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We consider two-machine flow shop problems with exact delays. In this model, there are two machines, the upstream machine and the downstream machine. Each job j has two operations: the first operation has to be processed on the upstream machine and the second operation has to be processed on the downstream machine, subject to the constraint that the time interval between the completion time of the first operation and the start time of the second operation is exactly [Formula: see text]. We concentrate on the objectives of makespan and total completion time. For the makespan objective, we first show that the problem is strongly NP-hard even if there are only two possible delay values. We then show that some special cases of the problem are solvable in polynomial time. Finally, we design efficient approximation algorithms for the general case and some special cases. For the total completion time objective, we give optimal polynomial-time algorithm for a special case and an efficient approximation algorithm for another one.

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Abstract. We present new approximation schemes for various classical problems of finding the minimum-weight spanning subgraph in edge-weighted undirected planar graphs that are resistant to edge or vertex removal. We first give a PTAS for the problem of finding minimum-weight 2-edge-connected spanning subgraphs where duplicate edges are allowed. Then we present a new greedy spanner construction for edge-weighted planar graphs, which augments any connected subgraph A of a weighted planar graph G to a (1 + ε)-spanner of G with total weight bounded by weight(A)/ε. From this we derive quasi-polynomial time approximation schemes for the problems of finding the minimum-weight 2-edge-connected or biconnected spanning subgraph in planar graphs. We also design approximation schemes for the minimum-weight 1-2-connectivity problem, which is the variant of the survivable network design problem where vertices have non-uniform (1 or 2) connectivity constraints. Prior to our work, for all these problems no polynomial or quasi-polynomial time algorithms were known to achieve an approximation ratio better than 2.

We present two new results about vertex and edge fault-tolerant spanners in Euclidean spaces.We describe the first construction of vertex and edge fault-tolerant spanners having optimal bounds for maximum degree and total cost. We present a greedy algorithm that for any t > 1 and any non-negative integer k, constructs a k-fault-tolerant t-spanner in which every vertex is of degree O(k) and whose total cost is O(k 2 ) times the cost of the minimum spanning tree; these bounds are asymptotically optimal.Our next contribution is an efficient algorithm for constructing good fault-tolerant spanners. We present a new, sufficient condition for a graph to be a k-fault-tolerant spanner. Using this condition, we design an efficient algorithm that finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost.