore t s qute obvous to epect that respectg ths codto should ot volate the balace of the recevables/debts. Assume that before the compesato process starts we have b epressed the form of Eq. 3. Subsequetl addg zero the form of the rght-had sde mus the left-had sde of Eq. 8 leaves b uchaged:. b = The orgal debts 9 10 are ow epressed the compesated form. From ths t follows that the mmzato of debts becomes equvalet to the detfcato of mamal crculato [ ] E ma 11 2 The dgraph G = E s strogl coected f for ever par of vertces a drect path ests from to as well as from to. 12

7 Acta Poltechca Hugarca ol. 12 No subect to the costrats epressed b Eq. 7 ad Eq. 8. The ma reaso wh we decded to pass to the mamum crculato problem s that t ca be tackled b meas of the stadard lear programmg methods as well as stadardzed specal-purpose graph algorthms. For eample the applcato of Kle s ccle cacellg see [12] orgall appled to fd the mmum cost crculato mght be possble 3. The algorthm s based o the detfcato of all the ccles ad the determato of the optmal sequece of the elmato the dgraph respectg the trade-off amog the ccle capactes see [7]. Based o Eq. 7 we ca coclude that f = 0 the = 0. Thus the total debt mmzato does ot cause the creato of ew credtor debtor relatos whch s cosdered as a mportat aspect of the proposed model. 4.1 Eample: Ccle Cacellg from a Cclcal to a Acclcal Graph Fgure 2 descrbes the mutual debts of fve frms. The total amout of mutual debts s equal to the sum of arc weghts.e. $25 ml. There are three ccles the dgraph. The frst oe s gve b the debt sequece 1 [12] 2 [23] 3 [34] 4 [45] 5 [51] 1 the secod oe b the sequece 1 [12] 2 [25] 5 [51] 1 ad the thrd oe b the sequece 1 [13] 3 [34] 4 [45] 5 [51] 1. All these ccles cota a commo arc [51] whch determes the mmum capact the secod ad the thrd ccle. Cosequetl the elmato of the secod ccle causes the elmato of arc [51] ad thus the elmato of the frst ad thrd ccles too. O the other had the elmato of the thrd ccle causes the elmato of the frst ad the secod ccle see Fgure 3. I both approaches the dgraph becomes acclcal meag that o more debts ma be compesated b the ccle cacellg. The total amouts of ucompesated debts are dfferet the two cases see Fgure 3A Fgure 3B $19 ml. the frst case ad $17 ml. the secod oe. It s qute obvous that the most approprate method s to elmate ccle 1 [12] 2 [23] 3 [34] 4 [45] 5 [51] 1 frst ad the to elmate 1 [13] 3 [34] 4 [45] 5 [51] 1. We ca see that elmato appled a sutable order causes the total amout of the ucompesated debts to rema at the level of $16 ml. see Fgure 4. 3 We choose Kle s algorthm for demostratve purposes; more effcet crculato algorthms were proposed later Goldberg ad Tara [10] s a good eample. 13

8 . Gazda et al. A Applcato of Graph Theor the Process of Mutual Debt Compesato Fgure 2 The structure of debts before ther compesato Fgure 3 The remag debt structure f: A ccle 1 [12] 2 [25] 5 [51] 1 s elmated; B ccle 1 [13] 3 [34] 4 [45] 5 [51] 1 s elmated Fgure 4 The structure of debts that stems from the applcato of ccle cacellg 14

9 Acta Poltechca Hugarca ol. 12 No The above-formulated MDC heurstc s rather tutve ad does ot provde the optmal orderg of the ccle elmato. That s wh t s applcable the case of large debt etworks wth a hgh level of dest. Here we decded to mamze the graph crculato b the modfed Kle s ccle-cacellg method see [12] whch s etremel effectve eablg us to solve the eteded graph problems. However we avod the usage of large graphs ths artcle because our pctures are teded to be eas to uderstad. We focus mal o the ecoomc aspects of the debt problems. We dd ot wat to sped too much tme o some of the fer detals of the stadard algorthm. 5 Retur of the Mutual Debt Compesato Orgazer Compaes themselves are mostl uable to solve the problem of debts whch eeds etesve mult-compa proects that ca help to solve the problem b tatg ad orgazg the process of debt compesato. I practce the orgazer of MDC epects to be rewarded. Its retur fucto c : E 0;1 assgs a ut prce per moetar ut of each compesated debt. More detaled formato ma be captured b dgraph G = E c. The total come of the orgazer ca be wrtte as c [ ] E. The mamal retur due to applcato of the crculato ma be defed as follows: 12 ma c [ ] E. 13 Aga the optmzato s cosdered commo wth the costrats defed b Eq. 7 ad Eq The Subsd Ceter ad the Problem of Restrcted Subsdes I practce there are a few eamples whch a o-partcpatg subect eters a debt structure to edow a hghl debted frm. It eables t to cover ts debts whch cosequetl allows the elmato of the whole cha of duced debts caused b the solvec of a hghl debted frm. Now we formulate a model 15

