32 research outputs found
A Greedy Algorithm for Constructing a Low-Width Generalized Hypertree Decomposition
ABSTRACT We propose a greedy algorithm which, given a hypergraph H and a positive integer k, produces a hypertree decomposition of width less than or equal to 3k â 1, or determines that H does not have a generalized hypertree-width less than k. The running time of this algorithm is O(m k+2 n), where m is the number of hyperedges and n is the number of vertices. If k is a constant, it is polynomial. The concepts of (generalized) hypertree decomposition and (generalized) hypertree-width were introduced by Gottlob et al. Many important NP-complete problems in database theory or artificial intelligence are polynomially solvable for classes of instances associated with hypergraphs of bounded hypertree-width. Gottlob et al. also developed a polynomial time algorithm det-k-decomp which, given a hypergraph H and a constant k, computes a hypertree decomposition of width less than or equal to k if the hypertree-width of H is less than or equal to k. The running time of det-k-decomp is O(m 2k n 2 ) in the worst case, where m and n are the number of hyperedges and the number of vertices, respectively. The proposed algorithm is faster than this. The key step of our algorithm is checking whether a set of hyperedges is an obstacle to a hypergraph having low generalized hypertree-width. We call such a local hypergraph structure a k-hyperconnected set. If a hypergraph contains a k-hyperconnected set with a size of at least 2k, it has hypertreewidth of at least k. Adler et al. propose another obstacle called a k-hyperlinked set. We discuss the difference between the two concepts with examples
Optimal Algorithms for Ranked Enumeration of Answers to Full Conjunctive Queries
We study ranked enumeration of join-query results according to very general
orders defined by selective dioids. Our main contribution is a framework for
ranked enumeration over a class of dynamic programming problems that
generalizes seemingly different problems that had been studied in isolation. To
this end, we extend classic algorithms that find the k-shortest paths in a
weighted graph. For full conjunctive queries, including cyclic ones, our
approach is optimal in terms of the time to return the top result and the delay
between results. These optimality properties are derived for the widely used
notion of data complexity, which treats query size as a constant. By performing
a careful cost analysis, we are able to uncover a previously unknown tradeoff
between two incomparable enumeration approaches: one has lower complexity when
the number of returned results is small, the other when the number is very
large. We theoretically and empirically demonstrate the superiority of our
techniques over batch algorithms, which produce the full result and then sort
it. Our technique is not only faster for returning the first few results, but
on some inputs beats the batch algorithm even when all results are produced.Comment: 50 pages, 19 figure
Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)
The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..
Relational Machine Learning Algorithms
The majority of learning tasks faced by data scientists involve relational data, yet most standard algorithms for standard learning problems are not designed to accept relational data as input. The standard practice to address this issue is to join the relational data to create the type of geometric input that standard learning algorithms expect. Unfortunately, this standard practice has exponential worst-case time and space complexity. This leads us to consider what we call the Relational Learning Question: "Which standard learning algorithms can be efficiently implemented on relational data, and for those that cannot, is there an alternative algorithm that can be efficiently implemented on relational data and that has similar performance guarantees to the standard algorithm?"
In this dissertation, we address the relational learning question for the well-known problems of support vector machine (SVM), logistic regression, and -means clustering. First, we design an efficient relational algorithm for regularized linear SVM and logistic regression using sampling methods. We show how to implement a variation of gradient descent that provides a nearly optimal approximation guarantee for stable instances. For the -means problem, we show that the -means++ algorithm can be efficiently implemented on relational data, and that a slight variation of adaptive k-means algorithm can be efficiently implemented on relational data while maintaining a constant approximation guarantee. On the way to developing these algorithms, we give an efficient approximation algorithm for certain sum-product queries with additive inequalities that commonly arise
Planning in constraint space for multi-body manipulation tasks
Robots are inherently limited by physical constraints on their link lengths, motor torques, battery
power and structural rigidity. To thrive in circumstances that push these limits, such as in search
and rescue scenarios, intelligent agents can use the available objects in their environment as
tools. Reasoning about arbitrary objects and how they can be placed together to create useful
structures such as ramps, bridges or simple machines is critical to push beyond one's physical
limitations. Unfortunately, the solution space is combinatorial in the number of available objects
and the configuration space of the chosen objects and the robot that uses the structure is high
dimensional.
To address these challenges, we propose using constraint satisfaction as a means to test the
feasibility of candidate structures and adopt search algorithms in the classical planning literature
to find sufficient designs. The key idea is that the interactions between the components of a
structure can be encoded as equality and inequality constraints on the configuration spaces of the
respective objects. Furthermore, constraints that are induced by a broadly defined action, such as
placing an object on another, can be grouped together using logical representations such as Planning
Domain Definition Language (PDDL). Then, a classical planning search algorithm can reason about
which set of constraints to impose on the available objects, iteratively creating a structure that
satisfies the task goals and the robot constraints. To demonstrate the effectiveness of this
framework, we present both simulation and real robot results with static structures such as ramps,
bridges and stairs, and quasi-static structures such as lever-fulcrum simple machines.Ph.D
Diversity of Answers to Conjunctive Queries
Enumeration problems aim at outputting, without repetition, the set of
solutions to a given problem instance. However, outputting the entire solution
set may be prohibitively expensive if it is too big. In this case, outputting a
small, sufficiently diverse subset of the solutions would be preferable. This
leads to the Diverse-version of the original enumeration problem, where the
goal is to achieve a certain level d of diversity by selecting k solutions. In
this paper, we look at the Diverse-version of the query answering problem for
Conjunctive Queries and extensions thereof. That is, we study the problem if it
is possible to achieve a certain level d of diversity by selecting k answers to
the given query and, in the positive case, to actually compute such k answers.Comment: 34 pages, accepted to ICDT 202