arXiv subjects

Norbert Blum

Publications and source records attributed to Norbert Blum.

3 recordsLinked to original sources

arXiv2020

On the Approximation Method and the P versus NP Problem

First of all we give some reasons that "natural proofs" built not a barrier to prove P $\not=$ NP using Boolean complexity. Then we investigate the approximation method for its extension to prove super-polynomial lower bounds for the non-monotone complexity of suitable Boolean functions in NP or to understand why this is not possible. It is given some evidence that the approximation method alone cannot be used to prove a super-linear lower bound for any function $f \in {\cal B}_n$. Additionally, an overview on the methods for proving lower bounds of the non-monotone and the monotone complexity of Boolean functions is given. Finally, a personal opinion how to proceed the research on the P versus NP problem and also on proving a super-linear lower bound for the non-monotone complexity of a Boolean function in NP is given.

cs.CC↗

arXiv2015

On LR(k)-parsers of polynomial size

Usually, a parser for an $LR(k)$-grammar $G$ is a deterministic pushdown transducer which produces backwards the unique rightmost derivation for a given input string $x \in L(G)$. The best known upper bound for the size of such a parser is $O(2^{|G||Σ|^k+k\log |Σ| + \log |G|})$ where $|G|$ and $|Σ|$ are the sizes of the grammar $G$ and the terminal alphabet $Σ$, respectively. If we add to a parser the possibility to manipulate a directed graph of size $O(|G|n)$ where $n$ is the length of the input then we obtain an extended parser. The graph is used for an efficient parallel simulation of all potential leftmost derivations of the current right sentential form such that the unique rightmost derivation of the input can be computed. Given an arbitrary $LR(k)$-grammar $G$, we show how to construct an extended parser of $O(|G| + \#LA |N|2^k k \log k)$ size where $|N|$ is the number of nonterminal symbols and $\#LA$ is the number of relevant lookaheads with respect to the grammar $G$. As the usual parser, this extended parser uses only tables as data structure. Using some ingenious data structures and increasing the parsing time by a small constant factor, the size of the extended parser can be reduced to $O(|G| + \#LA|N|k^2)$. The parsing time is $O(ld(input) + k|G|n)$ where $ld(input)$ is the length of the derivation of the input. Moreover, we have constructed a one pass parser.

cs.FL↗

arXiv2015

Maximum Matching in General Graphs Without Explicit Consideration of Blossoms Revisited

We reduce the problem of finding an augmenting path in a general graph to a reachability problem in a directed bipartite graph. A slight modification of depth-first search leads to an algorithm for finding such paths. Although this setting is equivalent to the traditional terminology of blossoms due to Edmonds, there are some advantages. Mainly, this point of view enables the description of algorithms for the solution of matching problems without explicit analysis of blossoms, nested blossoms, and so on. Exemplary, we describe an efficient realization of the Hopcroft-Karp approach for the computation of a maximum cardinality matching in general graphs and a variant of Edmonds' primal-dual algorithm for the maximum weighted matching problem.

cs.DS↗