Daniel E. Severin

A polyhedral approach for the equitable coloring problem

In this work we study the polytope associated with a 0,1-integer programming formulation for the Equitable Coloring Problem. We find several families of valid inequalities and derive sufficient conditions in order to be facet-defining inequalities. We also present computational evidence that shows the efficacy of these inequalities used in a cutting-plane algorithm.

A DSATUR-based algorithm for the Equitable Coloring Problem

This paper describes a new exact algorithm for the Equitable Coloring Problem, a coloring problem where the sizes of two arbitrary color classes differ in at most one unit. Based on the well known DSatur algorithm for the classic Coloring Problem, a pruning criterion arising from equity constraints is proposed and analyzed. The good performance of the algorithm is shown through computational experiments over random and benchmark instances.

A Tabu Search Heuristic for the Equitable Coloring Problem

The Equitable Coloring Problem is a variant of the Graph Coloring Problem where the sizes of two arbitrary color classes differ in at most one unit. This additional condition, called equity constraints, arises naturally in several applications. Due to the hardness of the problem, current exact algorithms can not solve large-sized instances. Such instances must be addressed only via heuristic methods.

Balanced Partition of a Graph for Football Team Realignment in Ecuador

In the second category of the Ecuadorian football league, a set of football teams must be grouped into \(k\) geographical zones according to some regulations, where the total distance of the road trips that all teams must travel to play a Double Round Robin Tournament in each zone is minimized. This problem can be modeled as a \(k\)-clique partitioning problem with constraints on the sizes and weights of the cliques. An integer programming formulation and a heuristic approach were developed to provide a solution to the problem which has been implemented in the 2015 edition of the aforementioned football championship.

Polyhedral results for the Equitable Coloring Problem

In this work we study the polytope associated with a 0/1 integer programming formulation for the Equitable Coloring Problem. We find several families of valid inequalities and derive sufficient conditions in order to be f acet-defining inequalities. We also present computational evidence of the effectiveness of including these inequalities as cuts in a Branch & Cut algorithm.

An exact approach for the balanced k-way partitioning problem with weight constraints and its application to sports team realignment

In this work a balanced k-way partitioning problem with weight constraints is defined to model the sports team realignment. Sports teams must be partitioned into a fixed number of groups according to some regulations, where the total distance of the road trips that all teams must travel to play a double round robin tournament in each group is minimized. Two integer programming formulations for this problem are introduced, and the validity of three families of inequalities associated to the polytope of these formulations is proved. The performance of a tabu search procedure and a branch and cut algorithm, which uses the valid inequalities as cuts, is evaluated over simulated and real-world instances. In particular, an optimal solution for the realignment of the Ecuadorian football league is reported and the methodology can be suitable adapted for the realignment of other sports leagues.

Cross-identification of stellar catalogs with multiple stars: Complexity and Resolution

Abstract In this work, I present an optimization problem which consists of assigning entries of a stellar catalog to multiple entries of another stellar catalog such that the probability of such assignment is maximum. I show a way of modeling it as a Maximum Weighted Stable Set Problem which is further used to solve a real astronomical instance and I partially characterize the forbidden subgraphs of the resulting family of graphs given by that reduction. Finally, I prove that the problem is N P-Hard.

Facets of the polytope of legal sequences

Abstract A sequence of vertices in a graph is called a (total) legal dominating sequence if every vertex in the sequence (totally) dominates at least one vertex not dominated by the ones that precedes it, and at the end all vertices of the graph are (totally) dominated. The Grundy (total) domination number of a graph is the size of the largest (total) legal dominating sequence. In this work, we present integer programming formulations for obtaining the Grundy (total) domination number of a graph, we study some aspects of the polyhedral structure of one of them and we test the performance of some new valid inequalities as cuts.