On the speed of constraint propagation and the time complexity of arc consistency testing

Establishing arc consistency on two relational structures is one of the most popular heuristics for the constraint satisfaction problem. We aim at determining the time complexity of arc consistency testing. The input structures $G$ and $H$ can be supposed to be connected colored graphs, as the general problem reduces to this particular case. We first observe the upper bound $O(e(G)v(H)+v(G)e(H))$, which implies the bound $O(e(G)e(H))$ in terms of the number of edges and the bound $O((v(G)+v(H))^3)$ in terms of the number of vertices. We then show that both bounds are tight up to a constant factor as long as an arc consistency algorithm is based on constraint propagation (like any algorithm currently known). Our argument for the lower bounds is based on examples of slow constraint propagation. We measure the speed of constraint propagation observed on a pair $G,H$ by the size of a proof, in a natural combinatorial proof system, that Spoiler wins the existential 2-pebble game on $G,H$. The proof size is bounded from below by the game length $D(G,H)$, and a crucial ingredient of our analysis is the existence of $G,H$ with $D(G,H)=\Omega(v(G)v(H))$. We find one such example among old benchmark instances for the arc consistency problem and also suggest a new, different construction.Comment: 19 pages, 5 figure

Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning under structural restrictions. All these problems involve two tasks: (i) identifying the structure in the input as required by the restriction, and (ii) using the identified structure to solve the reasoning task efficiently. We show that for most of the considered problems, task (i) admits a polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, in contrast to task (ii) which does not admit such a reduction to a problem kernel of polynomial size, subject to a complexity theoretic assumption. As a notable exception we show that the consistency problem for the AtMost-NValue constraint admits a polynomial kernel consisting of a quadratic number of variables and domain values. Our results provide a firm worst-case guarantees and theoretical boundaries for the performance of polynomial-time preprocessing algorithms for the considered problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541, arXiv:1104.556

Solving Functional Constraints by Variable Substitution

Functional constraints and bi-functional constraints are an important constraint class in Constraint Programming (CP) systems, in particular for Constraint Logic Programming (CLP) systems. CP systems with finite domain constraints usually employ CSP-based solvers which use local consistency, for example, arc consistency. We introduce a new approach which is based instead on variable substitution. We obtain efficient algorithms for reducing systems involving functional and bi-functional constraints together with other non-functional constraints. It also solves globally any CSP where there exists a variable such that any other variable is reachable from it through a sequence of functional constraints. Our experiments on random problems show that variable elimination can significantly improve the efficiency of solving problems with functional constraints

Stability and resource allocation in project planning.

The majority of resource-constrained project scheduling efforts assumes perfect information about the scheduling problem to be solved and a static deterministic environment within which the pre-computed baseline schedule is executed. In reality, project activities are subject to considerable uncertainty, which generally leads to numerous schedule disruptions. In this paper, we present a resource allocation model that protects a given baseline schedule against activity duration variability. A branch-and-bound algorithm is developed that solves the proposed resource allocation problem. We report on computational results obtained on a set of benchmark problems.Constraint satisfaction; Information; Model; Planning; Problems; Project management; Project planning; Project scheduling; Resource allocati; Scheduling; Stability; Uncertainty; Variability;

Models for robust resource allocation in project scheduling.

The vast majority of resource-constrained project scheduling efforts assumes complete information about the scheduling problem to be solved and a static deterministic environment within which the pre-computed baseline schedule will be executed. In reality, however, project activities are subject to considerable uncertainty which generally leads to numerous schedule disruptions. In this paper, we present a resource allocation model that protects the makespan of a given baseline schedule against activity duration variability. A branch-and-bound algorithm is developed that solves the proposed robust resource allocation problem in exact and approximate formulations. The procedure relies on constraint propagation during its search. We report on computational results obtained on a set of benchmark problems.Model; Resource allocation; Scheduling;

Solving Set Constraint Satisfaction Problems using ROBDDs

In this paper we present a new approach to modeling finite set domain constraint problems using Reduced Ordered Binary Decision Diagrams (ROBDDs). We show that it is possible to construct an efficient set domain propagator which compactly represents many set domains and set constraints using ROBDDs. We demonstrate that the ROBDD-based approach provides unprecedented flexibility in modeling constraint satisfaction problems, leading to performance improvements. We also show that the ROBDD-based modeling approach can be extended to the modeling of integer and multiset constraint problems in a straightforward manner. Since domain propagation is not always practical, we also show how to incorporate less strict consistency notions into the ROBDD framework, such as set bounds, cardinality bounds and lexicographic bounds consistency. Finally, we present experimental results that demonstrate the ROBDD-based solver performs better than various more conventional constraint solvers on several standard set constraint problems