The Multi-Inter-Distance Constraint

Transcription

1 The Mult-Inter-Dstance Constrant Perre Ouellet Unversté Laval Département d nformatque et de géne logcel perre.ouellet.4@ulaval.ca Claude-Guy Qumper Unversté Laval Département d nformatque et de géne logcel claude-guy.qumper@ft.ulaval.ca Abstract We ntroduce the MULTI-INTER-DISTANCE constrant that ensures no more than m varables are assgned to values lyng n a wndow of p consecutve values. Ths constrant s useful for modelng schedulng problems where tasks of processng tme p compete for m dentcal resources. We present a propagator that acheves bounds consstency n cubc tme. Experments show that ths new constrant offers a much stronger flterng than an edge-fnder and that t allows to solve larger nstances of the runway schedulng problem. Introducton Constrant solvers offer many ways to model schedulng problems. The constrant ALL-DIFFERENT (Régn 1994) ensures that at most one task starts at a gven tme. When there are m resources, the GCC (Régn 1996) lmts the number of tasks startng at a gven tme to m. These two constrants are useful when tasks have a unt processng tme. When processng tmes are greater than one unt, the INTER-DISTANCE (Artouchne & Baptste 2005; Qumper, López-Ortz, & Pesant 2006) constrant makes sure that the startng tmes are at least p unts of tme apart. When processng tmes dffer for each task, the CUMULATIVE (Mercer & Hentenryck 2008; Vlím 2009) constrant allows to model ths stuaton. All constrants, except the CUMULATIVE constrant, offer a complete flterng of the release tmes and deadlnes of the task n polynomal tme. Ths flterng s NP-Hard for the CUMULATIVE constrant. However, there exsts no flterng algorthm that prunes the release tmes and deadlnes when the processng tme of each task s p and when there are m > 1 resources. We ntroduce the MULTI-INTER-DISTANCE constrant and ts frst flterng algorthm achevng bounds consstency. Ths constrant complements the offerng of constrant solvers. We show that ths constrant s very effcent to solve problems such as the runway schedulng problem. The MULTI-INTER-DISTANCE Constrant The MULTI-INTER-DISTANCE constrant lmts the number of varables assgned n a wndow of p consecutve values to m. Defnton 1 formally presents the constrant. Copyrght c 2011, Amercan Assocaton for Artfcal Intellgence ( All rghts reserved. Defnton 1 MULTI-INTER-DISTANCE([X 1,..., X n ], m, p) s satsfed f and only f at most m varables X are assgned to a value n any wndow of p consecutve values. More formally, we have the followng logcal dentty. MULTI-INTER-DISTANCE([X 1,..., X n ], m, p) v { X [v, v + p)} m The MULTI-INTER-DISTANCE s a generalzaton of well known constrants. When p = 1 and m = 1, the MULTI-INTER-DISTANCE constrant specalzes nto an ALL-DIFFERENT constrant. When p = 1 and m > 1, the MULTI-INTER-DISTANCE encodes a GCC where each value can be assgned to at most m varables. Fnally, wth m = 1 and p > 1, the MULTI-INTER-DISTANCE becomes an INTER-DISTANCE constrant. On the other hand, the MULTI-INTER-DISTANCE constrant s a specalzaton of the CUMULATIVE constrant. Each varable can be seen as the startng tme of a task that consumes one unt of resource durng a perod of tme p where the cumulatve capacty of the resources s m. Two results emerge from these relatons among the constrants. Snce t s NP-Hard to enforce doman consstency on the INTER-DISTANCE constrant (Artouchne, Baptste, & Dürr 2004), t s necessarly NP-Hard to enforce doman consstency on the MULTI-INTER-DISTANCE. Because the edge-fnder (Mercer & Hentenryck 2008; Vlím 2009) does not enforce bounds consstency on the INTER-DISTANCE (Artouchne & Baptste 2005), t does not enforce bounds consstency on the MULTI-INTER-DISTANCE ether. We show how to detect feasblty of the MULTI-INTER-DISTANCE constrant n quadratc tme. We then show how bounds consstency can be enforced n cubc tme. General Background We consder constrant satsfacton problems where varables are assgned to nteger values. The doman of a varable X s denoted dom(x ). We assume that the doman of a varable dom(x ) = [l, u ) s a sem-open nterval where l and u are called the lower bound and upper bound of the doman. The value of the varable X must be greater than or equal to l and must be strctly smaller than u.

