1,895 research outputs found
Tractable Optimization Problems through Hypergraph-Based Structural Restrictions
Several variants of the Constraint Satisfaction Problem have been proposed
and investigated in the literature for modelling those scenarios where
solutions are associated with some given costs. Within these frameworks
computing an optimal solution is an NP-hard problem in general; yet, when
restricted over classes of instances whose constraint interactions can be
modelled via (nearly-)acyclic graphs, this problem is known to be solvable in
polynomial time. In this paper, larger classes of tractable instances are
singled out, by discussing solution approaches based on exploiting hypergraph
acyclicity and, more generally, structural decomposition methods, such as
(hyper)tree decompositions
Tree Projections and Structural Decomposition Methods: Minimality and Game-Theoretic Characterization
Tree projections provide a mathematical framework that encompasses all the
various (purely) structural decomposition methods that have been proposed in
the literature to single out classes of nearly-acyclic (hyper)graphs, such as
the tree decomposition method, which is the most powerful decomposition method
on graphs, and the (generalized) hypertree decomposition method, which is its
natural counterpart on arbitrary hypergraphs. The paper analyzes this
framework, by focusing in particular on "minimal" tree projections, that is, on
tree projections without useless redundancies. First, it is shown that minimal
tree projections enjoy a number of properties that are usually required for
normal form decompositions in various structural decomposition methods. In
particular, they enjoy the same kind of connection properties as (minimal) tree
decompositions of graphs, with the result being tight in the light of the
negative answer that is provided to the open question about whether they enjoy
a slightly stronger notion of connection property, defined to speed-up the
computation of hypertree decompositions. Second, it is shown that tree
projections admit a natural game-theoretic characterization in terms of the
Captain and Robber game. In this game, as for the Robber and Cops game
characterizing tree decompositions, the existence of winning strategies implies
the existence of monotone ones. As a special case, the Captain and Robber game
can be used to characterize the generalized hypertree decomposition method,
where such a game-theoretic characterization was missing and asked for. Besides
their theoretical interest, these results have immediate algorithmic
applications both for the general setting and for structural decomposition
methods that can be recast in terms of tree projections
Hypergraph Acyclicity and Propositional Model Counting
We show that the propositional model counting problem #SAT for CNF- formulas
with hypergraphs that allow a disjoint branches decomposition can be solved in
polynomial time. We show that this class of hypergraphs is incomparable to
hypergraphs of bounded incidence cliquewidth which were the biggest class of
hypergraphs for which #SAT was known to be solvable in polynomial time so far.
Furthermore, we present a polynomial time algorithm that computes a disjoint
branches decomposition of a given hypergraph if it exists and rejects
otherwise. Finally, we show that some slight extensions of the class of
hypergraphs with disjoint branches decompositions lead to intractable #SAT,
leaving open how to generalize the counting result of this paper
Tree Projections and Constraint Optimization Problems: Fixed-Parameter Tractability and Parallel Algorithms
Tree projections provide a unifying framework to deal with most structural
decomposition methods of constraint satisfaction problems (CSPs). Within this
framework, a CSP instance is decomposed into a number of sub-problems, called
views, whose solutions are either already available or can be computed
efficiently. The goal is to arrange portions of these views in a tree-like
structure, called tree projection, which determines an efficiently solvable CSP
instance equivalent to the original one. Deciding whether a tree projection
exists is NP-hard. Solution methods have therefore been proposed in the
literature that do not require a tree projection to be given, and that either
correctly decide whether the given CSP instance is satisfiable, or return that
a tree projection actually does not exist. These approaches had not been
generalized so far on CSP extensions for optimization problems, where the goal
is to compute a solution of maximum value/minimum cost. The paper fills the
gap, by exhibiting a fixed-parameter polynomial-time algorithm that either
disproves the existence of tree projections or computes an optimal solution,
with the parameter being the size of the expression of the objective function
to be optimized over all possible solutions (and not the size of the whole
constraint formula, used in related works). Tractability results are also
established for the problem of returning the best K solutions. Finally,
parallel algorithms for such optimization problems are proposed and analyzed.
Given that the classes of acyclic hypergraphs, hypergraphs of bounded
treewidth, and hypergraphs of bounded generalized hypertree width are all
covered as special cases of the tree projection framework, the results in this
paper directly apply to these classes. These classes are extensively considered
in the CSP setting, as well as in conjunctive database query evaluation and
optimization
Challenges for Efficient Query Evaluation on Structured Probabilistic Data
Query answering over probabilistic data is an important task but is generally
intractable. However, a new approach for this problem has recently been
proposed, based on structural decompositions of input databases, following,
e.g., tree decompositions. This paper presents a vision for a database
management system for probabilistic data built following this structural
approach. We review our existing and ongoing work on this topic and highlight
many theoretical and practical challenges that remain to be addressed.Comment: 9 pages, 1 figure, 23 references. Accepted for publication at SUM
201
On Low Treewidth Approximations of Conjunctive Queries
We recently initiated the study of approximations of conjunctive queries within classes that admit tractable query evaluation (with respect to combined complexity). Those include classes of acyclic, bounded treewidth, or bounded hypertreewidth queries. Such approximations are always guaranteed to exist. However, while for acyclic and bounded hypertreewidth queries we have shown a number of examples of interesting approximations, for queries of bounded treewidth the study had been restricted to queries over graphs, where such approximations usually trivialize. In this note we show that for relations of arity greater than two, the notion of low treewidth approximations is a rich one, as many queries possess them. In fact we look at approximations of queries of maximum possible treewidth by queries of minimum possible treewidth (i.e., one), and show that even in this case the structure of approximations remain rather rich as long as input relations are not binary
Compressed Representations of Conjunctive Query Results
Relational queries, and in particular join queries, often generate large
output results when executed over a huge dataset. In such cases, it is often
infeasible to store the whole materialized output if we plan to reuse it
further down a data processing pipeline. Motivated by this problem, we study
the construction of space-efficient compressed representations of the output of
conjunctive queries, with the goal of supporting the efficient access of the
intermediate compressed result for a given access pattern. In particular, we
initiate the study of an important tradeoff: minimizing the space necessary to
store the compressed result, versus minimizing the answer time and delay for an
access request over the result. Our main contribution is a novel parameterized
data structure, which can be tuned to trade off space for answer time. The
tradeoff allows us to control the space requirement of the data structure
precisely, and depends both on the structure of the query and the access
pattern. We show how we can use the data structure in conjunction with query
decomposition techniques, in order to efficiently represent the outputs for
several classes of conjunctive queries.Comment: To appear in PODS'18; 35 pages; comments welcom
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