Consistency properties and algorithms for achieving them are at the heart of the success of Constraint Programming. In this paper, we study the relational consistency property R(∗,m)C, which is equivalent to m-wise consistency proposed in relational databases. We also define wR(∗,m)C, a weaker variant of this property. We propose an algorithm for enforcing these properties on a Constraint Satisfaction Problem by tightening the existing relations and without introducing new ones. We empirically show that wR(∗,m)C solves in a backtrackfree manner all the instances of some CSP benchmark classes, thus hinting at the tractability of those classes.

Max Restricted Path Consistency (maxRPC) is a local consistency for binary constraints that enforces a higher order of consistency than arc consistency. Despite the strong pruning that can be achieved, maxRPC is rarely used because existing maxRPC algorithms suffer from overheads and redundancies as they can repeatedly perform many constraint checks without triggering any value deletions. In this paper we propose and evaluate techniques that can boost the performance of maxRPC algorithms by eliminating many of these overheads and redundancies. These include the combined use of two data structures to avoid many redundant constraint checks, and the exploitation of residues to quickly verify the existence of supports. Based on these, we propose a number of closely related maxRPC algorithms. The first one, maxRPC3, has optimal O(end 3) time complexity, displays good performance when used stand-alone, but is expensive to apply during search. The second one, maxRPC3 rm,hasO(en 2 d 4) time complexity, but a restricted version with O(end 4) complexity can be very efficient when used during search. The other algorithms are simple modifications of maxRPC3 rm. All algorithms have O(ed) space complexity when used stand-alone. However, maxRPC3 has O(end) space

Abstract. Max Restricted Path Consistency (maxRPC) is a local consistency for binary constraints that can achieve considerably stronger pruning than arc consistency. However, existing maxRPC algorithms suffer from overheads and redundancies as they can repeatedly perform many constraint checks without triggering any value deletions. In this paper we propose techniques that can boost the performance of maxRPC algorithms. These include the combined use of two data structures to avoid many redundant constraint checks, and heuristics for the efficient ordering and execution of certain operations. Based on these, we propose two closely related maxRPC algorithms. The first one has optimal O(end 3) time complexity, displays good performance when used stand-alone, but is expensive to apply during search. The second one has O(en 2 d 4) time complexity, but a restricted version with O(end 4) complexity can be very efficient when used during search. Both algorithms have O(ed) space complexity when used stand-alone. However, the first algorithm has O(end) space complexity when used during search, while the second retains the O(ed) complexity. Experimental results demonstrate that the resulting methods constantly outperform previous algorithms for maxRPC, often by large margins, and constitute a more than viable alternative to arc consistency. 1

I prove that a widely used technique for scanning lists in backtracking search algorithms has excellent optimality properties. The result applies to a simple general framework, which I present. It applies to critical algorithms such as watched literal unit propagation in SAT and important algorithms for maintaining generalised arc consistency in constraint satisfaction. I now show that both these techniques have optimal run time per branch in big-O terms when amortized across a search tree. Moreover the constant factor overhead of the worst case is only 2 in the simplest case. It is widely known that these methods are highly space efficient and effective in practice. My results help to explain this from a theoretical point of view. 1.