2 An nterval support for a value v dom(x ) wth respect to a constrant C(X 1,..., X n ) s a tuple t satsfyng three condtons for j 1..n: 1) l j t[j] < u j, 2) t[] = v and 3) the tuple (t[1],..., t[n]) satsfes the constrant C. A constrant s bounds consstent f l and u 1 have an nterval support for all = 1..n. Enforcng bounds consstency on a constrant conssts of ncreasng the lower bounds and decreasng the upper bounds of the domans untl all bounds have an nterval support. Example 1 shows the result of enforcng bounds consstency on the MULTI-INTER-DISTANCE constrant. Example 1 We consder the constrant MULTI-INTER-DISTANCE([X 1, X 2, X 3, X 4, X 5 ], m, p) wth m = 2, p = 3, and the followng domans. dom(x 1 ) = [7, 9) dom(x 2 ) = [2, 4) dom(x 3 ) = [4, 7) dom(x 4 ) = [2, 7) dom(x 5 ) = [3, 5) The domans are sem-open ntervals whch mples that the value 1 belongs to the doman of X 1 but the value 9 does not belong to the doman of X 1. Enforcng bounds consstency on the MULTI-INTER-DISTANCE shrnks the domans as follows. dom(x 1 ) = [8, 9) dom(x 2 ) = [2, 3) dom(x 3 ) = [5, 7) dom(x 4 ) = [5, 7) dom(x 5 ) = [3, 4) Testng for Satsfacton n Polynomal Tme The MULTI-INTER-DISTANCE has a strong connecton to schedulng problems. Consder the constrant MULTI-INTER-DISTANCE([X 1,..., X n ], m, p) where the doman of the varable X s the sem-open nterval [l, u ). The varable X can be consdered as the startng tme of a task whose processng tme s p, whose release tme s l, and whose deadlne s u + p 1. A total of m dentcal resources execute the tasks concurrently wthout preempton. The MULTI-INTER-DISTANCE constrant s satsfable f and only f there exsts a schedule. Ths schedulng problem can be solved n polynomal tme. The best algorthm (López-Ortz & Qumper 2011) so far has a worst-case tme complexty of O(n 2 mn(1, p m )). The algorthm fnds a vald schedule by computng a shortest path n a schedulng graph (Dürr & Hurand 2009). Ths secton provdes a descrpton of ths graph and ts propertes. The schedulng graph s the key concept on whch the flterng algorthm s based. Let l mn = mn l and u max = max u be the smallest lower bound and the greatest upper bound of a varable doman. The schedulng graph G has a node v for every nteger l mn v u max. There s a forward edge of weght m from node v to node v + p for all l mn v u max p. For every par of lower bound and upper bound l < u j, there s a backward edge (u j, l ) of weght {k l l k u k u j }. The absolute value of the weght of the backward edge (u j, l ) s the number of varables whose doman s contaned n the nterval [l, u j ). A backward edge can have a null weght. There s a null edge from any node v + 1 to node v of weght 0 for all v [l mn, u max ). Fnally, there s α\β Table 1: The weght matrx of the schedulng graph of Example 1. Empty entres ndcate the lack of an edge between nodes α and β.. an edge (l mn, u max ) of weght n where n s the arty of the MULTI-INTER-DISTANCE constrant. Table 1 shows the weght matrx of the schedulng graph of the problem depcted n Example 1. The schedulng graph has mportant propertes that we state here. Theorem 1 provdes the suffcent and necessary condton to test whether the schedulng problem has a soluton. Theorem 1 ((López-Ortz & Qumper 2011)) The schedulng problem has a soluton f and only f the schedulng graph has no negatve cycles. Theorem 2 ((López-Ortz & Qumper 2011)) Testng whether there s a negatve cycle n the schedulng graph can be done n O(n 2 mn(1, p m )) tme. Theorems 1 and 2 allow to test the satsfablty of the constrant n quadratc tme. In the next sectons, we reuse ths result to desgn an effcent flterng algorthm. Condtons to Mantan Bounds Consstency Our flterng algorthm detects forbdden sets of values that cannot be assgned to a set of varables. For the ALL- DIFFERENT constrant, these sets are called Hall ntervals (Leconte 1996). For the MULTI-INTER-DISTANCE, we call them forbdden regons. We present lemmas that detect these forbdden regons. To test whether an nterval [a, b) s a forbdden regon to