This paper presents a new method and a constraint-based objective function to solve two problemsrelated to the design of optical telecommunication networks, namely the Synchronous Optical Net-work Ring Assignment Problem (SRAP) and the Intra-ring Synchronous Optical Network DesignProblem (IDP). These network topology problems can be represented as a graph partitioning withcapacity constraints as shown in previous works. We present here a new objective function and anew local search algorithm to solve these problems. Experiments conducted in C
Y. Lee, H. Sherali, J. Han and S. Kim in 2000 ( @cite_7 ), have studied the IDP problem with an additional constraint such that for each ring @math , @math . The authors present a mixed-integer programming model for the problem, and develop a branch-and-cut algorithm. They also introduce a heuristic to generate an initial feasible solution, and another one to improve the initial solution. To initialize a ring, the heuristic first, adds the node @math with the maximum graph degree, with respect to unassigned edges, and then adds to the partition the edge @math such that the graph degree of @math is maximum. It iteratively increases the partition by choosing a node such that the total traffic does not exceed the limit @math . A set of 40 instances is generated to test these heuristics and the branch-and-cut.
@cite_7
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In this paper, we deal with a network design problem arising from the deployment of synchronous optical networks (SONET), a standard of transmission using optical fiber technology. The problem is to find an optimal clustering of traffic demands in the network such that the total number of node assignments (and, hence, add-drop multiplexer equipment requirements) is minimized, while satisfying the ring capacity and node cardinality constraints. This problem can be conceptualized as an edge-capacitated graph partitioning problem with node cardinality constraints. We formulate the problem as a mixed-integer programming model and develop a new branch-and-cut algorithm along with preprocessing routines for optimally solving the problem. We also prescribe an effective heuristic procedure. Promising computational results are obtained using the proposed method. We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset linear programming relaxation for this problem. For directed graphs and @math -out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node @math in the solution is at most @math , where @math is the degree upper bound on @math . For undirected graphs and element-connectivity requirements with maximum connectivity requirement @math , our algorithm computes a solution that is a @math -approximation on the cost, and the degree of each node @math in the solution is at most @math . These ratios improve the previous @math -approximation on the cost and @math -approximation on the degrees. Our algorithms can be used to improve approximation ratios for other node-connectivity problems such as undirected $k... We describe a new approximation algorithm for Max Cut. Our algorithm runs in @math time, where @math is the number of vertices, and achieves an approximation ratio of @math . In instances in which an optimal solution cuts a @math fraction of edges, our algorithm finds a solution that cuts a @math fraction of edges. Our main result is a variant of spectral partitioning, which can be implemented in nearly linear time. Given a graph in which the Max Cut optimum is a @math fraction of edges, our spectral partitioning algorithm finds a set @math of vertices and a bipartition @math of @math such that at least a @math fraction of the edges incident on @math have one endpoint in @math and one endpoint in @math . (This can be seen as an analogue of Cheeger's inequality for the smallest eigenvalue of the adjacency matrix of a graph.) Iterating this procedure yields the approximation results stated above. A different, more complicated, var... Let @math be a @math -regular triangle-free graph with @math edges. We present an algorithm which finds a cut in @math with at least @math edges in expectation, improving upon Shearer's classic result. In particular, this implies that any @math -regular triangle-free graph has a cut of at least this size, and thus, we obtain a new lower bound for the maximum number of edges in a bipartite subgraph of @math . Our algorithm is simpler than Shearer's classic algorithm and it can be interpreted as a very efficient randomised distributed (local) algorithm : each node needs to produce only one random bit, and the algorithm runs in one round. The randomised algorithm itself was discovered using computational techniques . We show that for any fixed @math , there exists a weighted neighbourhood graph @math such that there is a one-to-one correspondence between heavy cuts of @math and randomised local algorithms that find large cuts in any @math -regular input graph. This turns out to be a useful tool for analysing the existence of cuts in @math -regular graphs: we can compute the optimal cut of @math to attain a lower bound on the maximum cut size of any @math -regular triangle-free graph.