The main objective of the polynomial approximation is the development of polynomial time algorithms for NP-hard problems, these algorithms guaranteeing feasible solutions lying “as near as possible” to the optimal ones. This work is the fist part of a couple of papers where we introduce the key-concepts of the polynomial approximation and present the main lines of a new formalism. Our purposes are, on the one hand, to present this theory and its objectives and, on the other hand, to discuss the...

Cet article est la suite de l’article «Autour de nouvelles notions pour l’analyse des algorithmes d’approximation : formalisme unifié et classes d’approximation» où nous avons présenté et discuté, dans le cadre d’un nouveau formalisme pour l’approximation polynomiale (algorithmique polynomiale à garanties de performances pour des problèmes NP-difficiles), des outils permettant d’évaluer, dans l’absolu, les proporiétés d’approximation de problèmes difficiles. Afin de répondre pleinement à l’objectif...

This paper is the continuation of the paper “Autour de nouvelles notions pour l'analyse des algorithmes d'approximation: Formalisme unifié et classes d'approximation” where a new formalism for polynomial approximation and its basic tools allowing an “absolute” (individual) evaluation the approximability properties of NP-hard problems have been presented and discussed. In order to be used for exhibiting a structure for the class NPO (the optimization problems of NP), these tools must be enriched...

Cet article est le premier d'une série de deux articles où nous présentons les principales caractéristiques d'un nouveau formalisme pour l'approximation polynomiale (algorithmique polynomiale à garanties de performances pour les problèmes NP-difficiles). Ce travail est l'occasion d'un regard critique sur ce domaine et de discussions sur la pertinence des notions usuelles. Il est aussi l'occasion de se familiariser avec l'approximation polynomiale, de comprendre ses enjeux et ses méthodes. Ces deux...

In 1978, Courcelle asked for a complete set of axioms and rules for the equational theory of (discrete regular) words equipped with the operations of product, omega power and omega-op power. In this paper we find a simple set of equations and prove they are complete. Moreover, we show that the equational theory is decidable in polynomial time.

In 1978, Courcelle asked for a complete set of axioms and rules for the equational theory of (discrete regular) words equipped with the operations of product, omega power and omega-op power. In this paper we find a simple set of equations and prove they are complete. Moreover, we show that the equational theory is decidable in polynomial time.

In this paper we will deal with the balance properties of the infinite binary words associated to β-integers when β is a quadratic simple Pisot number. Those words are the fixed points of the morphisms of the type $\varphi \left(A\right)=A^{p}B$, $\varphi \left(B\right)=A^q$ for $p\in \mathbb{N}$, $q\in \mathbb{N}$, $p\ge q$, where $\beta =\frac{p+\sqrt{p^2+4q}}{2}$. We will prove that such word is t-balanced with $t=1+\left[(p-1)/(p+1-q)\right]$. Finally, in the case that p < q it is known [B. Adamczewski, Theoret. Comput. Sci.273 (2002) 197–224] that the fixed point of the substitution $\varphi \left(A\right)=A^{p}B$, $\varphi \left(B\right)=A^q$ is not m-balanced for any m. We exhibit an infinite sequence...

Suppose a graph G = (V,E) with edge weights w(e) and edges partitioned into disjoint categories S₁,...,Sₚ is given. We consider optimization problems on G defined by a family of feasible sets (G) and the following objective function: $L₅\left(D\right)=ma{x}_{1\le i\le p}(ma{x}_{e\in S_i\cap D}w\left(e\right)-mi{n}_{e\in S_i\cap D}w\left(e\right))$ For an arbitrary number of categories we show that the L₅-perfect matching, L₅-a-b path, L₅-spanning tree problems and L₅-Hamilton cycle (on a Halin graph) problem are NP-complete. We also summarize polynomiality results concerning above objective functions for arbitrary...

A word u defined over an alphabet $𝒜$ is c-balanced (c∈$\mathbb{N}$) if for all pairs of factors v, w of u of the same length and for all letters a∈$𝒜$, the difference between the number of letters a in v and w is less or equal to c. In this paper we consider a ternary alphabet $𝒜$ = {L, S, M} and a class of substitutions ${\varphi }_p$ defined by ${\varphi }_p$(L) = LpS, ${\varphi }_p$(S) = M, ${\varphi }_p$(M) = Lp–1S where p> 1. We prove that the fixed point of ${\varphi }_p$, formally written as ${\varphi }_p^{\infty }$(L), is 3-balanced and that its Abelian complexity is bounded above by...