whch a varable X s subject, one can modfy the doman of X to be [a, b) and test the satsfablty of the problem. If the schedulng graph of the transformed problem has a negatve cycle, then [a, b) s a forbdden regon for X. We defne the altered schedulng graph G v lke the schedulng graph wth one excepton, we alter the upper bound of the doman of X such that the new doman becomes [l, v). If G v has a negatve cycle, the constrant has no soluton and we conclude that X v. Table 1 gves the weght matrx of the graph G. Decrementng bold entres by one results n the altered schedulng graph G 5 3. In the graph G 5 3, the edges (5, 2) and(2, 5) form a cycle of weght -1 whch proves that X 3 5. The followng theorem shows how a forbdden regon for one varable can also be a forbdden regon for another varable.

3 Theorem 3 Let X and X j be two varables whose domans satsfy u u j. If the values n [l, v) have no nterval support n dom(x ), then they do not have an nterval support n dom(x j ) ether. Proof: We prove the contraposton: f there exsts a value a [l, v) that has an nterval support n the doman of X j then there exsts a value b [l, v) that has an nterval support n the doman of X. Suppose there exsts an nterval support t such that t[j] [l, v), we want to prove that there exsts an nterval support t such that t [] [l, v). Two cases can occur. If t[] t[j] then the nequaltes l t[] t[j] < v hold and t[] [l, v). The theorem holds wth t = t. However, f t[] > t[j] the nequaltes l j t[j] < t[] u u j hold. We can permute the values t[] and t[j] to obtan the support t. We have t [] = t[j] [l, v) and t [j] = t[] [l j, u j ) hence t s a vald support. In Example 1, the graph G 5 3 has a negatve cycle whch proves that [4, 5) s a forbdden regon for X 3. Theorem 3 also makes [4, 5) a forbdden regon for the varable X 4. The dstance matrx D G of a graph G s a square matrx such that D G [a, b] s the shortest dstance between node a and node b n graph G. Ths matrx exsts only f there are no negatve cycles n the graph G. When comparng the schedulng graph G of Example 1 and the altered schedulng graph G 5 3, one sees that the weghts of a subset of the edges are decremented by one. Ths can only lead to shorter paths n G 5 3 than n G. Lemma 1 captures ths property. Lemma 1 Let G be the schedulng graph of a problem where the domans are gven by the ntervals [l, u ) for 1 n. Let G be the schedulng graph of the same problem except that the doman of X s changed to [l, u ) where l l < u u holds. Thus D G [a, b] D G [a, b] holds for all pars of nodes (a, b). Proof: The weght of a backward edge (u j, l k ) s mnus the number of varable domans contaned n the nterval [l k, u j ). If the doman of a varable s shrunk, the weght of the backward edges can be decremented by one or reman unchanged. These changes to the schedulng graph can only favor shorter paths. However, the backward edges outgong from u or ncomng to l may not exst n G as u and l may not be doman bounds after shrnkng the doman of X. Let (u f, l e ) be a backward edge n G such that f = e =. We prove that the edge (u f, l e ) n G can be replaced by an equvalent path n G. Let u f = max({u k u k u f k } {u }) and let l e = mn({l k l e k } {l }). The weght of the backward edge (u f, l e) n G s the same as the weght of the backward edge (u f, l e ) n G. Moreover, we have l e l e and u f u f. Any path n G passng by the edge (u f, l e ) can be replaced by a sequence of null edges from u f to u f, then passng by the edge (u f, l e), and completng wth a sequence of null edges from l e to l e. Ths path n G has the same weght as the edge (u f, l e ) n G. Let U = {u 1..n} be the set of the doman upper bounds and let u = mn{u U u > l } be the smallest upper bound that does not make the nterval [l, u ) empty. The altered schedulng graph G u gves nformaton about how the lower bound of X can be fltered. If G u has a negatve cycle, then X u and the lower bound l can be safely ncreased to u. If G u has no negatve cycle, then Theorem 4 and Theorem 5 ndcates