Degree-constrained decompositions of graphs: Bounded treewidth and planarity

We study the problem of decomposing the vertex set V$V$ of a graph into two nonempty parts _V1,_V2$V_1,V_2$ which induce subgraphs where each vertex v∈_V1$v\in V_1$ has degree at least a(v)$a(v)$ inside _V1$V_1$ and each v∈_V2$v\in V_2$ has degree at least b(v)$b(v)$ inside _V2$V_2$. We give a polynomial-time algorithm for graphs with bounded treewidth which decides if a graph admits a decomposition, and gives such a decomposition if it exists. This result and its variants are then applied to designing polynomial-time approximation schemes for planar graphs where a decomposition does not necessarily exist but the local degree conditions should be met for as many vertices as possible.     

A palindromization map for the free group

We define a self-map Pal:_F2→_F2$Pal:F_2\to F_2$ of the free group on two generators a,b$a,b$, using automorphisms of _F2$F_2$ that form a group isomorphic to the braid group _B3$B_3$. The map Pal$Pal$ restricts to de Luca’s right iterated palindromic closure on the submonoid generated by a,b$a,b$. We show that Pal$Pal$ is continuous for the profinite topology on _F2$F_2$; it is the unique continuous extension of de Luca’s right iterated palindromic closure to _F2$F_2$. The values of Pal$Pal$ are palindromes and coincide with the elements g∈_F2$g\in F_2$ such that abg$abg$ and bag$bag$ are conjugate.     

Divergence bounded computable real numbers

A real x$x$ is called h$h$-bounded computable  , for some function h:N→N$h:\mathbb{N}\to \mathbb{N}$, if there is a computable sequence (_xs)$(x_s)$ of rational numbers which converges to x$x$ such that, for any n∈N$n\in \mathbb{N}$, at most h(n)$h(n)$ non-overlapping pairs of its members are separated by a distance larger than 2^-n$2^{-n}$. In this paper we discuss properties of h$h$-bounded computable reals for various functions h$h$. We will show a simple sufficient condition for a class of functions h$h$ such that the corresponding h$h$-bounded computable reals form an algebraic field. A hierarchy theorem for h$h$-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h$h$-bounded computability for special functions h$h$.     

Adaptive timeliness of consensus in presence of crash and timing faults

The Δ$\mathrm{\Delta }$-timed uniform consensus is a stronger variant of the traditional consensus and it satisfies the following additional property: every correct process terminates its execution within a constant time Δ$\mathrm{\Delta }$ (Δ$\mathrm{\Delta }$-timeliness), and no two processes decide differently (uniformity). In this paper, we consider the Δ$\mathrm{\Delta }$-timed uniform consensus problem in presence of _fc$f_{\mathrm{c}}$ crash processes and _ft$f_{\mathrm{t}}$ timing-faulty processes, and propose a Δ$\mathrm{\Delta }$-timed uniform consensus algorithm. The proposed algorithm is adaptive in the following sense: it solves the Δ$\mathrm{\Delta }$-timed uniform consensus when at least _ft+1$f_{\mathrm{t}}+1$ correct processes exist in the system. If the system has less than _ft+1$f_{\mathrm{t}}+1$ correct processes, the algorithm cannot solve the Δ$\mathrm{\Delta }$-timed uniform consensus. However, as long as _ft+1$f_{\mathrm{t}}+1$ processes are non-crashed, the algorithm solves (non-timed) uniform consensus. We also investigate the maximum number of faulty processes that can be tolerated. We show that any Δ$\mathrm{\Delta }$-timed uniform consensus algorithm tolerating up to _ft$f_{\mathrm{t}}$ timing-faulty processes requires that the system has at least _ft+1$f_{\mathrm{t}}+1$ correct processes. This impossibility result implies that the proposed algorithm attains the maximum resilience about the number of faulty processes. We also show that any Δ$\mathrm{\Delta }$-timed uniform consensus algorithm tolerating up to _ft$f_{\mathrm{t}}$ timing-faulty processes cannot solve the (non-timed) uniform consensus when the system has less than _ft+1$f_{\mathrm{t}}+1$ non-crashed processes. This impossibility result implies that our algorithm attains the maximum adaptiveness.     

