Sándor Z. Kiss

Counting and sampling SCJ small parsimony solutions

The Single Cut or Join (SCJ) operation on genomes, generalizing chromosome evolution by fusions and fissions, is the computationally simplest known model of genome rearrangement. While most genome rearrangement problems are already hard when comparing three genomes, it is possible to compute in polynomial time a most parsimonious SCJ scenario for an arbitrary number of genomes related by a binary phylogenetic tree. Here we consider the problems of sampling and counting the most parsimonious SCJ scenarios. We show that both the sampling and counting problems are easy for two genomes, and we relate SCJ scenarios to alternating permutations. However, for an arbitrary number of genomes related by a binary phylogenetic tree, the counting and sampling problems become hard. We prove that if a Fully Polynomial Randomized Approximation Scheme or a Fully Polynomial Almost Uniform Sampler exist for the most parsimonious SCJ scenario, then RP=NP. The proof has a wider scope than genome rearrangements: the same result holds for parsimonious evolutionary scenarios on any set of discrete characters.

Groups, partitions and representation functions

Let X be a semigroup written additively and h ≥ 2 a fixed integer. Let x be an element of X and A1, . . . , Ah be nonempty subsets of X. Let RA1+···+Ah(x) denote the number of solutions of the equation a1 + · · · + ah = x, where ai ∈ Ai. In this paper for X = N we give a necessary and sufficient condition such that the equality RA1+A2(n) = RX\A1+X\A2(n) holds from a certain point on. We study similar questions when X = Zm and in general when X = G, where G is a finite additive group.

On the maximum values of the additive representation functions

Let A and B be infinite sequences of nonnegative integers. For a positive integer n, let RA(n) denote the number of representations of n as the sum of two terms from A. Let sA(x) denote the maximum value of RA(n) up to x and dA,B(x) denote the distance of the sequences A and B. In this paper, we study the connection between sA(x), sB(x) and dA,B(x). We improve a result of Haddad and Helou about the Erdős–Turan conjecture.

On generalized Stanley sequences

Let

k

{\mathbb{N}}

On a parity based group testing algorithm

N denote the set of all nonnegative integers. Let

On additive complement of a finite set

{k \ge 3}