We introduce Polytopic Matrix Factorization (PMF) as a novel data decomposition approach. In this new framework, we model input data as unknown linear transformations of some latent vectors drawn from a polytope. In this sense, the article considers a semi-structured data model, in which the input matrix is modeled as the product of a full column rank matrix and a matrix containing samples from a polytope as its column vectors. The choice of polytope reflects the presumed features of the latent components and their mutual relationships. As the factorization criterion, we propose the determinant maximization (Det-Max) for the sample autocorrelation matrix of the latent vectors. We introduce a sufficient condition for identifiability, which requires that the convex hull of the latent vectors contains the maximum volume inscribed ellipsoid of the polytope with a particular tightness constraint. Based on the Det-Max criterion and the proposed identifiability condition, we show that all polytopes that satisfy a particular symmetry restriction qualify for the PMF framework. Having infinitely many polytope choices provides a form of flexibility in characterizing latent vectors. In particular, it is possible to define latent vectors with heterogeneous features, enabling the assignment of attributes such as nonnegativity and sparsity at the subvector level. The article offers examples illustrating the connection between polytope choices and the corresponding feature representations.

Answer Set Planning refers to the use of Answer Set Programming (ASP) to compute plans, i.e., solutions to planning problems, that transform a given state of the world to another state. The development of efficient and scalable answer set solvers has provided a significant boost to the development of ASP-based planning systems. This paper surveys the progress made during the last two and a half decades in the area of answer set planning, from its foundations to its use in challenging planning domains. The survey explores the advantages and disadvantages of answer set planning. It also discusses typical applications of answer set planning and presents a set of challenges for future research.

Weighted Counting for Constraint Satisfaction with Default Values (#CSPD) is a powerful special case of the sum-of-products problem that admits succinct encodings of #CSP, #SAT, and inference in probabilistic graphical models. We investigate #CSPD under the fundamental parameter of incidence treewidth (i.e., the treewidth of the incidence graph of the constraint hypergraph). We show that if the incidence treewidth is bounded, #CSPD can be solved in polynomial time. More specifically, we show that the problem is fixed-parameter tractable for the combined parameter incidence treewidth, domain size, and support size (the maximum number of non-default tuples in a constraint). This generalizes known results on the fixed-parameter tractability of #CSPD under the combined parameter primal treewidth and domain size. We further prove that the problem is not fixed-parameter tractable if any of the three components is dropped from the parameterization.

We pose the identified classes of problems within the general framework of Weighted Constraint Satisfaction Problems (WCSPs), reformulated as minimum weighted vertex cover problems. We examine the Constraint Composite Graphs (CCGs) associated with these WCSPs and provide simple arguments for establishing their tractability. We construct simple - almost trivial - bipartite graph representations for submodular cost functions, and reformulate these WCSPs as max-flow problems on bipartite graphs. By doing this, we achieve better time complexities than state-of-the-art algorithms. We also use CCGs to exploit planarity in variable interaction graphs, and provide algorithms with significantly improved time complexities for classes of submodular constraints. Moreover, our framework for exploiting planarity is not limited to submodular constraints. Our work confirms the usefulness of studying CCGs associated with combinatorial problems modeled as WCSPs.

In this paper, we present an efficient algorithm for verifying path-consistency on a binary constraint network. The complexities of our algorithm beat the previous conjectures on the lower bounds for verifying path-consistency. We therefore defeat the proofs for several published results that incorrectly rely on these conjectures. Our algorithm is motivated by the idea of reformulating path-consistency verification as fast matrix multiplication. Further, for a computational model that counts arithmetic operations (rather than bit operations), a clever use of the properties of prime numbers allows us to design an even faster variant of the algorithm. Based on our algorithm, we hope to inspire a new class of techniques for verifying and even establishing varying levels of local-consistency on given constraint networks.

Many real-world applications require the successful combination of spatial and temporal reasoning. In this paper, we study the general framework of the Traveling Salesman Problem with Simple Temporal Constraints. Representationally, this framework subsumes the Traveling Salesman Problem, Simple Temporal Problems, as well as many of the frameworks described in the literature. We analyze the theoretical properties of the combined problem providing strong inapproximability results for the general problem, and positive results for some special cases.

Many works have studied the properties of CSPs which are based on the structures of constraint networks, or based on the features of compatibility relations. Studies on structures rely generally on properties of graphs for binary CSPs and on properties of hypergraphs for the general case, that is CSPs with constraints of arbitrary arity. In the second case, using the dual representation of hypergraphs, that is a reformulation of the instances, we can exploit notions and properties of graphs. For the studies of compatibility relations, the exploitation of properties of graphs is possible studying a graph called microstructure which allows to reformulate instances of binary CSP. Unfortunately, this approach is limited to CSPs with binary constraints.

Unlike mathematical programming and SAT solving, ConstraintProgramming (CP) is based on the idea that both modeling and solving of combinatorial optimization problems can be based on conjunctions of loosely coupled, recurring, combinatorial sub-problems (aka "global constraints"). This rich representational approach means that, for better or for worse, pretty much anything can be expressed as a global constraint. Much of CP's success, however, has come from exploiting only one aspect of the rich constraint definition: global constraint propagation. In this talk, I will investigate how work in CP, SAT, AI planning, and mathematical programming can be understood as more seriously pursuing the implications of a rich constraint definition and how the interplay between local and global information can lead to dynamic problem reformulations and a flexible hybrid solver architecture.

A constraint satisfaction problem has compactness if any infinite set of constraints is satisfiable whenever all its finite subsets are satisfiable. We prove a sufficient condition for compactness, which holds for a range of problems including those based on the well-known Interval Algebra (IA) and RCC8. Furthermore, we show that compactness leads to a useful necessary and sufficient condition for the recently introduced patchwork property, namely that patchwork holds exactly when every satisfiable finite network (i.e., set of constraints) has a canonical solution, that is, a solution that can be extended to a solution for any satisfiable finite extension of the network. Applying these general theorems to qualitative reasoning, we obtain important new results as well as significant strengthenings of previous results regarding IA, RCC8, and their fragments and extensions. In particular, we show that all the maximal tractable fragments of IA and RCC8 (containing the base relations) have patchwork and canonical solutions as long as networks are algebraically closed.

The size of these networks is quadratic in the number of variables, which has severely limited the real-world application of QSR. In this paper, we propose another representation of spatial scenarios, in which each variable is associated with one or more rectangles. Instead of requiring these rectangles to define a solution of the corresponding constraint network, we construct sequences of rectangles that define partial solutions to progressively weaker constraint networks. We present experimental results that illustrate the effectiveness of this strategy.