32 research outputs found
Counting Problems in Parameterized Complexity
This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity. After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large graphs, and counting perfect matchings and Hamiltonian cycles in well-structured graphs.
While this survey presupposes familiarity with parameterized algorithms and complexity, we aim at explaining all relevant notions from counting complexity in a self-contained way
Supplement to the Collatz Conjecture
For any natural number was created the supplement sequence, that is convergent together with the original sequence. The parameter - index was defined, that is the same tor both sequences. This new method provides the following results:
All natural numbers were distributed into different classes according to the corresponding indexes;
The analytic formulas ( not by computer performed routine calculations) were produced, the formulas for groups of consecutive natural numbers of different lengths, having the same index;
The new algorithm to find index for any natural number was constructed and proved
On the value set of small families of polynomials over a finite field, I
We obtain an estimate on the average cardinality of the value set of any
family of monic polynomials of Fq[T] of degree d for which s consecutive
coefficients a_{d-1},..., a_{d-s} are fixed. Our estimate holds without
restrictions on the characteristic of Fq and asserts that
V(d,s,\bfs{a})=\mu_d.q+\mathcal{O}(1), where V(d,s,\bfs{a}) is such an average
cardinality, \mu_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!} and \bfs{a}:=(a_{d-1},..,
d_{d-s}). We provide an explicit upper bound for the constant underlying the
\mathcal{O}--notation in terms of d and s with "good" behavior. Our approach
reduces the question to estimate the number of Fq--rational points with
pairwise--distinct coordinates of a certain family of complete intersections
defined over Fq. We show that the polynomials defining such complete
intersections are invariant under the action of the symmetric group of
permutations of the coordinates. This allows us to obtain critical information
concerning the singular locus of the varieties under consideration, from which
a suitable estimate on the number of Fq--rational points is established.Comment: 30 page
A Methodological Framework for the Reconstruction of Contiguous Regions of Ancestral Genomes and Its Application to Mammalian Genomes
The reconstruction of ancestral genome architectures and gene orders from homologies between extant species is a long-standing problem, considered by both cytogeneticists and bioinformaticians. A comparison of the two approaches was recently investigated and discussed in a series of papers, sometimes with diverging points of view regarding the performance of these two approaches. We describe a general methodological framework for reconstructing ancestral genome segments from conserved syntenies in extant genomes. We show that this problem, from a computational point of view, is naturally related to physical mapping of chromosomes and benefits from using combinatorial tools developed in this scope. We develop this framework into a new reconstruction method considering conserved gene clusters with similar gene content, mimicking principles used in most cytogenetic studies, although on a different kind of data. We implement and apply it to datasets of mammalian genomes. We perform intensive theoretical and experimental comparisons with other bioinformatics methods for ancestral genome segments reconstruction. We show that the method that we propose is stable and reliable: it gives convergent results using several kinds of data at different levels of resolution, and all predicted ancestral regions are well supported. The results come eventually very close to cytogenetics studies. It suggests that the comparison of methods for ancestral genome reconstruction should include the algorithmic aspects of the methods as well as the disciplinary differences in data aquisition
Polynomial-delay Enumeration Kernelizations for Cuts of Bounded Degree
Enumeration kernelization was first proposed by Creignou et al. [TOCS 2017]
and was later refined by Golovach et al. [JCSS 2022] into two different
variants: fully-polynomial enumeration kernelization and polynomial-delay
enumeration kernelization. In this paper, we consider the d-CUT problem from
the perspective of (polynomial-delay) enumeration kenrelization. Given an
undirected graph G = (V, E), a cut F = E(A, B) is a d-cut of G if every u in A
has at most d neighbors in B and every v in B has at most d neighbors in A.
Checking the existence of a d-cut in a graph is a well-known NP-hard problem
and is well-studied in parameterized complexity [Algorithmica 2021, IWOCA
2021]. This problem also generalizes a well-studied problem MATCHING CUT (set d
= 1) that has been a central problem in the literature of polynomial-delay
enumeration kernelization. In this paper, we study three different enumeration
variants of this problem, ENUM d-CUT, ENUM MIN-d-CUT and ENUM MAX-d-CUT that
intends to enumerate all the d-cuts, all the minimal d-cuts and all the maximal
d-cuts respectively. We consider various structural parameters of the input and
provide polynomial-delay enumeration kernels for ENUM d-CUT and ENUM MAX-d-CUT
and fully-polynomial enumeration kernels of polynomial size for ENUM MIN-d-CUT.Comment: 25 page