how to prune the lower bound of varable X. Theorem 4 Let X be a varable subject to a MULTI-INTER-DISTANCE constrant. Let u be the smallest upper bound that s greater than l, and let G u be ts assocated altered schedulng graph. Suppose ths graph has no negatve cycle. Let t be the largest value such that the dstance D G u [l, t] s null. Values smaller than t n the doman of X have no nterval support. Proof: Snce G u has no negatve cycles, there exsts at least one soluton wth X [l, u ) where u = mn{u k u k > l }. Let t be the largest value for whch the shortest dstance n G u from l to t s null. We prove by contradcton that t < u. Suppose t u. The weght of the edge (u, l ) s at most -1 snce t fully contans the altered doman of X. Contradctorly, there s a negatve cycle n G u startng from l, gong to t through a path of weght zero (by defnton of t), passng by the null edges (t, t 1), (t 1, t 2),..., (u + 1, u ) wth weght zero, and then gong back to l usng the edge (u, l ) wth weght at most -1. Therefore, t < u. Knowng that t < u, we construct the altered schedulng graph G t usng the orgnal varable domans except for X for whch we use the doman dom(x ) = [l, t). The edge (t, l ) has a negatve weght snce the nterval [l, t) contans the doman of X. The dstance from l to t s null n G u and, by Lemma 1, no greater than 0 n G t. Therefore, there s a negatve cycle n G t passng by the edge (t, l ) whch mples that no values n [l, t) have an nterval support n dom(x ). To llustrate Theorem 4, consder the varable X 1 n Example 1. The smallest upper bound that s greater than the lower bound 7 s u = 9. The altered schedulng graph G 9 1 s dentcal to the schedulng graph G. The path (7, 2), (2, 5), (5, 8) has a null weght n G 9 1. By Theorem 4, there s no vald assgnment wth X 1 < 8. Theorem 4 shows there s no nterval support X [l, t). Theorem 5 shows that there s an nterval support for X = t. Theorem 5 Let X be a varable subject to a MULTI-INTER-DISTANCE constrant. Let u be the smallest upper bound that s greater than l, and let G u be ts assocated altered schedulng graph. Suppose ths graph has no negatve cycle. Let t be the largest value such that the dstance D G u [l, t] s null. The value t has an nterval support n the doman of X. Proof: We construct the schedulng graph G t+1 usng the orgnal varable domans except for X for whch we use the doman [l, t + 1). If one proves that G t+1 has

4 no negatve cycle, then one proves there s a soluton wth X [l, t + 1). Theorem 4 ensures that X t leavng X = t as the unque vald assgnment for the varable X. The graph G t+1 has no negatve cycles when t + 1 = u. Indeed, n that specfc case, the graphs G t+1 and G u are by constructon dentcal. From the proof of Theorem 4, we know that t u. We therefore need to prove that G t+1 has no negatve cycles when t + 1 < u. Let u q be the greatest upper bound that s strctly smaller than u. By constructon, there s no upper bounds greater than l and smaller than u, thus u q < l. The null edges form a path from l to u q whch mples D G t+1[l, u q ] 0. Consder any lower bound l p that s smaller than or equal to l. The doman dom(x ) = [l, t + 1) s the only one that s not contaned n the nterval [l p, u q ) but s contaned n the nterval [l p, t + 1), thus w(u q, l p ) = w(t + 1, l p ) 1. The graphs G u and G t+1 are very smlar. They only dffer by the addtons of backward edges (t + 1, l q ) for all lower bound l q l. It s therefore suffcent to prove that any cycle passng by these edges has a non-negatve weght. Snce the addton of the edges leavng t + 1 does not affect the paths leadng to t + 1, we have D G t+1[l, t + 1] = D G u [l, t + 1]. By the defnton of t, we obtan 0 < D G u [l, t + 1] = D G t+1[l, t + 1]. The consequence of ths nequalty s that any path gong from l to t + 1 has a postve weght and n partcular, a path gong from l to t + 1 passng by the edge (u q, l p ) has postve weght. 0 < D G t+1 [l, u q ] + w(u q, l p ) + D G t+1[l p, t + 1] (1) By substtutng w(u q, l p ) = w(t + 1, l p ) 1 and D G t+1[l, u q ] 0 n (1) we prove that all cycles passng by the edge (t + 1, l p ) are non-negatve. 