J. Jaffar, M. Maher, P. Stuckey, and R. Yap. Beyond finite domains. In PPCP'94: Second Workshop on Principles and Practice of Constraint Programming, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....h( Un ; y 1 ) 2 Also, for some 0 degree proofs the lower bound of the derivation length is proportional to the size of the domains of the formulas. This is a well known phenomenon well known in the context of finite domain and temporal constraint satisfaction problems (see for example [DMP91, JMSY94] In the following we give axioms, that, combined with the transitivity of , will improve the derivation lengths for a class of arithmetic problems. x y x y (A 1 ) x x (A 2 ) x y y z x z (T ) Proposition 2.3.9 For each n, there exists Delta and A such that any 0 degree ....

J. Jaffar, M. Maher, P. Stuckey, and R. Yap. Beyond finite domains. In PPCP'94: Second Workshop on Principles and Practice of Constraint Programming, 1994.

....h( Un ; y 1 ) 2 Also, for some 0 degree proofs the lower bound of the derivation length is proportional to the size of the domains of the formulas. This is a well known phenomenon well known in the context of finite domain and temporal constraint satisfaction problems (see for example [DMP91, JMSY94] In the following we give axioms, that, combined with the transitivity of , will improve the derivation lengths for a class of arithmetic problems. x y x y (A 1 ) x x (A 2 ) x y y z x z (T ) Proposition 2.3.9 For each n, there exists Delta and A such that any 0 degree ....

J. Jaffar, M. Maher, P. Stuckey, and R. Yap. Beyond finite domains. In PPCP'94: Second Workshop on Principles and Practice of Constraint Programming, 1994.

....of variables from C in O(dn 4 ) time where d is the number of disequations and n is the number of variables in C. 9 UTVPI is an acronym for linear inequalities with Unit coefficients and at most Two Variables Per Inequality. 10 UTVPI constraints over the integers have also been studied [20, 17]. 17] presents a constraint logic programming system with UTVPI constraints and several applications that would otherwise be solved using finite domain constraints. It is natural now to ask the following question. Can we extend the above theorem to the class of two variables per inequality or ....

Joxan Jaffar, Michael J. Maher, Peter Stuckey, and Ronald Yap. Beyond Finite Domains. In A. Borning, editor, Proceedings of PPCP'94, volume 874 of Lecture Notes in Computer Science, pages 86--94. Springer Verlag, 1994.

....1 12; x 1 x 2 2; x 3 Gamma x 2 0:5; x 3 x 2 6= 6 HDL constraints were defined in (Koubarakis 1996; Jonsson, P. and Backstrom, C. 1996) They subsume many interesting classes of temporal constraints. UTVPI 6= constraints (without disequations) have been considered in (Shostak 1981; Jaffar et al. 1994). UTVPI 6= constraints subsume the temporal constraints studied in (Koubarakis 1997a) They can also be used in spatial applications to define many kinds of polygons (e.g. arbitrary rectangles, several kinds of triangles and octagons and so on) The disequations allowed by this class give us ....

Jaffar, J.; Maher, M. J.; Stuckey, P.; and Yap, R. 1994. Beyond Finite Domains. In Borning, A., ed., Proceedings of PPCP'94, volume 874 of Lecture Notes in Computer Science, 86--94. Springer Verlag.

....ax by c where a; b 2 f Gamma1; 0; 1g and is or 6= HDL constraints were defined in (Koubarakis 1996; Jonsson, P. and Backstrom, C. 1996) They subsume many interesting classes of temporal constraints. UTVPI 6= constraints (without disequations) have been considered in (Shostak 1981; Jaffar et al. 1994). UTVPI 6= constraints subsume the temporal constraints studied in (Koubarakis 1997a) They can also be used in spatial applications to define many kinds of polygons (e.g. arbitrary rectangles, several kinds of triangles and octagons and so on) The disequations allowed by this class give us ....

Jaffar, J.; Maher, M. J.; Stuckey, P.; and Yap, R. 1994. Beyond Finite Domains. In Borning, A., ed., Proceedings of PPCP'94, vol. 874 of LNCS, 86--94. Springer Verlag.

