Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

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1 Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria {maheswar, tambe, bowrig, jppearce, DCOP Formulatios for DiMES: Proofs of Propositios TSA Time Slots as ariables): This method reflects a atural first step whe cosiderig schedulig issues. Let us defie a DCOP where a variable x t) represets the -th resource s t-th time slot. Thus, we have N T variables. Each variable ca a tae o a value of the idex of a evet for which it is a required resource, or the value 0 to idicate that o evet will be assiged for that particular time slot: x t) {0} { {1,..., K} : R A }. It is atural to distribute the variables i a maer such that {x 1),..., x T)} belog to a aget represetig the schedule of the -th resource. Propositio 1 The DCOP formulatio with time slots as variables, where the costrait betwee variables x 1 t 1 ) ad x 2 t 2 ) whe x 1 t 1 ) = ad x 2 t 2 ) = 2 taes o the utility f 1, t 1, ; 2, t 2, 2 ) = ) ) ) I {1 2 } 1 I{t1 t 2 } 1 I{P1 P 2 A {1,...,K}} fiter, 2 ) + I {1 = 2 } 1 I{t1 =t 2 } fitra 1 ; t 1, ; t 2, 2 ) where the iter-aget costrait utilities are f iter, 2 ) = MI {1 2 }I {A A 2 } ad itra-aget costraits fully coected amog the time slots of a sigle aget) for t 1 < t 2 w.l.o.g. are f itra ; t 1, ; t 2, 2 ) = M 0, t 2 t 1 < L, 2 M 0, t 2 t 1 L, 2 = g; t 1, ; t 2, 2 ) otherwise where g; t 1, ; t 2, 2 ) = 1 [ 0 t 1 ) ) I {1 0} + 2 i 0 T 1 t 2) ) ] I {2 0} with M > NT max where N is the umber of participats, T is the umber of time slots ad max = max,, yields the optimal solutio to the Distributed Multi-Evet Schedulig DiMES) problem. ) Proof. The fuctio f ca be aalyzed as follows. The first term i the factor I {1 2 } 1 I{t1 t 2 } idetifies a iteraget costrait ad the secod term reflects that iter-aget costraits exist oly across idetical time slots. The factor 1 I {t1 t 2 }) implies that a iter-aget li exists betwee x1 t) ad x 2 t), oly if there exists a evet that requires both P 1 ad P 2 as attedees. The factor I {1 = 2 } idetifies a itra-aget costrait ad subsequetly ) 1 I {t1 =t 2 } expresses that there is o costrait from a variable to itself. The fuctio f iter ) idicates that a pealty of M is assessed if the agets assig differet evets for the same time 2 ) ad the evets force a participat to be at two evets at the same time A A 2 ). 1 1 For otatioal cosistecy i the situatio where a agets decides ot to assig a evet to a particular time slot, we defie A 0 :=. 1

