On global warming: Flow-based soft global constraints

Abstract

We describe soft versions of the global cardinality constraint and the regular constraint, with efficient filtering algorithms maintaining domain consistency. For both constraints, the softening is achieved by augmenting the underlying graph. The softened constraints can be used to extend the meta-constraint framework for over-constrained problems proposed by Petit, Régin and Bessière.

Sequencing and Counting with the multicost-regular Constraint

“…3. Consider now the depicted DFA applied to X = (x1, x2) ∈ {a, b, c} × {a, b} and z ∈ [2,2]. Enforcing AC on the decomposed model (on the right) achieves GAC.…”

“…The allowed patterns are gathered in an acyclic digraph whose paths coincide with the valid sequences of activities. This approach was later extended to optimization constraints soft-regular [2] and cost-regular [3] for enforcing bounds on the global cost -a violation cost or any financial cost -of the sequence of assignments. The underlying problem is now to compute shortest and longest paths in the acyclic graph of patterns.…”

Including Soft Global Constraints in DCOPs

“…Costs coming from non-global constraints are calculated as usual, aggregating all non-global constraint costs evaluated on self value and the assignments of the current context (lines [5][6][7][8][9][10][11][12][13]). Costs coming from global constraints are calculated in lines [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Although there is no need to separate non-global from global cost aggregation, (1) procedure CalculateCost(value) (2) cost = cost + NonGlobalCostWithValue(value); (3) cost = cost + GlobalCostWithValue(value); (4) return cost; (5) function NonGlobalCostWithValue(value) (6) cost = 0; (7) for each nonGlobal ∈ nonGlobalConstraintSet do (8) assignments = new list(); assignments.add(self, value); (9) for each (xi , di ) ∈ context do (10) if xi ∈ nonGlobal.vars then assignments.add(xi, di); we have presented them in separate procedures for a better understanding of the new modifications.…”

“…A graph for every soft-alldifferent constraint is constructed following [15]. This graph is stored by the agent and updated during execution.…”

Open Constraints in a Closed World

“…A number of soft global constraints can be represented by a flow in a graph, similar to the gcc, see [6]. In this case, rather than associating a cost to an arc (x, d), for all x ∈ X i (i = 1, .…”