0 < 0 + w(t + 1, l p ) D G t+1[l p, t + 1] (2) 0 w(t + 1, l p ) + D G t+1[l p, t + 1] (3) In Example 1, the path (7, 2), (2, 5), (5, 8) n G 9 1 has a null weght whch certfes that there exsts an nterval support for 8 n the doman of X 1. Ths nterval support s t = [8, 2, 5, 6, 3]. A Flterng Algorthm Achevng Bounds Consstency The flterng algorthm for the MULTI-INTER-DISTANCE s a drect applcaton of Theorem 3, Theorem 4, and Theorem 5. The algorthm on Fgure 1 processes the varables n non-decreasng order of doman upper bounds. When processng X, the algorthm updates the lower bound l accordng to prevously dscovered fobdden regons. Then t computes the value u that s the smallest doman upper bound greater than l. Based on ths value, the algorthm tests whether the schedulng graph G u has a negatve cycle. If so, the algorthm stores n the data structure F the newly detected forbdden regon [l, u ). Because the algorthm processes varables n non-decreasng order of upper bounds, Theorem 3 ensures that the nterval [l, u ) s a forbdden regon for all unprocessed varables. On ts next teraton, the lower bound l s ncreased to avod all forbdden regons, ncludng the newly dscovered one. Ths process contnues untl the schedulng graph G u contans no negatve cycles. When the schedulng graph G u contans no negatve cycles, Theorem 4 and Theorem 5 apply. The algorthm computes the shortest path from node l to all the other nodes and fnds the node t whch s the largest one whose dstance from node l s null. From Theorem 4, we know that values smaller than t have no nterval support n the doman of the current varable. The algorthm removes them from the doman by ncreasng the value of l. Fnally, Theorem 5 guarantees that the lower bound of each varable doman has an nterval support upon completon of the algorthm. Algorthm 1: PruneLowerBounds([X 1,..., X n ]) Sort the varables such that u u +1 U {u 1,..., u n } F for 1..n do repeat l mn([l, u ) \ F ) u mn((l, u ] U) Let G u be the altered schedulng graph wth dom(x ) = [l, u ) Compute the shortest dstances D G u [l, t] from node l to all the other nodes t f there s a negatve cycle n G u then F F [l, u ) untl there s no negatve cycles n G u l max({t D G u [l, t] = 0}) Computng the sngle-source shortest path n the schedulng graph should be done wth care. Indeed, the schedulng graph has as many nodes as the number of values n the domans. These domans can be very large especally n schedulng problems where domans represent tme ntervals. (López-Ortz & Qumper 2011) showed how to compute the shortest path from the rght-most node to all the other nodes n O(n 2 mn(1, p m )) steps whch s strongly polynomal and nvarant n the cardnalty of the varable domans. The algorthm s an adaptaton of the Bellman- Ford algorthm (Cormen et al. 2001) whch ntalzes a dstance vector that keeps track of the dstances from the node l to all the other nodes. Snce the rght-most node can reach all the other nodes va the null edges, the dstance vector s ntalzed to zero and as new paths are explored, the dstance vector s updated wth smaller values. To compute the shortest path from l to all the other nodes, the dstance vector should be ntalzed to zero for all the nodes smaller than or equal to l and n for all the nodes that are greater than l. Indeed, any node t can be reached by gong to l mn through the null edges, then gong to u max usng the edge (l mn, u max ), and fnally usng the null edges to go to t. Ths

