We consider spatio-temporal databases supporting spatial objects with continuously changing position and extent, termed moving objects databases . We formally define a data model for such databases that includes complex evolving spatial structures such as line networks or multi-component regions with holes. The data model is given as a collection of data types and operations which can be plugged as attribute types into any DBMS data model (e.g. relational, or object-oriented) to obtain a complete model and query language. A particular novel concept is the sliced representation which represents a temporal development as a set of units , where unit types for spatial and other data types represent certain “simple” functions of time. We also show how the model can be mapped into concrete physical data structures in a DBMS environment.

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Abstract. Given an instance of an optimization problem together with an optimal solution, we consider the scenario in which this instance is modified locally. In graph problems, e. g., a singular edge might be removed or added, or an edge weight might be varied, etc. For a problem U and such a local modification operation, let lm-U (local-modification-U ) denote the resulting problem. The question is whether it is possible to exploit the additional knowledge of an optimal solution to the original instance or not, i. e., whether lm-U is computationally more tractable than U . Here, we give non-trivial examples both of problems where this is and problems where this is not the case. Our main results are these:1. The local modification to change the cost of a singular edge turns the traveling salesperson problem (TSP) into a problem lm-TSP which is as hard as TSP itself, i. e., unless P = N P , there is no polynomial-time p(n)-approximation algorithm for lm-TSP for any polynomial p. Moreover, lm-TSP where inputs must satisfy the β-triangle inequality (lm-∆ β -TSP) remains NP-hard for all β > 1 2 . 2. For lm-∆-TSP (i. e., metric lm-TSP), an efficient 1.4-approximation algorithm is presented. In other words, the additional information enables us to do better than if we simply used Christofides' algorithm for the modified input. 3. Similarly, for all 1 < β < 3.34899, we achieve a better approximation ratio for lm-∆ β -TSP than for ∆ β -TSP. 4. Metric TSP with deadlines (time windows), if a single deadline or the cost of a single edge is modified, exhibits the same lower bounds on the approximability in these local-modification versions as those currently known for the original problem.

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Given an n-vertex non-negatively real-weighted graph G, whose vertices are partitioned into a set of k clusters, a clustered network design problem on G consists of solving a given network design optimization problem on G, subject to some additional constraint on its clusters. In particular, we focus on the classic problem of designing a single-source shortest-path tree, and we analyze its computational hardness when in a feasible solution each cluster is required to form a subtree. We first study the unweighted case, and prove that the problem is NP-hard. However, on the positive side, we show the existence of an approximation algorithm whose quality essentially depends on few parameters, but which remarkably is an O(1)-approximation when the largest out of all the diameters of the clusters is either O(1) or Θ(n). Furthermore, we also show that the problem is fixed-parameter tractable with respect to k or to the number of vertices that belong to clusters of size at least 2. Then, we focus on the weighted case, and show that the problem can be approximated within a tight factor of O(n), and that it is fixed-parameter tractable as well. Finally, we analyze the unweighted single-pair shortest pathThe results presented in this work have been announced in a preliminary form in [6].