491 research outputs found
On rectangular covering problems
Many applications like picture processing, data compression or pattern recognition require a covering of a set of points most often located in the (discrete) plane by rectangles due to specific cost constraints. In this paper we provide exact dynamic programming algorithms for covering point sets by regular rectangles, that have to obey certain (parameterized) boundary conditions. The concrete representative out of a class of objective functions that is studied is to minimize sum of area, circumference and number of patches used. This objective function may be motivated by requirements of numerically solving PDE's by discretization over (adaptive multi-)grids. More precisely, we propose exact deterministic algorithms for such problems based on a (set theoretic) dynamic programming approach yielding a time bound of O(n^23^n) . In a second step this bound is (asymptotically) decreased to O(n^62^n) by exploiting the underlying rectangular and lattice structures. Finally, a generalization of the problem and its solution methods is discussed for the case of arbitrary (finite) space dimension
On variable-weighted exact satisfiability problems
We show that the NP-hard optimization problems minimum and maximum weight exact satisfiability (XSAT) for a CNF formula C over n propositional variables equipped with arbitrary real-valued weights can be solved in O(|C|2^{0.2441n}) time. To the best of our knowledge, the algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time. We also investigate the corresponding weighted counting problems, namely we show that the number of all minimum, resp. maximum, weight exact satisfiability solutions of an arbitrarily weighted formula can be determined in O(n^2cdot |C|+2^{0.40567n}) time. In recent years only the unweighted counterparts of these problems have been studied cite{dahl,dahl2,porschen}
On rectangular covering problems
Many applications like picture processing, data compression or pattern recognition require a covering of a set of points most often located in the (discrete) plane by rectangles due to specific cost constraints. In this paper we provide exact dynamic programming algorithms for covering point sets by regular rectangles, that have to obey certain (parameterized) boundary conditions. The concrete representative out of a class of objective functions that is studied is to minimize sum of area, circumference and number of patches used. This objective function may be motivated by requirements of numerically solving PDE's by discretization over (adaptive multi-)grids. More precisely, we propose exact deterministic algorithms for such problems based on a (set theoretic) dynamic programming approach yielding a time bound of O(n^23^n) . In a second step this bound is (asymptotically) decreased to O(n^62^n) by exploiting the underlying rectangular and lattice structures. Finally, a generalization of the problem and its solution methods is discussed for the case of arbitrary (finite) space dimension
On variable-weighted exact satisfiability problems
We show that the NP-hard optimization problems minimum and maximum weight exact satisfiability (XSAT) for a CNF formula C over n propositional variables equipped with arbitrary real-valued weights can be solved in O(|C|2^{0.2441n}) time. To the best of our knowledge, the algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time. We also investigate the corresponding weighted counting problems, namely we show that the number of all minimum, resp. maximum, weight exact satisfiability solutions of an arbitrarily weighted formula can be determined in O(n^2cdot |C|+2^{0.40567n}) time. In recent years only the unweighted counterparts of these problems have been studied cite{dahl,dahl2,porschen}
Visual sociology for research and concept development: graphic and film material - additional ways for gaining knowledge ; paper on the occasion of the Conference of the International Visual Sociology Association (IVSA)
Algorithms for Variable-Weighted 2-SAT and Dual Problems
In this paper we study NP-hard weighted satisfiability optimization problems for the class 2-CNF providing worst-case upper time bounds. Moreover we consider the monotone dual class consisting of clause sets where all variables occur at most twice. We show that weighted SAT, XSAT and NAESAT optimization problems for this class are polynomial time solvable using appropriate reductions to specific polynomial time solvable graph problems
Improving a fixed parameter tractability time bound for the shadow problem
AbstractConsider a forest of k trees and n nodes together with a (partial) function σ mapping leaves of the trees to non-root nodes of other trees. Define the shadow of a leaf ℓ to be the subtree rooted at σ(ℓ). The shadow problem asks whether there is a set S of leaves exactly one from each tree such that none of these leaves lies in the shadow of another leaf in S. This graph theoretical problem as shown in Franco et al. (Discrete Appl. Math. 96 (1999) 89) is equivalent to the falsifiability problem for pure implicational Boolean formulas over n variables with k occurences of the constant false as introduced in: Heusch J. Wiedermann, P. Hajek (Eds.), Proceedings of the Twentieth International Symposium on Mathematical Foundations of Computer Science (MFCS’95), Prague, Czech Republic, Lecture Notes in Computer Science, Vol. 969, Springer, Berlin, 1995, pp. 221–226, where its NP-completeness is shown for arbitrary values of k and a time bound of O(nk) for fixed k was obtained. In Franco et al. (1999) this bound is improved to O(n2kk) showing the problem's fixed parameter tractability (Congr. Numer. 87 (1992) 161). In this paper the bound O(n33k) is achieved by dynamic programming techniques thus significantly improving the fixed parameter part
Clause set structures and satisfiability
We propose a new perspective on propositional clause sets and on that basis we investigate (new) polynomial time SAT-testable classes. Moreover, we study autarkies using a closure concept. A specific simple type of closures the free closures leads to a further formula class called hyperjoins that is studied w.r.t. SAT
Clause Set Structures and Polynomial-Time SAT-Decidable Classes
Proposing a fibre view on propositional clause sets, we investigate satisfiability testing for several CNF subclasses. Specifically, we show how to decide SAT in polynomial time for formulas where each pair of different clauses intersect either in all or in one variable
Algorithms for Variable-Weighted 2-SAT and Dual Problems
In this paper we study NP-hard weighted satisfiability optimization problems for the class 2-CNF providing worst-case upper time bounds. Moreover we consider the monotone dual class consisting of clause sets where all variables occur at most twice. We show that weighted SAT, XSAT and NAESAT optimization problems for this class are polynomial time solvable using appropriate reductions to specific polynomial time solvable graph problems
- …