10 . Gazda et al. A Applcato of Graph Theor the Process of Mutual Debt Compesato whch the subsd cetre does ot kow a pror whch frm should be subsdzed. Coversel the lst of frms wll be determed b the algorthm. I addto we assume that the sources avalable for provdg the subsdes are restrcted. The verte 1 s added to substtute the effect of the subsd cetre. All of ts relatos wll be represeted b [ 1 ] arcs. Assumg ths structure the lear programmg model from Eq. 13 takes the form ma 1 [ ] E c 14 = 0 for = {12 } 15 1 B 16 0 E for [ ] for. 18 Accordg to Eq. 16 the total amout of the subsdes provded s restrcted b costat B. The subsd provded to the th frm s restrcted b ts upper boud epressed b Eq. 18. Now we fsh the specfc MDC formulato for the purposes of lear programmg. Ths approach s however less effcet for the solvg of ma verte graph problems. Greater effcec ma be gaed b Kle s ccle-cacellg method. I the et secto we wll dscuss ts adapto to the gve problem. 6.1 Kle s Ccle Cacellg Adapted to the Model of the Subsd Cetre erte 1 does ot clude a cdet put edges; o crculato through ths verte s possble. As a soluto we propose a modfed debt dgraph G = E c wth modfed topolog cludg subsd verte 1 as well as aular verte 0. Formall = { 1} {0} All the edges {[ 1 ] } weghted b represet the possble support provded b the subsd cetre localzed at the verte 1. The arc [0 1] eables us to create the flow codto see costrat Eq. 8 16

11 Acta Poltechca Hugarca ol. 12 No balacg the subsd cetre. erte 0 s balaced va addtoal edges {[ 0] }. The the edges of G ma be specfed as follows: E = E {[ 0] } {[0 1]} {[ 1 ] }. 20 : We ow specf fucto wth respect to the orgal E R whch cossts of the addtoal rules = for [ ] E 21 0 = 1 for = m 1 B = Eq. 21 shows that the dstrbuto of debts remas the same as the orgal debt dgraph G whle Eq. 23 descrbes the total amout of dsposable subsdes. The remag costtuet of G represets the retur c : E R. The costs of debt reducto per moetar ut are epressed b c 1 c = c for [ ] E 25 c = 0 for 26 0 c = c 1 = 1 for. 28 After the above modfcatos we obta dgraph G whch eables the applcato of the ccle-cacellg algorthm for the mamum retur crculato. 6.2 Eample: The Role of the Subsd Cetre The structure of the mutual debts amog the frms s represeted the dgraph Fgure 2. We cosdered $25 ml of the total mutual debts. The coeffcets c are set to be equal to 0.5 for all the arcs returs. We assume that the subsd cetre 1 = verte 6 s allowed to provde a subsd b a amout equal 17

12 . Gazda et al. A Applcato of Graph Theor the Process of Mutual Debt Compesato to $2 ml. to the frms 135. The total sum of the subsdes s restrcted b the upper boud B = $3 ml. The dagram of ths stuato s preseted Fgure 5. Fgure 5 The tal stuato before the compesato of debts The optmal soluto of the above-metoed optmzato model s gve Fgure 6A. The sum of the compesated debts s $15 ml. The subsd ceter provdes a subsd to frm 1 to the amout of $2 ml. The resdual debts are represeted Fgure 6B. It shows that the subsdes avalable were ot completel used ad $1 ml. remaed. The MDC orgazer acheves a proft of $5.5 ml. Fgure 6 A The crculato of compesated debts; B The structure of mutual debts that results from the process of ther compesato 18

