We propose an online form of the cake cutting problem. This models situations where agents arrive and depart during the process of dividing a resource. We show that well known fair division procedures like cut-and-choose and the Dubins-Spanier moving knife procedure can be adapted to apply to such online problems. We propose some fairness properties that online cake cutting procedures can possess like online forms of proportionality and envy-freeness. We also consider the impact of collusion between agents. Finally, we study theoretically and empirically the competitive ratio of these online cake cutting procedures. Based on its resistance to collusion, and its good performance in practice, our results favour the online version of the cut-and-choose procedure over the online version of the moving knife procedure.

The performance of anytime algorithms can be improved by simultaneously solving several instances of algorithm-problem pairs. These pairs may include different instances of a problem (such as starting from a different initial state), different algorithms (if several alternatives exist), or several runs of the same algorithm (for non-deterministic algorithms). In this paper we present a methodology for designing an optimal scheduling policy based on the statistical characteristics of the algorithms involved. We formally analyze the case where the processes share resources (a single-processor model), and provide an algorithm for optimal scheduling. We analyze, theoretically and empirically, the behavior of our scheduling algorithm for various distribution types.

VHPOP is a partial order causal link (POCL) planner loosely based on UCPOP. It draws from the experience gained in the early to mid 1990's on flaw selection strategies for POCL planning, and combines this with more recent developments in the field of domain independent planning such as distance based heuristics and reachability analysis. We present an adaptation of the additive heuristic for plan space planning, and modify it to account for possible reuse of existing actions in a plan. We also propose a large set of novel flaw selection strategies, and show how these can help us solve more problems than previously possible by POCL planners. VHPOP also supports planning with durative actions by incorporating standard techniques for temporal constraint reasoning. We demonstrate that the same heuristic techniques used to boost the performance of classical POCL planning can be effective in domains with durative actions as well. The result is a versatile heuristic POCL planner competitive with established CSP-based and heuristic state space planners.

Unit resolution is a key feature of state of the art sat solvers [13] [7] [5], where it speeds up the search for solutions and inconsistencies. It is well known that different cnf representations of a given problem do not always allow unit resolution to deduce the same information. For example, the cnf encoding for pseudo Boolean constraints proposed in [3] allows unit resolution to restore generalized arc consistency. This is not the case with the encoding proposed in [16], which does not allow unit resolution to deduce as much information as the former encoding does. As a manner of speaking, the expressive power of unit resolution is best exploited using the encoding proposed in [3], with notable consequences on the resolution time.

Enforcing local consistencies is one of the main features of constraint reasoning. Which level of local consistency should be used when searching for solutions in a constraint network is a basic question. Arc consistency and partial forms of arc consistency have been widely studied, and have been known for sometime through the forward checking or the MAC search algorithms. Until recently, stronger forms of local consistency remained limited to those that change the structure of the constraint graph, and thus, could not be used in practice, especially on large networks. This paper focuses on the local consistencies that are stronger than arc consistency, without changing the structure of the network, i.e., only removing inconsistent values from the domains. In the last five years, several such local consistencies have been proposed by us or by others. We make an overview of all of them, and highlight some relations between them. We compare them both theoretically and experimentally, considering their pruning efficiency and the time required to enforce them.

In the recent literature of Artificial Intelligence, an intensive research effort has been spent, for various algebras of qualitative relations used in the representation of temporal and spatial knowledge, on the problem of classifying the computational complexity of reasoning problems for subsets of algebras. The main purpose of these researches is to describe a restricted set of maximal tractable subalgebras, ideally in an exhaustive fashion with respect to the hosting algebras. In this paper we introduce a novel algebra for reasoning about Spatial Congruence, show that the satisfiability problem in the spatial algebra MC-4 is NP-complete, and present a complete classification of tractability in the algebra, based on the individuation of three maximal tractable subclasses, one containing the basic relations. The three algebras are formed by 14, 10 and 9 relations out of 16 which form the full algebra.

In recent years, many improvements to backtracking algorithms for solving constraint satisfaction problems have been proposed. The techniques for improving backtracking algorithms can be conveniently classified as look-ahead schemes and look-back schemes. Unfortunately, look-ahead and look-back schemes are not entirely orthogonal as it has been observed empirically that the enhancement of look-ahead techniques is sometimes counterproductive to the effects of look-back techniques. In this paper, we focus on the relationship between the two most important look-ahead techniques---using a variable ordering heuristic and maintaining a level of local consistency during the backtracking search---and the look-back technique of conflict-directed backjumping (CBJ). We show that there exists a "perfect" dynamic variable ordering such that CBJ becomes redundant. We also show theoretically that as the level of local consistency that is maintained in the backtracking search is increased, the less that backjumping will be an improvement. Our theoretical results partially explain why a backtracking algorithm doing more in the look-ahead phase cannot benefit more from the backjumping look-back scheme. Finally, we show empirically that adding CBJ to a backtracking algorithm that maintains generalized arc consistency (GAC), an algorithm that we refer to as GAC-CBJ, can still provide orders of magnitude speedups. Our empirical results contrast with Bessiere and Regin's conclusion (1996) that CBJ is useless to an algorithm that maintains arc consistency.

Robust search procedures are a central component in the design of black-box constraint-programming solvers. This paper proposes activity-based search, the idea of using the activity of variables during propagation to guide the search. Activity-based search was compared experimentally to impact-based search and the wdeg heuristics. Experimental results on a variety of benchmarks show that activity-based search is more robust than other heuristics and may produce significant improvements in performance.

We introduce a temporal model for reasoning on disjunctive metric constraints on intervals and time points in temporal contexts. This temporal model is composed of a labeled temporal algebra and its reasoning algorithms. The labeled temporal algebra defines labeled disjunctive metric point-based constraints, where each disjunct in each input disjunctive constraint is univocally associated to a label. Reasoning algorithms manage labeled constraints, associated label lists, and sets of mutually inconsistent disjuncts. These algorithms guarantee consistency and obtain a minimal network. Additionally, constraints can be organized in a hierarchy of alternative temporal contexts. Therefore, we can reason on context-dependent disjunctive metric constraints on intervals and points. Moreover, the model is able to represent non-binary constraints, such that logical dependencies on disjuncts in constraints can be handled. The computational cost of reasoning algorithms is exponential in accordance with the underlying problem complexity, although some improvements are proposed.

In many combinatorial problems one may need to model the diversity or similarity of assignments in a solution. For example, one may wish to maximise or minimise the number of distinct values in a solution. To formulate problems of this type, we can use soft variants of the well known AllDifferent and AllEqual constraints. We present a taxonomy of six soft global constraints, generated by combining the two latter ones and the two standard cost functions, which are either maximised or minimised. We characterise the complexity of achieving arc and bounds consistency on these constraints, resolving those cases for which NP-hardness was neither proven nor disproven. In particular, we explore in depth the constraint ensuring that at least k pairs of variables have a common value. We show that achieving arc consistency is NP-hard, however achieving bounds consistency can be done in polynomial time through dynamic programming. Moreover, we show that the maximum number of pairs of equal variables can be approximated by a factor 1/2 with a linear time greedy algorithm. Finally, we provide a fixed parameter tractable algorithm with respect to the number of values appearing in more than two distinct domains. Interestingly, this taxonomy shows that enforcing equality is harder than enforcing difference.