Article HistoryIn this paper we investigate two properties of some propositional systems of Intuitionistic, Johansson's and Monotone logics: 1) the relations between the proofs complexities of strongly equal tautologies (valid sequents) and 2) the relations between the proofs complexities of minimal tautologies (valid sequents) and of results of substitutions in them. We show that 1) strongly equal tautologies (valid sequents) can have essential different proof complexities in the same system and 2) the result of substitution can be proved easier, than corresponding minimal tautology (valid sequents), therefore the systems, which are considered in this paper, are no monotonous neither by lines nor by size.
In this paper, we present the results on Frege proof complexities of some DNFtautologies. At first we introduce the notion of complete DNFs and prove that complete DNFs are tautologies, we also show that if the proof complexities for the set of complete DNFs are polynomially bounded, then the set of DNF-tautologies D, for which the number of non-negated variables in every conjunct is O(log(D)), also has polynomially bounded proof complexities. Later we show that the set of all balanced DNF-tautologies has polynomial proof complexities..
In this paper, we discuss parameterized algorithms for variants of the partiall vertex cover problem. Recall that in the classical vertex cover problem (VC), we are given a graph G = (V, E) and a number K and asked if we can cover the edges e ∈ E, using at most K vertices from V . In the Partial vertex cover problem (PVC), in addition to the parameter K, we are given a second parameter K ′ and the question is whether we can cover at least K ′ edges e ∈ E using at most K vertices from V . The weighted generalizations of the VC and PVC problems are called the Weighted vertex cover (WVC) and the Partial weighted vertex cover problem (WPVC) respectively. In the WPCV problem, we are given two parameters R and L, associated respectively with the vertex set V and edge set E of the graph G. Additionally, we are given non-negative integral weight functions for the vertices and the edges. The goal then is to cover edges of total weight at least L, using vertices of total weight at most R. (In the WVC problem, the goal is to cover all the edges with vertices whose total cost is at most R). Observe that the variants of VC mentioned here, viz., PVC, WVC and WPVC are all generalizations of VC and hence their NP-completeness follows immediately from the NP-completeness of VC. One attack on NP-complete problems is to devise algorithms that are polynomial, if certain selected selected parameters are bounded. Such algorithms, if they exist are called parameterized algorithms and if they run in time polynomial in the size of the input (but exponential time in the size of the parameter), the problem is said to be fixed-parameter tractable. This paper studies several variants of the PVC problem and establishes new results from the perspective of fixed parameter tractability and W[1]-hardness. We also introduce a new problem called the Partial vertex cover with matching constraint and show that it is fixed-parameter tractable for a certain class of graphs.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.