Complexity of question/answer games

Question/Answer games (Q/A games for short) are a generalization of the Rényi–Ulam game and they are a model for information extraction in parallel. A Q/A game, G=(D,s,(_q1,…,_qk))$G=(D,s,(q_1,\dots ,q_k))$, is played on a directed acyclic graph, D=(V,E)$D=(V,E)$, with a distinguished start vertex s$s$. In the i$i$th round, Paul selects a set, _Qi⊆V$Q_i\subseteq V$, of at most _qi$q_i$ non-terminal vertices. Carole responds by choosing an outgoing edge from each vertex in _Qi$Q_i$. At the end of k$k$ rounds, Paul wins if Carole’s answers define a unique path from the root to one of the terminal vertices in D$D$.     

Large independent sets in random regular graphs

We present algorithmic lower bounds on the size _sd$s_d$ of the largest independent sets of vertices in random d$d$-regular graphs, for each fixed d≥3$d\ge 3$. For instance, for d=3$d=3$ we prove that, for graphs on n$n$ vertices, _sd≥0.43475n$s_d\ge 0.43475n$ with probability approaching one as n$n$ tends to infinity.     

Effective bounds for P-recursive sequences

We describe an algorithm that takes as input a complex sequence (_un)$\left(u_n\right)$ given by a linear recurrence relation with polynomial coefficients along with initial values, and outputs a simple explicit upper bound (_vn)$\left(v_n\right)$ such that |_un|≤_vn$\left|u_n\right|\le v_n$ for all n$n$. Generically, the bound is tight, in the sense that its asymptotic behaviour matches that of _un$u_n$. We discuss applications to the evaluation of power series with guaranteed precision.     

A quantitative Pólya’s Theorem with zeros

Let R[X]:=R[_X1,…,_Xn]$\mathbb{R}\left[X\right]:=\mathbb{R}[X_1,\dots ,X_n]$. Pólya’s Theorem says that if a form (homogeneous polynomial) p∈R[X]$p\in \mathbb{R}\left[X\right]$ is positive on the standard n$n$-simplex _Δn${\Delta }_n$, then for sufficiently large N$N$ all the coefficients of (_X1+?+_Xn)^Np${(X_1+?+X_n)}^{N}p$ are positive. The work in this paper is part of an ongoing project aiming to explain when Pólya’s Theorem holds for forms if the condition “positive on _Δn${\Delta }_n$” is relaxed to “nonnegative on _Δn${\Delta }_n$”, and to give bounds on N$N$. Schweighofer gave a condition which implies the conclusion of Pólya’s Theorem for polynomials f∈R[X]$f\in \mathbb{R}\left[X\right]$. We give a quantitative version of this result and use it to settle the case where a form p∈R[X]$p\in \mathbb{R}\left[X\right]$ is positive on _Δn${\Delta }_n$, apart from possibly having zeros at the corners of the simplex.     

On asymptotic extrapolation

Consider a power series f∈R[[z]]$f\in \mathbb{R}\left[\left[z\right]\right]$, which is obtained by a precise mathematical construction. For instance, f$f$ might be the solution to some differential or functional initial value problem or the diagonal of the solution to a partial differential equation. In cases when no suitable method is available beforehand for determining the asymptotics of the coefficients _fn$f_n$, but when many such coefficients can be computed with high accuracy, it would be useful if a plausible asymptotic expansion for _fn$f_n$ could be guessed automatically.     

Algorithms for near solutions to polynomial equations

Let F(x,y)$F(x,y)$ be a polynomial over a field K$K$ and m$m$ a nonnegative integer. We call a polynomial g$g$ over K$K$ an m$m$-near solution of F(x,y)$F(x,y)$ if there exists a c∈K$c\in K$ such that F(x,g)=cx^m$F(x,g)=c{x}^m$, and the number c$c$ is called an m$m$-value of F(x,y)$F(x,y)$ corresponding to g$g$. In particular, c$c$ can be 0. Hence, by viewing F(x,y)=0$F(x,y)=0$ as a polynomial equation over K[x]$K\left[x\right]$ with variable y$y$, every solution of the equation F(x,y)=0$F(x,y)=0$ in K[x]$K\left[x\right]$ is also an m$m$-near solution. We provide an algorithm that gives all m$m$-near solutions of a given polynomial F(x,y)$F(x,y)$ over K$K$, and this algorithm is polynomial time reducible to solving one variable equations over K$K$. We introduce approximate solutions to analyze the algorithm. We also give some interesting properties of approximate solutions.     