13 Acta Poltechca Hugarca ol. 12 No Mmzato of Recevables b the Mstr of Face The orgazer of the mutual debt compesato s ulkel to be terested dmshg the total debts the whole complcated structure of debts; stead he clams to decrease the recevables/debts of a sgle partcular subect. Ths motvated us to cosder a realstc stuato whch the role of the orgazer of debt elmato s the Mstr of Face. O oe had the Mstr collects ts come from frms e.g. b wa of taes but o the other had t speds ts budget b carrg pamets to the frms provdg servces. Here we do ot pa atteto to the facal relatos wth aother subects. Let = {12 } be a set of frms. B formulato terms of graphs the Mstr of Face represeted b verte 1 s teded for mutual debt compesato. The a set = { 1} volves all the subects cluded the compesato process. Let 1 represet a debt of frm towards the Mstr ad 1 represet the recevable of frm towards the Mstr. The comprehesve debt structure s represeted b E = 1 E {[ 1 ] > 0} {[ 1] 1 > 0} 29 where the structure of debts E s defed Secto 2. The fucto assgg a partcular debt to each debt relato s defed as follows: : E R 30 where = for [ ] E 31 1 = 1 for 32 1 = 1 for. 33 The dgraph G = E represets the mutual debt structure amog the frms ad the Mstr. The recevables/debts balace of the Mstr s epressed as = b

14 . Gazda et al. A Applcato of Graph Theor the Process of Mutual Debt Compesato 20 B followg the deas troduced above we defe crculato ; 0 ; : E R 35 = 36 as mutual debt compesato gve a dgraph of debts. It should be oted that f the recevables/debts balace of subects 12 dgraph = E G remas costat the process of the mutual debt compesato the the balace of subect 1 remas costat too. Ths prevets the Mstr from mprovg ts recevables/debts balace b orgazg the mutual debt compesato. The proof s based o the fact that the sum of the recevables/debts balaces of all the subects before debt compesato s see Secto 3 = The balace codto must also be satsfed after the compesato.e. for = If the faress codto of the compesated debts s satsfed for each of the partcpatg frms = 39 the followg codto must hold too. = 40 It drectl mples the faress of the compesato of the Mstr of Face:. =

15 Acta Poltechca Hugarca ol. 12 No The problem of the mmzato of Mstr recevables ma be modeled as the mamzato of the obectve fucto: ma = for 0 E 43 for [ ]. 44 Such formulato admts the applcato of the lear programmg methods. The are effcet for graphs of a small or moderate sze. O the other had Kle s ccle-cacellg method dscussed the et subsecto represets a more sutable tool for larger graphs. 7.1 Recevables of the Mstr of Face Tackled b Kle s Ccle-Cacellg Method The mplemetato of ew vrtual verte 0 order to appl Kle s cclecacellg method elds ~ = {0}. 45 The all the arcs eterg verte 1 are redrected to the vrtual verte 0.e. {[ 1] ; 1 > 0} {[ 0] } 46 ad a ew arc [0 1] s added. It elds the set of arcs ~ E = E {[ 0] > 0} {[0 1]} The fucto 1 1 ] > 0}. 47 {[ 1 ~ s defed as : E ~ R ~ = for [ ] E 48 ~ 0 = 1 for 1 49 ~ =

16 . Gazda et al. A Applcato of Graph Theor the Process of Mutual Debt Compesato ~ 1 = 1 for. 51 ~ ~ Due to the fact that dgraph G = E ~ ~ cotas verte 0 the mmzato of the recevables becomes equvalet to the fdg crculato wth the mamum value alog arc [0 1]. Ths leads us to the applcato of the aforemetoed mamum retur crculato method wth the retur fucto elemets c ~ E = 0 for [ ] 52 c ~0 = 0 for 53 c ~ = c~ 1 = 0 for Eample: The Mstr of Face as the MDC Orgazer Let the tal stuato of mutual debts be represeted b the dgraph Fgure 7A. The verte eumerated b 6 represets the Mstr of Face. The Mstr has recevables towards frm 5 to a amout of $2 ml. ad towards frm 4 to a amout of $3 ml. The Mstr owes $5 ml. to frm 1 ad $1 ml. to frm 3. We assume that the Mstr orgazes the compesato of mutual debts whereb the mmzg of ts recevables s ts obvous obectve fucto see Eq. 42. Fgure 7 The tal stuato before debt compesato A G = E to be solved b the lear programmg methods ~ ~ ~ ~ B G = E to be solved b a approprate ccle-cacellg method 22

