Using Graph Theory to Derive Inequalities for the Bell Numbers

The Bell numbers count the number of different ways to partition a set of $n$ elements while the graphical Bell numbers count the number of non-equivalent partitions of the vertex set of a graph into stable sets. This relation between graph theory and integer sequences has motivated us to study properties on the average number of colors in the non-equivalent colorings of a graph to discover new non trivial inequalities for the Bell numbers. Example are given to illustrate our approach

An improved tabu search approach for solving the job shop scheduling problem with tooling constraints

AbstractFlexible manufacturing systems (FMSs) are nowadays installed in the mechanical industry. In such systems, many different part types are produced simultaneously and it is necessary to take tooling constraints into account for finding an optimal schedule.A heuristic method is presented for solving the m-machine, n-job shop scheduling problem with tooling constraints. This method, named TOMATO, is based on an adaptation of tabu search techniques and is an improvement on the JEST algorithm proposed by Widmer in 1991

A note on r-equitable k-colorings of trees

A graph G = (V, E) is r-equitably k-colorable if there exists a partition of V into k independent sets V¹, V², ... , Vk such that | |Vi| − |Vj| | ≤ r for all i, j ∈ {1, 2, ... , k}. In this note, we show that if two trees T¹ and T² of order at least two are r-equitably k-colorable for r ≥ 1 and k ≥ 3, then all trees obtained by adding an arbitrary edge between T¹ and T² are also r-equitably k-colorable

A solution method for a car fleet management problem with maintenance constraints

The problem retained for the ROADEF'99 international challenge was an inventory management problem for a car rental company. It consists in managing a given fleet of cars in order to satisfy requests from customers asking for some type of cars for a given time period. When requests exceed the stock of available cars, the company can either offer better cars than those requested, subcontract some requests to other providers, or buy new cars to enlarge the available stock. Moreover, the cars have to go through a maintenance process at a regular basis, and there is a limited number of workers that are available to perform these maintenances. The problem of satisfying all customer requests at minimum cost is known to be NP-hard. We propose a solution technique that combines two tabu search procedures with algorithms for the shortest path, the graph coloring and the maximum weighted independent set problems. Tests on benchmark instances used for the ROADEF'99 challenge give evidence that the proposed algorithm outperforms all other existing methods (thirteen competitors took part to this contest

About equivalent interval colorings of weighted graphs

AbstractGiven a graph G=(V,E) with strictly positive integer weights ωi on the vertices i∈V, a k-interval coloring of G is a function I that assigns an interval I(i)⊆{1,…,k} of ωi consecutive integers (called colors) to each vertex i∈V. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0̸ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χint(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ωi=1 for all vertices i∈V).Two k-interval colorings I1 and I2 are said equivalent if there is a permutation π of the integers 1,…,k such that ℓ∈I1(i) if and only if π(ℓ)∈I2(i) for all vertices i∈V. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions

P5-free augmenting graphs and the maximum stable set problem

AbstractThe complexity status of the maximum stable set problem in the class of P5-free graphs is unknown. In this paper, we first propose a characterization of all connected P5-free augmenting graphs. We then use this characterization to detect families of subclasses of P5-free graphs where the maximum stable set problem has a polynomial time solution. These families extend several previously studied classes