The Frobenius Problem in a Free Monoid

Given positive integers c1,c2,...,ck with gcd(c1,c2,...,ck) = 1, the Frobenius problem (FP) is to compute the largest integer g(c1,c2,...,ck) that cannot be written as a non-negative integer linear combination of c1,c2,...,ck. The Frobenius problem in a free monoid (FPFM) is a non-commutative generalization of the Frobenius problem. Given words x1,x2,...,xk such that there are only finitely many words that cannot be written as concatenations of words in {x1,x2,...,xk}, the FPFM is to find the longest such words. Unlike the FP, where the upper bound g(c1,c2,...,ck)≤max 1≤i≤k ci2 is quadratic, the upper bound on the length of the longest words in the FPFM can be exponential in certain measures and some of the exponential upper bounds are tight. For the 2FPFM, where the given words over Σ are of only two distinct lengths m and n with 1<m<n, the length of the longest omitted words is ≤g(m, m|Σ|n-m + n - m). In Chapter 1, I give the definition of the FP in integers and summarize some of the interesting properties of the FP. In Chapter 2, I give the definition of the FPFM and discuss some general properties of the FPFM. Then I mainly focus on the 2FPFM. I discuss the 2FPFM from different points of view and present two equivalent problems, one of which is about combinatorics on words and the other is about the word graph. In Chapter 3, I discuss some variations on the FPFM and related problems, including input in other forms, bases with constant size, the case of infinite words, the case of concatenation with overlap, and the generalization of the local postage-stamp problem in a free monoid. In Chapter 4, I present the construction of some essential examples to complement the theory of the 2FPFM discussed in Chapter 2. The theory and examples of the 2FPFM are the main contribution of the thesis. In Chapter 5, I discuss the algorithms for and computational complexity of the FPFM and related problems. In the last chapter, I summarize the main results and list some open problems. Part of my work in the thesis has appeared in the papers

The Average State Complexity of Rational Operations on Finite Languages

Submitted, 21 pages.International audienceConsidering the uniform distribution on sets of m non-empty words whose sum of lengths is n, we establish that the average state complexities of the rational operations are asymptotically linear

Word Equations and Related Topics. Independence, Decidability and Characterizations

The three main topics of this work are independent systems and chains of word equations, parametric solutions of word equations on three unknowns, and unique decipherability in the monoid of regular languages. The most important result about independent systems is a new method giving an upper bound for their sizes in the case of three unknowns. The bound depends on the length of the shortest equation. This result has generalizations for decreasing chains and for more than three unknowns. The method also leads to shorter proofs and generalizations of some old results. Hmelevksii’s theorem states that every word equation on three unknowns has a parametric solution. We give a significantly simplified proof for this theorem. As a new result we estimate the lengths of parametric solutions and get a bound for the length of the minimal nontrivial solution and for the complexity of deciding whether such a solution exists. The unique decipherability problem asks whether given elements of some monoid form a code, that is, whether they satisfy a nontrivial equation. We give characterizations for when a collection of unary regular languages is a code. We also prove that it is undecidable whether a collection of binary regular languages is a code.Siirretty Doriast

Distances to lattice points in rational polyhedra

Let a ∈ Z n >0 , n ≥ 2 , gcd(a) := gcd(a1 , . . . , an ) = 1, b ∈ Z≥0 and denote by k · k∞ the ℓ∞-norm. Consider the knapsack polytope P(a, b) = { x ∈ R n ≥0 : a T x = b and assume that P(a, b) ∩ Z n 6= ; holds. The main result of this thesis states that for any vertex x ∗ of the knapsack polytope P(a, b) there exists a feasible integer point z ∗ ∈ P(a, b) such that, denoting by s the size of the support of z ∗ , i.e. the number of nonzero components in z ∗ and upon assuming s > 0 , the inequality kx ∗ − z ∗ k∞ 2 s−1 s < kak∞ holds. This inequality may be viewed as a transference result which allows strengthening the best known distance (proximity) bounds if integer points are not sparse and, vice versa, strengthening the best known sparsity bounds if feasible integer points are sufficiently far from a vertex of the knapsack polytope. In particular, this bound provides an exponential in s improvement on the previously best known distance bounds in the knapsack scenario. Further, when considering general integer linear programs, we show that a resembling inequality holds for vertices of Gomory’s corner polyhedra [49, 96]. In addition, we provide several refinements of the known distance and support bounds under additional assumption

Generalised Frobenius numbers: geometry of upper bounds, Frobenius graphs and exact formulas for arithmetic sequences

Given a positive integer vector {\ve a}=(a_{1},a_{2}\dots,a_k)^t with \bea 1< a_{1}<\cdots<a_{k}\, \quad \text{and}\quad \gcd(a_{1},\ldots,a_{k})=1 \,. \eea The Frobenius number of the vector {\ve a}, \frob_k({\ve a}), is the largest positive integer that cannot be represented as $\sum\limits_{i=1}^{k}a_{i}x_{i}$, where $x_{1},\ldots,x_{k}$ are nonnegative integers. We also consider a generalised Frobenius number, known in the literature as the $s$-Frobenius number, \frob_{s}(a_{1},a_{2},\ldots,a_{k}), which is defined to be the largest integer that cannot be represented as $\sum\limits_{i=1}^{k}a_{i}x_{i}$ in at least $s$ distinct ways. The classical Frobenius number corresponds to the case $s=1$. The main result of the thesis is the new upper bound for the $2$-Frobenius number, \be \label{equ:UB} \frob_2(a_{1},\ldots,a_{k})\leq \frob_1(a_{1},\ldots,a_{k}) +2\left(\frac {(k-1)!}{{2(k-1) \choose k-1}}\right)^{1/(k-1)} \left(a_{1}\cdots a_{k}\right)^{1/(k-1)}\,, \ee that arises from studying the bounds for the quantity \big(\frob_s({\ve a})-\frob_1({\ve a})\big)\left(a_{1}\cdots a_{k}\right)^{-1/(k-1)}\,. The bound (\ref{equ:UB}) is an improvement, for $s=2$, on a bound given by Aliev, Fukshansky and Henk \cite{aliev2011generalized}. Our proofs rely on the geometry of numbers. By using graph theoretic techniques, we also obtain an explicit formula for the $2$-Frobenius number of the arithmetic progression $a,a+d,\ldots a+nd$ (i.e. the $a_{i}$'s are in an arithmetic progression) with $\gcd(a,d)=1$ and $1\leq d<a$. \be \label{2} \frob_{2}(a,a+d,\ldots a+nd)=a\left\lfloor\frac{a}{n}\right\rfloor+d(a+1)\,, \quad n \in \{2,3\}. \ee % This result generalises Roperts's result \cite{Roberts} for the Frobenius number of general arithmetic sequences. In the course of our investigations we derive a formula for the shortest path and the distance between any two vertices of a graph associated with the positive integers $a_{1},\ldots,a_{k}$. Based on our results, we observe a new pattern for the $2$-Frobenius number of general arithmetic sequences $a,a+d,\dots,a+nd$, $\gcd(a,d)=1$, which we state as a conjecture. Part of this work has appeared in \cite{Alievdistance}

THE FROBENIUS PROBLEM IN A FREE MONOID

The classical Frobenius problem over N is to compute the largest integer g not representable as a non-negative integer linear combination of non-negative integers x1, x2,..., xk, where gcd(x1, x2,..., xk) = 1. In this paper we consider novel generalizations of the Frobenius problem to the noncommutative setting of a free monoid. Unlike the commutative case, where the bound on g is quadratic, we are able to show exponential or subexponential behavior for several analogues of g, with the precise bound depending on the particular measure chosen