17 Acta Poltechca Hugarca ol. 12 No I order to eable the applcato of the crculato algorthm we eted the dgraph that cludes verte 0 see Fgure 7B. Accordgl the arcs [46] ad [56] wth weghts 2 ad 3 Fgure 7A are redrected to the 0 see Fgure 7B. I addto a ew arc [06] wth debt weght 5 s created. Ths modfed dgraph becomes read for the applcato of the crculato mamzato alog [06]. After the applcato of the ccle-cacellg method the resdual debts dmsh to 14 see Fgure 8B. All the ccles are elmated from the fal form of the dgraph. Cocluso Fgure 8 A The crculato of the compesated debts; B The resultg structure of debts I ths paper we dscuss the prcples of the graph theor that are relevat to performg the MDC procedure. Prcpall ew formulatos are preseted cludg models wth a subsd ceter ad ther coverso to a form tractable b Kle s ccle-cacellg algorthm. Besdes the solvg of the geeral task of the ccle cacellg we demostrate the method s applcablt to the problem of recevables of a sgle selected frm. Our research shows that both topc ad methodolog are stll far from beg ehausted. The approach ca be adapted straghtforwardl to dgraphs of the Iteratoal relatoshps amog the coutres vertces equpped wth the mport eport lks. The mpulses for ogog research ma come from studes o the alterg ature of the ecoomc world. Damc graph models of debt structures should be troduced that formalze the cocept of the fleble lks. These ca be trasformed uder the fluece of the stochastcall varg eteral codtos represeted b the reevaluatg/devaluatg domestc currec. Ackowledgemet The paper was elaborated wth the proect EGA No 1/1195/12. Refereces [1] Ahua R. K. Magat T. L. Orl J. B.: Network Flows: Theor Algorthms ad Applcatos. Pretce Hall Uverst of Mchga 1993 [2] Alle F. Gale D.: Optmal Facal Crses. Joural of Face :

18 . Gazda et al. A Applcato of Graph Theor the Process of Mutual Debt Compesato [3] Alle F. Gale D.: Facal Cotago. Joural of Poltcal Ecoom : 1-33 [4] Boerer L. Hatfeld J. W.: The Ecoomcs of Debt Clearg Mechasms. Workg Paper No. 2010/27 School of Busess Ecoomcs Dscusso Paper 2010 [5] Chartrad G. Oellerma O. R.: Appled ad Algorthmc Graph Theor. McGraw-Hll Uverst of Calfora 1993 [6] Damod D. W. Dbvg P. H.: Bak Rus Depost Isurace ad Lqudt. Joural of Poltcal Ecoom : [7] Destel R.: Graph Theor. Sprger-erlag Lodo 1997 [8] Feceko J.: About Optmsato of Recevables ad Paables Compesato. Ekoomcký časops : [9] Gazda.: Mutual Debts Compesato as Graph Theor Problem. I: Mathematcal Face 2001 [10] Goldberg A.. Tara R. E.: Fdg Mmum-Cost Crculatos b Cacelg Negatve Ccles. Joural of the ACM : [11] Harar F.: Graph Theor. Addso-Wesle Publshg Compa Lodo 1969 [12] Kle M.: A Prmal Method for Mmum Cost Flow wth Applcato to the Assgmet ad Trasportato Problem. Maagemet Scece : [13] Ju Q. Wg L. N.: Asmmetrc Lqudt Rsk Prema Itrada Hgh Frequec Tradg. Iteratoal J. of Appled Mathematcs [14] Ner E. Yag J. Yorulmazer T. Aletor A.: Network Models ad Facal Stablt. Joural of Ecoomc Damcs ad Cotrol : [15] Rotemberg J. J.: Lqudt Needs Ecoomes wth Itercoected Facal Oblgatos. Natoal Bureau of Ecoomc Research Workg Paper 2008 [16] Tara R. E.: Depth Frst Search ad Lear Graph Algorthms. SIAM J. Computg : [17] Upper C.: Smulato Methods to Assess the Dager of Cotago Iterbak Markets. Joural of Facal Stablt : [18] erhoeff T.: Settlg Multple Debts Effcetl: A Ivtato to Computg Scece. Iformatcs Educato : [19] ver-lrmot S.: Cotago Iterbak Debt Networks. Workg Paper