2 The fuctio f itra ) assesses a pealty to esure that at least L slots must be assiged cotiguously whe evet E is scheduled 0, t 2 t 1 < L, 2 ). Also, a pealty uder the secod coditio 0, t 2 t 1 L, 2 = ) is assessed to esure that o more tha L slots are assiged ad also to esure that the same evet is ot scheduled twice. If oe of these coditios are met, the utility o the costrait is the differece i the values for attedig the evets ad the value of eepig the time slots uassiged divided by T 1 the umber of outgoig itra-aget lis). The sum of all costrait utilities excludig the pealties) is N T 1 =1 s=1 t=s+1 T g; s, x i s); t, x t)) N T 1 T =1 s=1 t=s+1 2 max N T 1 = =1 N T 1 T =1 s=1 t=s+1 1 [ x i s) T 1 ] + x it) TT 1) 2 max 2 T 1 = NT max = M Thus, the total utility if a pealty is icurred is o-positive. Cosequetly, it would be better to assig a value of 0 to all variables ad obtai a global utility of zero tha cosider a solutio with a pealty. This implies that ay optimal solutios to the DCOP will ot icur ay pealties. The absece of iter-aget pealties imply, for ay, {t : x 1 t) = E } = {t : x 2 t) = E }, 1, 2 such that P 1, P 2 A. This expresses that if a evet is scheduled all participats will have assiged the idetical set of time slots to that evet. The absece of itra-aget pealties implies that if P A, {t : x t) = E } = {t0,..., t 0 + L 1} where t0 = mi{t : x t) = E }. This expresses that a evet will be scheduled i cotiguous slots totalig the exact legth of the evet. If we defie S E ) := {t : x t) = E } where P A, the because two evets caot be assiged to the same time slot, we have S E ) S E 2 ) =, 2 {1,..., K}, 2, A A 2. 1) Summig over the lis, we have the followig global utility: N T 1 T =1 s=1 t=s+1 1 [ x s) 0 s) ) I {x s) 0} + x t) 0 t) ) ] I {x t) 0} T 1 Because the itra-aget variables are fully coected, each variable has T 1 outgoig lis. Rewritig the previous expressio as a sum over time slots, we have the global utility as: N T x t) 0 t) ) N K I {x t) 0} = I {i A } 0 t) ) K = R t) ). 2) =1 t=1 =1 =1 t S E ) =1 A t S E ) The first equality i 2) is obtaied by partitioig the time domai for each participat ito scheduled evets ad the secod equality by switchig the order of the first two summatios. The solutio to the DCOP will determie a schedule that maximizes the fial expressio i 2) which, whe coupled with the coditios i 1) implied by the absece of pealties, is idetical to the multi-evet schedulig problem MESP). EA Evets as ariables): We ote that the graph structure of TSA grows as the time rage cosidered icreases or the size of the time quatizatio iterval decreases, leadig to a deser graph. A alterate approach is to cosider the evets as the decisio variables. Let us defie a DCOP where the variable x represets the startig time for evet E. Each of the K variables ca tae o a value from the time slot rage that is sufficietly early to allow for the required legth of the evet or 0 which idicates that a evet is ot scheduled: x {0, 1,..., T L + 1}, = 1,..., K. If a variable x taes o a value t 0, the it is assumed that for all required resources of that evet A ), the time slots {t,..., t + L 1} must be assiged to the evet E. It would be logical to assig each variable/evet to the aget of oe of the required resources for the evet. Propositio 2 The DCOP formulatio with evets as variables where a costrait betwee variables x ad x 2 whe x = t 1 ad x 2 = t 2 exists if the two evets have a commo required attedee A A 2 for 2 ), ad 2

3 cosequetly taes o the utility f, t 1 ; 2, t 2 ) = M t 1 0, t 2 0, t 1 t 2 t 1 + L M t 1 0, t 2 0, t 2 t 1 t 2 + L 2 g, t 1 ; 2, t 2 ) otherwise. where g, t 1 ; 2, t 2 ) = β I {t1 0} L 0 t 1 + l 1) ) + β 2 I {t2 0} A l=1 L 2 A 2 l=1 2 0 t 2 + l 1) ) for β i = 1/ K=1 I {A A i } 1 ) multiplicative iverse of the umber of outgoig lis) with M > NT max where N is the umber of participats, T is the umber of time slots ad max = max,, yields the optimal solutio to the Distributed Multi-Evet Schedulig DiMES) problem. Proof. The first two coditios of the fuctio f state that a pealty of M is assessed if the variable assigmets cause a schedulig coflict. If o pealties are imposed, the utility gai for schedulig the evet the differece betwee the rewards for all the attedees a the values for eepig the utilized times uassiged) is distributed uiformly over the outgoig lis from the variable. Let us assume that a pealty is icurred because of a schedulig coflict betwee the variables x = t 1 0 ad x 2 = t 2 0. By settig x = 0 decidig ot to schedule evet E ), the global utility will chage by at least β 2 L 2 2 A 2 l=1 0 t 2 + l 1) ) K=1 I {A A i } 2 L K=1 I {A A i } 1 A l=1 0 t 1 + l 1) ) M where the term i the first set of bracets represets the value of the li whe x = 0 ad the term i the secod set of bracets represets the value of the lie whe x = t 1 0. We ca see that the utility chage is greater tha L M > M NT max > 0. A l=1 Thus, the optimal solutio to the DCOP caot have ay pealties o the lis as we ca always mae a utility gai by ot schedulig a evet that leads to a pealty. Let us defie S E ) = {t,..., t + L 1} if t 0 ad S E ) = if t = 0. Because t + L 1 T which S E ) T. Furthermore, we ow that the optimal solutio will have o coflicts, i.e. S E ) S E 2 ) =, 2 {1,..., K}, 2, A A 2. 3) Sice we have o pealties, the uiformly distributed gais from schedulig ca be aggregated. The global utility will be the sum of the rewards for the scheduled evets mius the values of the utilized times: K L 0 t + l 1) ) = K 0 t) ). =1 A l=1 =1 A t S E ) The DCOP solutio maximizes the previous expressio. This alog with coditio 3), is idetical to the MESP problem. PEA Private Evets as ariables): We ote that i EA, if a aget is to mae a decisio for a evet as a variable, it must be edowed with both the authority to mae assigmets for multiple resources as well as have valuatio iformatio for all required resources. There are settigs where resources, though part of a team, are uwillig or uable to cede this authority or iformatio. To address this, we cosider a modificatio of EA that protects these iterests. Let us defie a set of variables X := {x : A } where x {0, 1,..., T L +1} deotes the startig time for 3

