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.     

Minimum cost source location problem with local 3-vertex-connectivity requirements

Let G=(V,E)$G=(V,E)$ be a simple undirected graph with a set V$V$ of vertices and a set E$E$ of edges. Each vertex v∈V$v\in V$ has a demand d(v)∈_Z+$d\left(v\right)\in Z_{+}$ and a cost c(v)∈_R+$c\left(v\right)\in R_{+}$, where _Z+$Z_{+}$ and _R+$R_{+}$ denote the set of nonnegative integers and the set of nonnegative reals, respectively. The source location problem with vertex-connectivity requirements in a given graph G$G$ requires finding a set S$S$ of vertices minimizing _∑v∈Sc(v)${\sum }_{v\in S}c\left(v\right)$ such that there are at least d(v)$d\left(v\right)$ pairwise vertex-disjoint paths from S$S$ to v$v$ for each vertex v∈V−S$v\in V-S$. It is known that if there exists a vertex v∈V$v\in V$ with d(v)≥4$d\left(v\right)\ge 4$, then the problem is NP-hard even in the case where every vertex has a uniform cost. In this paper, we show that the problem can be solved in O(|V|⁴log²|V|)$O\left({\left|V\right|}^4{log}^2\left|V\right|\right)$ time if d(v)≤3$d\left(v\right)\le 3$ holds for each vertex v∈V$v\in V$.     

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.     

Characteristic set algorithms for equation solving in finite fields

Efficient characteristic set methods for computing zeros of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of zeros of a proper and monic triangular set is given. An improved zero decomposition algorithm is proposed to reduce the zero set of an equation system to the union of zero sets of monic proper triangular sets. The bitsize complexity of this algorithm is shown to be O(lⁿ)$O\left(l^n\right)$ for Boolean polynomials, where n$n$ is the number of variables and l≥2$l\ge 2$ is the number of equations. We also give a multiplication free characteristic set method for Boolean polynomials, where the sizes of the polynomials occurred during the computation do not exceed the sizes of the input polynomials and the bitsize complexity of algorithm is O(n^d)$O\left(n^d\right)$ for input polynomials with n$n$ variables and degree d$d$. The algorithms are implemented in the case of Boolean polynomials and extensive experiments show that they are quite efficient for solving certain classes of Boolean equations raising from stream ciphers.     

Online vector scheduling and generalized load balancing

We give a polynomial time reduction from the vector scheduling problem (VS) to the generalized load balancing problem (GLB). This reduction gives the first non-trivial online algorithm for VS where vectors come in an online fashion. The online algorithm is very simple in that each vector only needs to minimize the _Lln(md)$L_{ln\left(md\right)}$ norm of the resulting load when it comes, where m$m$ is the number of partitions and d$d$ is the dimension of vectors. It has an approximation bound of elog(md)$elog\left(md\right)$, which is in O(ln(md))$O(ln\left(md\right))$, so it also improves the O(ln²d)$O\left({ln}^{2}d\right)$ bound of the existing polynomial time algorithm for VS. Additionally, the reduction shows that GLB does not have constant approximation algorithms that run in polynomial time unless P=NP$P=NP$.     

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$h$.     

Approximate string matching with address bit errors

A string S∈Σ^m$S\in {\Sigma }^m$ can be viewed as a set of pairs S={(_σi,i):i∈{0,…,m−1}}$S=\{({\sigma }_i,i):i\in \{0,\dots ,m-1\}\}$. We consider approximate pattern matching problems arising from the setting where errors are introduced to the location component (i$i$), rather than the more traditional setting, where errors are introduced into the content itself (_σi${\sigma }_i$). In this paper, we consider the case where bits of i$i$ may be erroneously flipped, either in a consistent or transient manner. We formally define the corresponding approximate pattern matching problems, and provide efficient algorithms for their resolution, while introducing some novel techniques.     

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.     

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$f$.     

A bound for the Rosenfeld–Gröbner algorithm

We consider the Rosenfeld–Gröbner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n$n$ indeterminates. For a set of ordinary differential polynomials F$F$, let M(F)$M\left(F\right)$ be the sum of maximal orders of differential indeterminates occurring in F$F$. We propose a modification of the Rosenfeld–Gröbner algorithm, in which for every intermediate polynomial system F$F$, the bound M(F)?(n−1)!M(_F0)$M\left(F\right)?(n-1)!M\left(F_0\right)$ holds, where _F0$F_0$ is the initial set of generators of the radical ideal. In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in a characteristic decomposition of a radical differential ideal.     

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.     

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.     

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.     

Non-linear polynomial selection for the number field sieve

We present an algorithm to find two non-linear polynomials for the Number Field Sieve integer factorization method. This algorithm extends Montgomery’s “two quadratics” method; for degree 3$3$, it gives two skewed polynomials with resultant O(N^5/4)$O\left(N^{5/4}\right)$, which improves on the Williams O(N^4/3)$O\left(N^{4/3}\right)$ result (Williams, 2010).     

On the decomposition of rational functions

Let f:=p/q∈K(x)$f:=p/q\in \mathbb{K}\left(x\right)$ be a rational function in one variable. By Lüroth’s theorem, the collection of intermediate fields K(f)?L?K(x)$\mathbb{K}\left(f\right)?\mathbb{L}?\mathbb{K}\left(x\right)$ is in bijection with inequivalent proper decompositions f=g°h$f=g°h$, with g,h∈K(x)$g,h\in \mathbb{K}\left(x\right)$ of degrees ≥2$\ge 2$. In [Alonso, Cesar, Gutierrez, Jaime, Recio, Tomas, 1995. A rational function decomposition algorithm by near-separated polynomials. J. Symbolic Comput. 19, 527–544] an algorithm is presented to calculate such a function decomposition. In this paper we describe a simplification of this algorithm, avoiding expensive solutions of linear equations. A MAGMA implementation shows the efficiency of our method. We also prove some indecomposability criteria for rational functions, which were motivated by computational experiments.     

Algorithms for a class of infinite permutation groups

Motivated by the famous 3n+1$3n+1$ conjecture, we call a mapping from Z$\mathbb{Z}$ to Z$\mathbb{Z}$residue-class-wise affine   if there is a positive integer m$m$ such that it is affine on residue classes (mod m$m$). This article describes a collection of algorithms and methods for computation in permutation groups and monoids formed by residue-class-wise affine mappings.