Modular Las Vegas algorithms for polynomial absolute factorization

Let f(X,Y)∈Z[X,Y]$f(X,Y)\in \mathbb{Z}[X,Y]$ be an irreducible polynomial over Q$\mathbb{Q}$. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of f$f$, or more precisely, of f$f$ modulo some prime integer p$p$. The same idea of choosing a p$p$ satisfying some prescribed properties together with LLL$LLL$ is used to provide a new strategy for absolute factorization of f(X,Y)$f(X,Y)$. We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to construct the algebraic extension containing one absolute factor of a polynomial of degree up to 400.     

Computing phylogenetic roots with bounded degrees and errors is NP-complete

In this paper we study the computational complexity of the following optimization problem: given a graph G=(V,E)$G=(V,E)$, we wish to find a tree T such that (1) the degree of each internal node of T   is at least 3 and at most Δ$\Delta$, (2) the leaves of T are exactly the elements of V, and (3) the number of errors, that is, the symmetric difference between E   and {{u,v}:u,v$\{\{u,v\}:u,v$ are leaves of T   and _dT(u,v)≤k}$d_T(u,v)\le k\}$, is as small as possible, where _dT(u,v)$d_T(u,v)$ denotes the distance between u$u$ and v$v$ in tree T  . We show that this problem is NP-hard for all fixed constants Δ,k≥3$\Delta ,k\ge 3$.     

Stratification associated with local b-functions

The local b$b$-function _bf,p(s)$b_{f,p}\left(s\right)$ of an n$n$-variate polynomial f∈C[x]$f\in \mathbb{C}\left[x\right]$ (x=(_x1,…,_xn)$x=(x_1,\dots ,x_n)$) at a point p∈Cⁿ$p\in {\mathbb{C}}^n$ is constant on each stratum of a stratification of Cⁿ${\mathbb{C}}^n$. We propose a new method for computing such a stratification and _bf,p(s)$b_{f,p}\left(s\right)$ on each stratum. In the existing method proposed in Oaku (1997b), a primary ideal decomposition of an ideal in C[x,s]$\mathbb{C}[x,s]$ is needed and our experiment shows that the primary decomposition can be a bottleneck for computing the stratification. In our new method, the computation can be done by just computing ideal quotients and examining inclusions of algebraic sets. The precise form of a stratum can be obtained by computing the decomposition of the radicals of the ideals in C[x]$\mathbb{C}\left[x\right]$ defining the stratum. We also introduce various techniques for improving the practical efficiency of the implementation and we show results of computations for some examples.     

The number of roots of a lacunary bivariate polynomial on a line

We prove that a polynomial f∈R[x,y]$f\in \mathbb{R}[x,y]$ with t$t$ non-zero terms, restricted to a real line y=ax+b$y=ax+b$, either has at most 6t−4$6t-4$ zeros or vanishes over the whole line. As a consequence, we derive an alternative algorithm for deciding whether a linear polynomial y−ax−b∈K[x,y]$y-ax-b\in K[x,y]$ divides a lacunary polynomial f∈K[x,y]$f\in K[x,y]$, where K$K$ is a real number field. The number of bit operations performed by the algorithm is polynomial in the number of non-zero terms of f$f$, in the logarithm of the degree of f$f$, in the degree of the extension K/Q$K/\mathbb{Q}$ and in the logarithmic height of a$a$, b$b$ and f$f$.     

Choosing starting values for certain Newton–Raphson iterations

We aim at finding the best possible seed values when computing a^1/p$a^{1/p}$ using the Newton–Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value can be quite far from this natural choice. When we evaluate a monotone function f(a)$f(a)$ in the interval [_amin,_amax]$[a_{\min },a_{\max }]$, by building the sequence _xn$x_n$ defined by the Newton–Raphson iteration, the natural choice consists in choosing _x0$x_0$ equal to the arithmetic mean of the endpoint values. This minimizes the maximum possible distance between _x0$x_0$ and f(a)$f(a)$. And yet, if we perform n$n$ iterations, what matters is to minimize the maximum possible distance between _xn$x_n$ and f(a)$f(a)$. In several examples, the value of the best starting point varies rather significantly with the number of iterations.