An uncertain database is defined as a relational database in which primary keys need not be satisfied. A repair (or possible world) of such database is obtained by selecting a maximal number of tuples without ever selecting two distinct tuples with the same primary key value. For a Boolean query q, the decision problem CERTAINTY(q) takes as input an uncertain database db and asks whether q is satisfied by every repair of db. Our main focus is on acyclic Boolean conjunctive queries without self-join. Previous work has introduced the notion of (directed) attack graph of such queries, and has proved that CERTAINTY(q) is first-order expressible if and only if the attack graph of q is acyclic. The current paper investigates the boundary between tractability and intractability of CERTAINTY(q). We first classify cycles in attack graphs as either weak or strong, and then prove among others the following. If the attack graph of a query q contains a strong cycle, then CERTAINTY(q) is coNP-complete. If the attack graph of q contains no strong cycle and every weak cycle of it is terminal (i.e., no edge leads from a vertex in the cycle to a vertex outside the cycle), then CERTAINTY(q) is in P. We then partially address the only remaining open case, i.e., when the attack graph contains some nonterminal cycle and no strong cycle. Finally, we establish a relationship between the complexities of CERTAINTY(q) and evaluating q on probabilistic databases.
Kolaitis and Pema @cite_3 recently showed that for every query @math with exactly two atoms, @math is either in or -complete, and it is decidable which of the two is the case. If @math is in and not first-order expressible, then it can be reduced in polynomial time to the problem of finding maximal (with respect to cardinality) independent sets of vertices in claw-free graphs. The latter problem can be solved in polynomial time by an ingenious algorithm of Minty @cite_6 . Unfortunately, the proposed reduction is not applicable on queries with more than two atoms.
@cite_6 @cite_3
2071423034 2037028865 2152035036 2772612740
We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2O(k2) whether a given set of m clauses of size r admits a truth assignment that satisfies at least ((2r - 1)m + k) 2r clauses. Thus Max-r-Sat is fixed-parameter tractable when parameterized by the number of satisfied clauses above the tight lower bound (1--2-r)m. This solves an open problem of Mahajan, Raman and Sikdar (J. Comput. System Sci., 75, 2009). Our algorithm is based on a polynomial-time data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with O(k2) variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size O(k2), then there is a truth assignment satisfying the required number of clauses. Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of Max-2-Sat with m clauses has at least 3k variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least (3m + k) 4 clauses. We also outline how the fixed-parameter tractability result on Max-r-Sat can be extended to a family of Boolean Constraint Satisfaction Problems. Papadimitriou introduced several classes of NP search problems based on combinatorial principles which guarantee the existence of solutions to the problems. Many interesting search problems not known to be solvable in polynomial time are contained in these classes, and a number of them are complete problems. We consider the question of the relative complexity of these search problem classes. We prove several separations which show that in a generic relativized world the search classes are distinct and there is a standard search problem in each of them that is not computationally equivalent to any decision problem. (Naturally, absolute separations would imply thatP?NP.) Our separation proofs have interesting combinatorial content and go to the heart of the combinatorial principles on which the classes are based. We derive one result via new lower bounds on the degrees of poly- nomials asserted to exist by Hilbert's nullstellensatz over finite fields. We consider the problem of finding a characterization for polynomial time computable queries on finite structures in terms of logical definability. It is well known that fixpoint logic provides such a characterization in the presence of a built-in linear order, but without linear order even very simple polynomial time queries involving counting are not expressible in fixpoint logic. Our approach to the problem is based on generalized quantifiers. A generalized quantifier isn-ary if it binds any number of formulas, but at mostnvariables in each formula. We prove that, for each natural numbern, there is a query on finite structures which is expressible in fixpoint logic, but not in the extension of first-order logic by any set ofn-ary quantifiers. It follows that the expressive power of fixpoint logic cannot be captured by adding finitely many quantifiers to first-order logic. Furthermore, we prove that, for each natural numbern, there is a polynomial time computable query which is not definable in any extension of fixpoint logic byn-ary quantifiers. In particular, this rules out the possibility of characterizing PTIME in terms of definability in fixpoint logic extended by a finite set of generalized quantifiers. Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph primitives. For m-edge and n-vertex graphs, it is well-known to be solvable in O(m n ) time; however, for several applications this running time is still too slow. We investigate how linear-time (and almost linear-time) data reduction (used as preprocessing) can alleviate the situation. More specifically, we focus on linear-time kernelization. We start a deeper and systematic study both for general graphs and for bipartite graphs. Our data reduction algorithms easily comply (in form of preprocessing) with every solution strategy (exact, approximate, heuristic), thus making them attractive in various settings.