Branch and Bound in a Data Parallel Setting Extended Abstract

Computers & Operations Research, 2011

This paper presents a new exact maximum clique algorithm which improves the bounds obtained in state of the art approximate coloring by reordering the vertices at each step. Moreover the algorithm can make full use of bit strings to sort vertices in constant time as well as to compute graph transitions and bounds efficiently, exploiting 10 the ability of CPUs to process bitwise operations in blocks of size the ALU register word. As a result it significantly outperforms a current leading algorithm.

Computers & Operations Research, 1992

Scope and Purpose-Finding a maximum clique of a graph is a well-known NP-hard problem, equivalent to finding a maximum independent set of the complement graph. Finding the maximum clique in an arbitrary graph is a very difficult computational problem. This paper deals primarily with a quadratic zero-one modeling of the maximum clique problem. A branch and bound algorithm based on this modeling, and different vertex selection heuristics (the greedy and the nongreedy vertex selection rules), are used to solve many instances of the maximum clique problem. It is demonstrated that the nongreedy vertex selection rule and the data structures obtained from the quadratic formulation, together in a branch and bound algorithm, allow us to solve relatively large graph problems.

1994

In this report, we propose new concurrent data structures and load balancing strategies for Branch-and-Bound (B&B)/A* algorithms in two models of parallel programming : shared and distributed memory. For the shared memory model (SMM), we present a general methodology which allows concurrent manipulations for most tree data structures, and show its usefulness for implementation on multiprocessors with global shared memory. Some priority queues which are suited for basic operations performed by B&B algorithms are described : the Skew-heaps, the funnels and the Splay-trees. We also detail a speciic data structure, called treap and designed for A* algorithm. These data structures are implemented on a parallel machine with shared memory : KSR1. For the distributed memory model (DMM), we show that the use of partial cost in the B&B algorithms is not enough to balance nodes between the local queues. Thus, we introduce another notion of priority, called potentiality, between nodes that take...

Applied Optimization, 1998

In this paper we present a portable exact parallel algorithm for the maximum clique problem on general graphs. Computational results with random graphs and some test graphs from applications are presented. The algorithm is parallelized using the Message Passing Interface (MPI) standard. The algorithm is based on the Carraghan-Pardalos exact algorithm (for unweighted graphs) and incorporates a variant of the greedy randomized adaptive search procedure (GRASP) for maximum independent set of Feo, Resende, and Smith (1994) to obtain good starting solutions.

Journal of Physics: Conference Series, 2011

A clique is a subgraph in a graph that is complete in the sense that each two of its nodes are connected by an edge. Finding cliques in a given graph is an important procedure in discrete mathematical modeling. The paper will show how concepts such as splitting partitions, quasi coloring, node and edge dominance are related to clique search problems. In particular we will discuss the connection with parallel clique search algorithms. These concepts also suggest practical guide lines to inspect a given graph before starting a large scale search.

BIT, 1984

Parallel algorithms for some graph-theoretic problems on a tree-structured computer are presented. In particular, if p denotes the number of processing elements, algorithms that run in O(nZ/p) time for finding connected components, transitive closure and the minimum spanning tree of an undirected graph with n vertices are obtained.

Information Processing Letters, 1988

Naor et al. (1987) proposed parallel algorithms for several problems on chordal graphs such as computing maximal cliques, a minimum coloring, a perfect elimination scheme and so on. They first solved the problem of computing maxima1 cliques in O(log'n) time with 0( n5+E ) processors and, using this result, they solved all the other problems. In this paper we propose ,.nother parallel algorithm for maximal cliques which can be executed in O(10g2n) time by using only 0( n3) processors. This . result reduces the processor bound from 0( n5+E ) to 0( n3) for all the problems solved by Naor et al. Based upon this result we propose another two algorithms for computing a clique tree and minimum coloring which are more efficient than those proposed by Naor et al.

SIAM Journal on Computing, 1984

In this paper, we present efficient parallel algorithms for the following graph problems: finding the lowest common ancestors for vertex pairs of a directed tree; finding all fundamental cycles, a directed spanning forest, all bridges, all bridge-connected components, all separation vertices, all biconnected components, and testing the biconnectivity of an undirected graph. All these algorithms achieve the O(lg n) time bound, with the first two algorithms using n[n/lg n] processors and the remaining algorithms using n[n/lg n] processors. In all cases, our algorithms are better than the previously known algorithms and in most cases reduce the number of processors used by a factor of n lg n. Moreover, our algorithms are optimal with respect to the time-processor product for dense graphs, with the exception of the first two algorithms. The machine model we use is the PRAM which is a SIMD model allowing simultaneous reads but not simultaneous writes to the same memory location.

1995

In this report, we propose the library BOB for an easy development of the Branch-and-Bound applications (min/maximization). This library has the double goal of allowing on the one hand the Combinatorial Optimization community to implement their applications without worrying about the architecture of the machines and bene ting the advantages provided by parallelism. On the other hand, BOB o ers to the community of Parallelism a set of benchmark composed by the e cient algorithms of Combinatorial Optimization for its parallelization methods and/or tools.