5 smple adaptaton to the shortest path algorthm s suffcent to compute the shortest path from the source node l to all the other nodes or to detect a negatve cycle. The runnng tme complexty remans O(n 2 mn(1, p m )). The computaton of shortest paths domnates the algorthms s complexty. Lemma 2 The algorthm PruneLowerBounds performs at least n and at most 3n 1 shortest path computatons where n s the number of varables. Proof: Computng the shortest path of an altered schedulng graph leads to two possble outcomes: the graph has negatve cycles or t does not. If the graph has no negatve cycles, the repeat loop termnates. Ths occurs exactly n tmes, once for each varable. If the graph has negatve cycles, the algorthm loops and create the forbdden regon [l, u ). The value u s a doman upper bound. Each tme a new forbdden regon s created, one of two events can occur: 1) a doman upper bound becomes for the frst tme the open upper bound of a forbdden regon 2) a doman upper bound gets ncluded n forbdden regon. These two events can occur only once for each upper bound at the excepton of the largest upper bound u max for whch only the frst event can occur. Therefore, the algorthm creates at most 2n 1 forbdden regons and performs at most 3n 1 shortest path computatons. The shortest path problem s solved n O(n 2 mn(1, p m )) steps when usng the algorthm of (López-Ortz & Qumper 2011) and modfyng the ntal state of the dstance vector as descrbed above. The total runnng tme complexty of the flterng algorthm s therefore O(n 3 mn(1, p m )) whch s never more than O(n 3 ). Prunng the upper bounds can be done by prunng the lower bounds n a symmetrc problem where dom(x ) = [ u, l ). When prunng the lower bounds, the value u s ncreased whch s equvalent to decrease the upper bound u n the orgnal problem. Experments We experment the MULTI-INTER-DISTANCE on the runway schedulng problem as presented n (Artouchne & Baptste 2005). Ths problem conssts of determnng the order n whch n arplanes should land on a runway. We consder an arport wth m runways allowng smultaneous landngs. Each arplane labeled from 1 to n has multple dsjont tme wndows [a 1, b 1],..., [a k, b k ] n whch t s n poston to land. The goal s to make each arplane land wthn one of ts tme wndows whle maxmzng the tme gap p between each landng on a same runway. We model the problem wth two varables per arplane. The varable S defnes at what tme the arplane lands. The varable W [1, k ] defnes n whch tme wndow the arplane lands. The two varables are connected wth the constrants W w S b w and W w S a w. Fnally, there s a MULTI-INTER-DISTANCE posted on the vector of varables S wth fxed parameter p and m. We solve the problem for p = 1, 2, 4, 8,... untl the problem becomes nfeasble. We then perform a bnary search between the largest p for whch there exsts a soluton and the smallest p for whch we detect nfeasblty. We branch on the varables W followed by the varables S. For m = 1, we use the benchmark descrbed as the second set n (Artouchne & Baptste 2005). There are 64 nstances for each n {15, 30, 45, 60, 75, 90}. The number of tme wndows per arplane randomly vares from 1 to 5. We generate nstances for m = 2 and m = 3 by mergng nstances from the orgnal benchmark. As a rule, we merge nstances havng the same sze of tme wndows or the same gap between the tme wndows. Merged nstances also have a smlar number of arplanes. As m ncreases, we obtan nstances wth more arplanes. We mplement our soluton (denoted MID) usng the Choco Solver and compare t aganst two mplementatons of the cumulatve constrant that the solver provdes. The frst mplementaton performs an overload checkng (denoted OC) n O(n log n) usng a Θ- tree as desgned by (Vlím, Barták, & Cepek 2004). The second mplementaton (denoted MVH) s an mplementaton of the edge-fnder (Mercer & Hentenryck 2008) n O(n 2 ). Experments are run arbtrarly on an Opteron GHz and an Opteron GHz. Runnng tmes may vary dependng on whch processor executes the program. Fgure 1 shows that MID solves more problems n a gven tme for m {1, 2, 3} than MVH and OC. The dfference n performance becomes more promnent as m ncreases. For m = 3, MID solves 134 nstances n 20 mnutes, MVH solves 3 nstances, and OC solves none. Table 2 shows that MID solves the problem wth sgnfcantly fewer backtracks. MVH performs fewer backtracks as n ncreases ndcatng that the flterng gets slower and fewer nodes are vsted wthn 20 mnutes. m = 1 m {2, 3} Fgure 1: Number of solved problems vs tme.

