Statistical mechanics of the vertex-cover problem

We review recent progress in the study of the vertex-cover problem (VC). VC belongs to the class of NP-complete graph theoretical problems, which plays a central role in theoretical computer science. On ensembles of random graphs, VC exhibits an coverable-uncoverable phase transition. Very close to this transition, depending on the solution algorithm, easy-hard transitions in the typical running time of the algorithms occur. We explain a statistical mechanics approach, which works by mapping VC to a hard-core lattice gas, and then applying techniques like the replica trick or the cavity approach. Using these methods, the phase diagram of VC could be obtained exactly for connectivities $c<e$, where VC is replica symmetric. Recently, this result could be confirmed using traditional mathematical techniques. For $c>e$, the solution of VC exhibits full replica symmetry breaking. The statistical mechanics approach can also be used to study analytically the typical running time of simple complete and incomplete algorithms for VC. Finally, we describe recent results for VC when studied on other ensembles of finite- and infinite-dimensional graphs.Comment: review article, 26 pages, 9 figures, to appear in J. Phys. A: Math. Ge

Load Balancing for Distributed Branch Bound Algorithms

In this paper, we present a new load balancing algorithm and its application to distributed branch &amp; bound algorithms. We demonstrate the efficiency of this scheme by solving some NP-complete problems on a network of up to 256 Transputers. The parallelization of our branch &amp; bound algorithm is fully distributed. Every processor performs the same algorithm but on a different part of the solution tree. In this case, it is necessary to distribute subproblems among the processors to achieve a well balanced workload. We present a load balancing method which overcomes the problem of search overhead and idle times by an appropriate load model and avoids trashing effects by a feedback control strategy. To show the performance of our strategy, we solved the Vertex Cover and the weighted Vertex Cover problem for graphs of up to 150 nodes, using highly efficient branch and bound algorithms. Although the computing times were very short on a 256 processor network, we were able to achieve a speedup ..

Solving the Traveling Salesman Problem with a Distributed Branch-and-Bound Algorithm on a 1024 Processor Network

This paper is the first to present a parallelization of a highly efficient best-first branch-and-bound algorithm to solve large symmetric traveling salesman problems on a massively parallel computer containing 1024 processors. The underlying sequential branch-and-bound algorithm is based on 1-tree relaxation. The parallelization of the branch-and-bound algorithm is fully distributed. Every processor performs the same sequential algorithm but on a different part of the solution tree. To distribute subproblems among the processors we use a new direct-neighbor dynamic load-balancing strategy. The general principle can be applied to all other branch-and-bound algorithms leading to an &quot;automatic&quot; parallelization. At present we can efficiently solve traveling salesman problems up to a size of 318 cities on networks of up to 1024 transputers. On hard problems we achieve an almost linear speed-up

A Study on Dynamic Load Balancing Algorithms

Dynamic load balancing techniques have proved to be the most critical part of an efficient implementation of various algorithms on large distributed computing systems. In this paper a classification of dynamic distributed load balancing algorithms for homogeneous multiprocessor systems is introduced and a general test bed, using a random branch &amp; bound load-generator, for evaluating load balancing strategies is described. With its help a number of well known load balancing strategies are compared with two new algorithms based on the gradient model method. The behavior of all algorithms on various networks when running different workload patterns is studied. By our simulations on a reconfigurable transputer system it is shown that all strategies perform better on networks with small diameter. The measurements indicate that even on large networks one of the randomized strategies and our extension of the gradient model method behaves very well when simulating data-migration, while under p..

Load Balancing in Large Networks: A Comparative Study

Dynamic load balancing techniques have been shown to be the most critical part of an efficient implementation of various algorithms on large distributed computing systems. In this paper a classification of dynamic distributed load balancing algorithms for homogeneous multiprocessor systems is introduced and a general test bed, using a random load generator, for the evaluation of load balancing strategies is described. With its help a number of well known load balancing strategies are compared with two new algorithms based on the gradient model method. The behavior of all algorithms under different workload characteristics on various networks is studied. By our simulations on a transputer network it is shown that all strategies perform better on networks with small diameter. The measurements indicate, that even on large networks one of the randomized strategies and a new extension of the gradient model method behaves very well when simulating data-migration, while under process-migratio..

Load Balancing in Large Networks: A Comparative Study (Extended Abstract)

) R. Luling, B. Monien, F. Ramme Department of Mathematics and Computer Science University of Paderborn, Germany e-mail : [email protected], [email protected], [email protected] Abstract In this paper we compare six well known and two new load balancing strategies on torus and ring topologies of different sizes and workload characteristics. Through simulations on a large transputer network, we show that all strategies behave differently under the workload of process and data migration. The two new algorithms based on the gradient model method are shown to be robust to both kinds of workloads. Thus, these new algorithms are good candidates for distributed operating systems running on large networks, where the workload characteristics can not be determined in advance. 1 Introduction We study load balancing algorithms on large MIMD multiprocessor systems. The systems we consider are homogeneous and consist of autonomous processing elements (324 transputers in our case), which..

Parallel Architectures: Design and Efficient Use

In this paper we want to demonstrate the large impact of theoretical considerations on the design and efficient use of parallel machines. We describe interconnection networks for parallel computers, tools for their efficient use (mapping, load balancing) and the parallelization of a problem which is hard to parallelize (chess programming)

An Improved Algorithm to Detect Communication Deadlocks in Distributed Systems

This paper presents a new algorithm for the detection and resolution of communication deadlocks in distributed systems. The algorithm is based on some well known concepts for distributed deadlock detection and adds some new features to reduce message- and space complexity. It was implemented on a transputer network and shown to be more efficient than previously published algorithms

Solving the traveling salesman problem with a parallel branch-and-bound algorithm on a 1024 processor network

This paper is the first to present a parallelization of a higly efficient best-first branch-and-bound algorithm to solve large symmetric traveling saleman problems on a massively parallel computer containing 1024 processors. The underlying sequential branch &amp;bound algorithm is based on 1-tree relaxation introduced by Held and Karp (Lagrangean approach) and improved by Volgenant and Jonker. The parallelization of the branch &amp; bound algorithm is fully distributed. Every processor performs the same sequential algorithm but on a different part of the solution tree. To distribute subproblems among the processors we use a new direct-neighbor dynamic load-balancing strategy. The general principle can be applied to all other branch-and-bound algorithms leading to an &amp;quot;automatic &amp;quot; parallelization. At present we can efficiently solve traveling salesman problems up to a size of 318 cities on networks of up to 1024 transputers. On hard problems we achieve an almost linear speedup