....propagation in some polynomially bounded way, we can arrive at a partial (i.e. incomplete) algorithm. How effective such an algorithm could be is a question for future research, but it seems that it shouldn t be too difficult to achieve more effectiveness than Jaffar et al. s Unit TVPI algorithm ([5]) applied to general TVPI. In the context of a CLP solver, limiting the propagation in this way is not the liability it might otherwise be. This is because at subsequent steps in the computation, further integer hull then propagate iterations occur, which serve to further disseminate information ....

J. Jaffar, M. J. Maher, P. J. Stuckey, and R. H. C. Yap, 1994. Beyond Finite Domains, in Proceedings of the Second Workshop on Principles and Practice of Constraint Programming, LNCS 874, 86--94.

.... v) c [3] D 0 : D [4] PriorityQueue : OE [5] InsertHeap(PriorityQueue, v, 0) 6] while PriorityQueue = OE do [7] x, dist x) FindAndDeleteMin(PriorityQueue) 8] if D(u) length(u v) dist x D(x) then [9] if (y = u) then [10] Infeasible system: reject new constraint [11] remove edge u v from E [12] return false [13] else [14] D 0 (x) D(u) length(u v) dist x [15] for every vertex y in Succ(x) do [16] scaledPathLength : dist x (D(x) length(x y) Gamma D(y) 17] if (scaledPathLength KeyOf(PriorityQueue; y) 18] AdjustHeap(PriorityQueue, y, ....

....Shostak s algorithm into a polynomial time algorithm. The most efficient algorithm currently known for this problem is an O(mn 2 log m) algorithm due to Hochbaum and Naor [9] Le Pape [16] shows how one can deal with difference constraints over any totally ordered Abelian Group. Jaffar et al. [11] considered the problem of two variable constraints of the form, ax by c, where a; b 2 f Gamma1; 0; 1g. They present an algorithm for computing a feasible solution to a system of two variable constraints that processes the constraints one by one. The algorithm takes O(n 2 ) time per ....

Joxan Jaffar, Michael J. Maher, Peter J. Stuckey, and Roland H. C. Yap. Beyond finite domains. In Proceedings of Workshop on Principles and Practice of Constraint Programming, pages 86--94, 1994.

....than deterministic rules. 2. 2 Background in Scheduling Scheduling is the practical problem of allocating resources over time to perform a collection of tasks [2] Other techniques which have been used, include Heuristic Search [20] Simulated Annealing [14] and Constraint Logic Programming [11, 18, 19]. The problem with scheduling, is that it is known to be NP complete, therefore it is impossible to find the optimal solution in reasonable time. That is why we require a stochastic or heuristic optimisation technique. There are three basic goals that are prevalent in scheduling [2] these are : ....

J. Jaffar, M. J. Maher, P. J. Stuckey, and R. H. C. Yap. Beyond finite domains. In Principles and Practice of Constraint Programming, pages 86--94, 1994.

....As for inequations, they are ubiquitous in several domains such as constraint logic programming (CLP) integer linear programming, and operational research. In the above literature, the algorithms for solving linear inequations over natural numbers are not complete but over finite domains [16] and they usually proceed by turning inequations into equations by introducing new variables generally called slack [24] Such methods yield voluminous problems, which is an handicap since the solving complexity is an exponential in the number of variables. It is therefore quite natural to ....

E. Joxan, M. J. Maher, P. J. Stuckey, and R. H. C. Yap. Beyond finite domains. In A. Borning, editor, Proceedings of the Second International Workshop on Principles and Practice of Constraint Programming, volume 874 of Lecture Notes in Computer Science, pages 86--94. Springer-Verlag, may 1994.

....search tree where the discriminator constraints are conjunctions of inequalities of the form ax i bx j d where x 1 ; xn are the n dimensions of objects to be stored, i 6= j, and fa; bg f0; 1; 1g. These constraints are known as unit two variable per inequality (UTVPI) constraints [4]. In 2 dimensions these are exactly the constraints from domain O. The O constraint keys for the polygons of Figure 6 are also shown (dashed) and here the lack of overlap is clear. One can think of the additional complexity of an O discriminator as representing another bounding box along the two ....