4 evet E i the schedule of R which is a required resource for the evet. If x = 0, the R is choosig ot to schedule E. We the costruct a DCOP with the variable set X := =1 K X. Let us ow defie a set X := {xm X : m = } X which is the collectio of variables pertaiig to the -th resource. Clearly, X > 0, otherwise the resource is ot required i ay evet. If X = 1, let X := X {x} 0 where xi 0 0 is a dummy variable. Otherwise, X := X. The DCOP partitios the variables i X to a aget represetig the -th resource s iterests. Let all the variables withi X itra-aget lis) be fully coected. The additio of the dummy variable to sets X with cardiality oe is to esure that itra-aget lis exist for all agets. Iter-aget lis exist betwee the variables for all participats of a give evet, i.e., all the variables i X are fully coected. Propositio 3 The DCOP formulatio with private evets as variables, where the costrait betwee the variables x 1 ad x 2 2 whe x 1 = t 1 ad x 2 2 = t 2 taes o the utility f 1,, t 1 ; 2, 2, t 2 ) = MI {1 2 }I {1 = 2 }I {t1 t 2 } + I {1 = 2 }I {1 2 } f iter 1 ;, t 1 ; 2, t 2 ). 4) where f iter ;, t 1 ; 2, t 2 ) = M t 1 0, t 2 0, t 1 t 2 t 1 + L M t 1 0, t 2 0, t 2 t 1 t 2 + L 2 g;, t 1 ; 2, t 2 ) otherwise ad g;, t 1 ; 2, t 2 ) = 1 Z t 1 ) + Z 2 t 2 ) ) where Z i t i ) = X 1 L i l=1 i 0 t i + l 1) ) I {ti 0} with M > NT max where N is the umber of participats, T is the umber of time slots ad max = max,, yields the optimal solutio to the Distributed Multi-Evet Schedulig DiMES) problem. Proof. The first term i 4) characterizes that a pealty of M is assessed o a iter-aget li 1 2 ) for a commo evet = 2 ) for which the same startig time is ot selected t 1 t 2 ) by the coected resources. The latter term i 4) addresses itra-aget costraits 1 = 2 ) betwee differet evets 2 ) where the li utility f iter ) esures that a pealty is icurred o a itra-aget costrait if a schedulig coflict is created. Otherwise, the utility gai for a resource assigig a viable time for a evet is uiformly distributed amog the outgoig itra-aget lis as deoted i g ). Let us assume that a pealty is icurred o a iter-aget costrait. This implies that the required resources for a particular evet could ot agree o a commo time to start. Sice the total utility gai excludig pealties) for holdig a evet E caot exceed L max NT max < M, A t=1 there exists a solutio for the DCOP where the evet is ot scheduled which is at least as good as that with the evet scheduled. Let us ow assume that a pealty is icurred o a itra-aget li. This implies that a aget has chose startig times for two evets that causes the same time slot to be assiged to two evets. By similar logic, the pealty M is sufficietly large such that by choosig ot to schedule oe of the evets ad allowig all other agets to choose ot to schedule that evet thereby avoidig icurrig a iter-aget pealty), we obtai a higher quality solutio. The above aalysis implies that the optimal DCOP solutio is void of assigmets that would activate a pealty. Thus, x = x m, m, A. Give E ad some A, let us defie S E ) = if x = 0, ad S E ) = {x,..., x + L 1} if x 0. The, we have S E ) S E 2 ) =, 2 {1,..., K}, 2, A A 2. 5) 4

5 Otherwise, a pealty would have bee assessed. The global utility is the the sum of all itra-aget lis devoid of pealties g )), which ca be represeted as N K =1 =1 = I { A } K L =1 A l=1 1 K X 1 Z x) =1 I { A } 0 x + l 1) ) I {x 0} = N K 1 = I { A }Zx ) K =1 =1 =1 A t S E ) 0 t) ). The solutio to the DCOP maximizes the previous expressio, which whe coupled with the o coflict coditio i 5) is idetical to the DiMES problem. 5

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