23 research outputs found
Understanding the complexity of #SAT using knowledge compilation
Two main techniques have been used so far to solve the #P-hard problem #SAT.
The first one, used in practice, is based on an extension of DPLL for model
counting called exhaustive DPLL. The second approach, more theoretical,
exploits the structure of the input to compute the number of satisfying
assignments by usually using a dynamic programming scheme on a decomposition of
the formula. In this paper, we make a first step toward the separation of these
two techniques by exhibiting a family of formulas that can be solved in
polynomial time with the first technique but needs an exponential time with the
second one. We show this by observing that both techniques implicitely
construct a very specific boolean circuit equivalent to the input formula. We
then show that every beta-acyclic formula can be represented by a polynomial
size circuit corresponding to the first method and exhibit a family of
beta-acyclic formulas which cannot be represented by polynomial size circuits
corresponding to the second method. This result shed a new light on the
complexity of #SAT and related problems on beta-acyclic formulas. As a
byproduct, we give new handy tools to design algorithms on beta-acyclic
hypergraphs
A formal context for closures of acyclic hypergraphs
Database constraints in the relational database model (RDBM) can be viewed as a set of rules that apply to a dataset, or as a set of axioms that can generate a (closed) set of those constraints. In this paper, we use Formal Concept Analysis to characterize the axioms of Acyclic Hypergraphs (in the RDBM they are called Acyclic Join Dependencies). This present paper complements and generalizes previous work on FCA and databases constraints.Peer ReviewedPostprint (author's final draft
Constant Delay Enumeration with FPT-Preprocessing for Conjunctive Queries of Bounded Submodular Width
Marx (STOC 2010, J. ACM 2013) introduced the notion of submodular width of a conjunctive query (CQ) and showed that for any class Phi of Boolean CQs of bounded submodular width, the model-checking problem for Phi on the class of all finite structures is fixed-parameter tractable (FPT). Note that for non-Boolean queries, the size of the query result may be far too large to be computed entirely within FPT time. We investigate the free-connex variant of submodular width and generalise Marx\u27s result to non-Boolean queries as follows: For every class Phi of CQs of bounded free-connex submodular width, within FPT-preprocessing time we can build a data structure that allows to enumerate, without repetition and with constant delay, all tuples of the query result. Our proof builds upon Marx\u27s splitting routine to decompose the query result into a union of results; but we have to tackle the additional technical difficulty to ensure that these can be enumerated efficiently
A polynomial-size extended formulation for the multilinear polytope of beta-acyclic hypergraphs
We consider the multilinear polytope defined as the convex hull of the set of
binary points satisfying a collection of multilinear equations. The complexity
of the facial structure of the multilinear polytope is closely related to the
acyclicity degree of the underlying hypergraph. We obtain a polynomial-size
extended formulation for the multilinear polytope of beta-acyclic hypergraphs,
hence characterizing the acyclic hypergraphs for which such a formulation can
be constructed
Synchronisation Games on Hypergraphs
We study a strategic game model on hypergraphs where players, modelled by nodes, try to coordinate or anti-coordinate their choices within certain groups of players, modelled by hyperedges. We show this model to be a strict generalisation of symmetric additively separable hedonic games to the hypergraph setting and that such games always have a pure Nash equilibrium, which can be computed in pseudo-polynomial time. Moreover, in the pure coordination setting, we show that a strong equilibrium exists and can be computed in polynomial time when the game possesses a certain acyclic structure.</jats:p
Structure and Complexity of Bag Consistency
Since the early days of relational databases, it was realized that acyclic
hypergraphs give rise to database schemas with desirable structural and
algorithmic properties. In a by-now classical paper, Beeri, Fagin, Maier, and
Yannakakis established several different equivalent characterizations of
acyclicity; in particular, they showed that the sets of attributes of a schema
form an acyclic hypergraph if and only if the local-to-global consistency
property for relations over that schema holds, which means that every
collection of pairwise consistent relations over the schema is globally
consistent. Even though real-life databases consist of bags (multisets), there
has not been a study of the interplay between local consistency and global
consistency for bags. We embark on such a study here and we first show that the
sets of attributes of a schema form an acyclic hypergraph if and only if the
local-to global consistency property for bags over that schema holds. After
this, we explore algorithmic aspects of global consistency for bags by
analyzing the computational complexity of the global consistency problem for
bags: given a collection of bags, are these bags globally consistent? We show
that this problem is in NP, even when the schema is part of the input. We then
establish the following dichotomy theorem for fixed schemas: if the schema is
acyclic, then the global consistency problem for bags is solvable in polynomial
time, while if the schema is cyclic, then the global consistency problem for
bags is NP-complete. The latter result contrasts sharply with the state of
affairs for relations, where, for each fixed schema, the global consistency
problem for relations is solvable in polynomial time
The pseudo-Boolean polytope and polynomial-size extended formulations for binary polynomial optimization
With the goal of obtaining strong relaxations for binary polynomial
optimization problems, we introduce the pseudo-Boolean polytope defined as the
convex hull of the set of binary points satisfying a collection of equations
containing pseudo-Boolean functions. By representing the pseudo-Boolean
polytope via a signed hypergraph, we obtain sufficient conditions under which
this polytope has a polynomial-size extended formulation. Our new framework
unifies and extends all prior results on the existence of polynomial-size
extended formulations for the convex hull of the feasible region of binary
polynomial optimization problems of degree at least three