....also immediately defines efficient index structures in terms of available constraint solvers. O trees are a simple form of HCSTs defining a spatial index structure. Our motivation for examining O trees arose from earlier work on constraint solving for unit two variable per inequality constraints [4]. Comparing O trees versus R trees on 2 d data it seems that the extra discriminating ability of O trees is usually overridden by the reduction of fan out in the resulting trees because of the larger size of O discriminators. For line intersection queries O trees are superior, and for join queries ....

J. Jaffar, M. J. Maher, P. Stuckey, and R. Yap. Beyond finite domains. In Proceedings of the International Workshop on Principle and Practices of Constraint Programming, number 874 in LNCS, pages 86--93, Orcas Island, Washington, May 1994. Springer-Verlag.

....be any such constraint, while discriminators are single UTVPI inequalities. Such constraints have an efficient incremental satisfiability test and implication test for discriminators (both (O(n 2 ) where n is the number of inequalities in the constraint item) based on transitive closure (see [8] 3 ) Analogous to the use of bounding boxes in spatial data structures we can use UTVPI constraints to define a bounding structure for an arbitrary k dimensional object. Note that UTVPI constraints can discriminate each of the constraints represented in Figure 5. We can then use it as a key ....

J. Jaffar, M. Maher, P. Stuckey and R. Yap. Beyond finite domains. In Procs of PPCP'94, LNCS 874, Seattle WA, May 1994, 86--94.

....real linear inequalities, but this is not the case for integers. Integer TVPI problems are, surprisingly, NP complete [5] Hence we need to restrict the class of constraints even further if we desire a practical solver. A class of constraints intermediate between SVPI and TVPI was introduced in [4]. This is the class of TVPI constraints ax by d with unit coefficients, that is, a; b 2 f Gamma1; 0; 1g. We call these unit TVPI constraints or UTVPI constraints. UTVPI constraints are expressive enough to describe all constraints occurring in many scheduling and temporal reasoning problems. And ....

....propagation approach we can begin from a complete UTVPI solver and add propagation methods to handle more complex constraints. In this paper we give a direct mathematical formulation of a constraint solver for integer UTVPI constraints based on transitive closure. As opposed to the algorithm in [4], in our formulation tightening and transitive closure are performed in a single step. We show that this algorithm produces a set of UTVPI constraints closed under transitivity and tightening. We describe how this can be used as the basis for constraint solvers able to handle larger classes of ....

[Article contains additional citation context not shown here]

J. Jaffar, M. Maher, P. Stuckey and R. Yap. Beyond finite domains. In Alan Borning (editor), PPCP'94: Second Workshop on Principles and Practice of Constraint Programming, Seattle WA, May 1994.

....instance, consider x 1 y x; 0 x; y k. Clearly this system is unsatisfiable, by a simple transitivity argument. However, detecting this by bounds propagation requires cost proportional to k. Note that this problem is compounded if good, natural bounds are not available. In an earlier paper [4], we suggested an alternative to the FD approach which is based on the class of two variable per inequality constraints with unit coefficients (UTVPI) In this paper we describe in detail a solver for this class of constraints, discuss how it can be used as the basis of a more general integer ....

.... still NP complete [7] However TVPI integer constraints seem more directly amenable than general integer constraints to transitivity based methods similar to those employed for solving real TVPI constraints [1, 2, 3, 13] A class of constraints intermediate between SVPI and TVPI was introduced in [4]. This is the class of TVPI constraints ax by d with unit coefficients, that is, a; b 2 f Gamma1; 0; 1g. We call these unit TVPI constraints. This class is considerably less expressive than the general class of TVPI constraints (for example, it cannot express modulo constraints) but is ....

J. Jaffar, M. Maher, P. Stuckey and R. Yap. Beyond finite domains. In Alan Borning (editor), PPCP'94: Second Workshop on Principles and Practice of Constraint Programming, Seattle WA, May 1994.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC