Ricardo C. Corrêa

On the asymmetric representatives formulation for the vertex coloring problem

We consider the vertex coloring problem, which can be stated as the problem of minimizing the number of labels that can be assigned to the vertices of a graph G such that each vertex receives at least one label and the endpoints of every edge are assigned different labels. In this work, the 0-1 integer programming formulation based on representative vertices is revisited to remove symmetry. The previous polyhedral study related to the original formulation is adapted and generalized. New versions of facets derived from substructures of G are presented, including cliques, odd holes and anti-holes and wheels. In addition, a new class of facets is derived from independent sets of G. Finally, a comparison with the independent sets formulation is provided.

Cliques, holes and the vertex coloring polytope

Certain subgraphs of a given graph G restrict the minimum number χ(G) of colors that can be assigned to the vertices of G such that the endpoints of all edges receive distinct colors. Some of such subgraphs are related to the celebrated Strong Perfect Graph Theorem, as it implies that every graph G contains a clique of size χ(G), or an odd hole or an odd anti-hole as an induced subgraph. In this paper, we investigate the impact of induced maximal cliques, odd holes and odd anti-holes on the polytope associated with a new 0-1 integer programming formulation of the graph coloring problem. We show that they induce classes of facet defining inequalities.

Gathering algorithms on paths under interference constraints

We study the problem of gathering information from the nodes of a multi-hop radio network into a pre-determined destination node under interference constraints which are modeled by an integer d ≥ 1, so that any node within distance d of a sender cannot receive calls from any other sender. A set of calls which do not interfere with each other is referred to as a round. We give algorithms and lower bounds on the minimum number of rounds for this problem, when the network is a path and the destination node is either at one end or at the center of the path. The algorithms are shown to be optimal for any d in the first case, and for 1 ≤ d ≤ 4, in the second case.

Optimal gathering protocols on paths under interference constraints

We study the problem of gathering information from the nodes of a multi-hop radio network into a predefined destination node under reachability and interference constraints. In such a network, a node is able to send messages to other nodes within reception distance, but doing so it might create interference with other communications. Thus, a message can only be properly received if the receiver is reachable from the sender and there is no interference from another message being transmitted simultaneously. The network is modeled as a graph, where the vertices represent the nodes of the network and the edges, the possible communications. The interference constraint is modeled by a fixed integer d>=1, which implies that nodes within distance d in the graph from one sender cannot receive messages from another node. In this paper, we suppose that each node has one unit-length message to transmit and, furthermore, we suppose that it takes one unit of time (slot) to transmit a unit-length message and during such a slot we can have only calls which do not interfere (called compatible calls). A set of compatible calls is referred to as a round. We give protocols and lower bounds on the minimum number of rounds for the gathering problem when the network is a path and the destination node is either at one end or at the center of the path. These protocols are shown to be optimal for any d in the first case, and for 1@?d@?4, in the second case.

A Distributed Implementation of Asynchronous Parallel Branch and Bound

Branch-and-bound is a popular method for searching an optimal solution in the scope of discrete optimization. It consists of a heuristic iterative tree search and its principle lies in successive decompositions of the original problem in smaller disjoint subproblems until an optimal solution is found. In this paper, we propose new models to implement branch-and-bound in parallel, where several subproblems are concurrently decomposed at each iteration. In our shared data-object model processors communicate through a distributed shared memory that allows subproblems to be totally or partially ordered. We show that a variation from the synchronous to asynchronous parallel branch-and-bound allows optimizations of the operations over the data structure that improve their performance and, possibly, the overall performance as well. Finally, some experiments are described that corroborate our theoretical previsions.

The design of a CCA framework with distribution, parallelism, and recursive composition

HPE is a platform of parallel components that complies to the # component model, whose components are intrinsically parallel. This paper describes the design of a new CCA framework based on HPE, aimed to reconcile distribution and parallelism of components. Besides exposing the essential differences between the two platforms, the new framework has a set of features that distinguishes it from other CCA frameworks.

A Polynomial-Time Branching Procedure for the Multiprocessor Scheduling Problem

We present and analyze a branching procedure suitable for best-first branch-and-bound algorithms for solving multiprocessor scheduling problems. The originality of this branching procedure resides mainly in its ability to enumerate all feasible solutions without generating duplicated subproblems. This procedure is shown to be polynomial in time and space complexities.

A parallel approximation scheme for the multiprocessor scheduling problem

Abstract In this paper, a parallel branch-and-bound approach for finding approximate solutions to a general version of the multiprocessor scheduling problem is presented and analyzed. In this approach, a list heuristic and a genetic algorithm are used to find solutions to the subproblems enumerated during the branch-and-bound search. The strategy used to guide the search is based on the lower bound and the best solution known to each subproblem. With this strategy, the subproblems are branched only according to the precedence constraints, without generating duplicated subproblems. This algorithm was implemented in sequential and parallel, and experiments were carried out with instances from a synthetic benchmark. In these experiments, the results show that this algorithm is better than each of its components (list, genetic and best-first branch-and-bound) used separately, in terms of the quality of the schedule found.

Partially ordered distributed computations on asynchronous point-to-point networks

Asynchronous executions of a distributed algorithm differ from each other due to the nondeterminism in the order in which the messages exchanged are handled. In many situations of interest, the asynchronous executions induced by restricting nondeterminism are more efficient, in an application-specific sense, than the others. In this work, we define partially ordered executions of a distributed algorithm as the executions satisfying some restricted orders of their actions in two different frameworks, those of the so-called event- and pulse-driven computations. The aim of these restrictions is to characterize asynchronous executions that are likely to be more efficient for some important classes of applications. Also, an asynchronous algorithm that ensures the occurrence of partially ordered executions is given for each case. Two of the applications that we believe may benefit from the restricted nondeterminism are backtrack search, in the event-driven case, and iterative algorithms for systems of linear equations, in the pulse-driven case. We provide some experimental evidence in these two cases.

Um algoritmo de planos-de-corte para o número cromático fracionário de um grafo

The fractional chromatic number χF(G) of a graph G is a well-known lower bound for its chromatic number χ(G). Experiments reported in the literature show that using χF(G), instead of the size of the maximum clique, can be much more efficient to drive a search in a branch-and-bound algorithm for finding χ(G). One difficulty though is to deal with the linear program that is known for χF(G). Such a formulation has an exponential number of variables and demands an expensive column generation process. In this work, we investigate the use of an alternative formulation to find a lower bound for χF(G), which has a linear number of variables but an exponential number of constraints. We use a cutting plane method to deal with this inconvenience. Some separation heuristics are proposed, and some computational experiments were carried out. They show that values very close to χF(G) (in many cases exact values) are found in less time than that required by the column generation.