6 m = 1 m = 2 m = 3 n OC MVH MID MVH MID MVH MID 15 94% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 775 Table 2: Percentage of problems that generated backtracks and average number of backtracks for a perod of 20 mnutes for each of these problems. Concluson We ntroduced the MULTI-INTER-DISTANCE constrant. We made the connecton between ths constrant and the propertes of the shortest paths n the schedulng graph. We appled these propertes n the desgn of the frst flterng algorthm achevng bounds consstency. Ths new constrant s very successful at solvng large nstances of the runway schedulng problem. The theorems we proved about the MULTI-INTER-DISTANCE constrant may lead n the future to the desgn of better flterng algorthms for related constrants such as the INTER-DISTANCE and the CUMULATIVE constrants. Acknowledgments We would lke to thank Arnaud Malapert for hs support wth the Choco solver and the revewers for helpng us smplfyng the proof of Theorem 3. Ths research s supported by a NSERC Dscovery Grant. References Artouchne, K., and Baptste, P Inter-dstance constrant: An extenson of the all-dfferent constrant for schedulng equal length jobs. In Proc. of the 11th Int. Conf. on Prncples and Practce of Constrant Programmng, Artouchne, K.; Baptste, P.; and Dürr, C Runway schedulng wth holdng loop. In Proc. of 2nd Int. Workshop on Dscrete Optmzaton Methods n Producton and Logstcs, Cormen, T. H.; Sten, C.; Rvest, R. L.; and Leserson, C. E Introducton to Algorthms. McGraw-Hll Hgher Educaton, 2nd edton. Dürr, C., and Hurand, M Fndng total unmodularty n optmzaton problems solved by lnear programs. Algorthmca. DOI /s Leconte, M A bounds-based reducton scheme for constrants of dfference. In Proc. of the Constrant Int. Workshop on Constrant-Based Reasonng, López-Ortz, A., and Qumper, C.-G A fast algorthm for mult-machne schedulng problems wth jobs of equal processng tmes. In In 28th Int. Symp. on Theoretcal Aspects of Computer Scence (to appear n STACS 11). Mercer, L., and Hentenryck, P. V Edge fndng for cumulatve schedulng. INFORMS Journal on Computng 20(1): Qumper, C.-G.; López-Ortz, A.; and Pesant, G A quadratc propagator for the nter-dstance constrant. In Proc. of the 21st Nat. Conf. on Artfcal Intellgence (AAAI 06), Régn, J.-C A flterng algorthm for constrants of dfference n CSPs. In Proc. of the 11th Natonal Conference on Artfcel Intellgence (AAAI-94), Régn, J.-C Generalzed arc consstency for global cardnalty constrant. In Proc. of the 8th Annual Conference on Innovatve Applcatons of Artfcal Intellgence, Vlím, P.; Barták, R.; and Cepek, O Unary resource constrant wth optonal actvtes. In Proc. of the 10th Int. Conference on Prncples and Practce of Constrant Programmng, Vlím, P Edge fndng flterng algorthm for dscrete cumulatve resources n o(kn log n). In Proc. of the 15th Int. Conference on Prncples and Practce of Constrant Programmng,