We show an O∗((l + 1)n)-time algorithm for the channel assignment problem, where l is the maximum edge weight. This improves on the previous O∗((l + 2)n)-time algorithm by Kral [4], as well as algorithms for important special cases, like L(2, 1)-labelling. For the latter problem, our algorithm works in O∗(3n) time. The progress is achieved by applying the fast zeta transform in combination with...

We consider a new way of looking at multisets, involving linear rather than multiplicative notation and operations which mirror those on natural numbers. The motivation is to create a framework for mathematics which more closely connects with scientific usage in which repetition and relative concepts of equality play an important role. The tropical calculus is introduced as an important tool. a...

We propose an algorithm to calculate the number of weak compositions, wherein each part is restricted a different range integers. This performs orders approximation up exact solution by using Inclusion-Exclusion Principle. The great advantage it with respect classical generating function technique that calculation exponentially faster as size numbers involved increases.

Probably Approximately Correct learning algorithms generalize a small number of examples about an unknown concept into a function that can predict a future observation. More formally, let X and Y be the instance and outcome spaces, respectively. Then a PAC algorithm observes randomly drawn examples (x; f(x)) about an unknown concept f : X ! Y . These examples are independently and identically d...

On multivariate analyses generally distributions of explanatory variable have deviation depending on each unique quality, and eliminating deviation often beneficially effective. We propose two algorithms for rescaling of raw data and verify the validity of them using real reliable big data.

The nnnrlnip of inclusion and exclusion and Consecutive Sets M. H. McIlwain )p'n,Clrj·'Yli'·nl of Mathematics Furman University Greenville, SC USA 29613 applied to numerous areas of discrete and combinatorial mathematics. One manifestation of this occurs in probability of the union of events an alternating sum of pn)o(HHjlIW~S of various intersections of events. If the constituent events are th...

We show that the number of satisfying assignments of a k-CNF formula is determined uniquely from the numbers of unsatisfying assignments for clause-sets of size up to ⌊log k⌋+ 2. The information of this size is also shown to be necessary. key words: combinatorial problems; SAT; k-CNF formula; counting; inclusion-exclusion

Let A 1 , A 2 ,. .. , A n be events in some probability space. The approximate inclusion-exclusion problem, due to Linial and Nisan (1990), is to estimate Pr[A 1 ∪· · ·∪A n ] given Pr[ i∈S A i ] for all |S | k. Kahn et al. (1996) solve this problem optimally for each k. We study the following more general question: given Pr[ i∈S A i ] for all |S | k, estimate Pr the number of events among A 1 ,...