## 90C47 Minimax problems [See also 49K35]

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We continue in this paper the study of k-adaptable robust solutions for combinatorial optimization problems with bounded uncertainty sets. In this concept not a single solution needs to be chosen to hedge against the uncertainty. Instead one is allowed to choose a set of k different solutions from which one can be chosen after the uncertain scenario has been revealed. We first show how the problem can be decomposed into polynomially many subproblems if k is fixed. In the remaining part of the paper we consider the special case where k=2, i.e., one is allowed to choose two different solutions to hedge against the uncertainty. We decompose this problem into so called coordination problems. The study of these coordination problems turns out to be interesting on its own. We prove positive results for the unconstrained combinatorial optimization problem, the matroid maximization problem, the selection problem, and the shortest path problem on series parallel graphs. The shortest path problem on general graphs turns out to be NP-complete. Further, we present for minimization problems how to transform approximation algorithms for the coordination problem to approximation algorithms for the original problem. We study the knapsack problem to show that this relation does not hold for maximization problems in general. We present a PTAS for the corresponding coordination problem and prove that the 2-adaptable knapsack problem is not at all approximable.

We extend the standard concept of robust optimization by the introduction of an alternative solution. In contrast to the classic concept, one is allowed to chose two solutions from which the best can be picked after the uncertain scenario has been revealed. We focus in this paper on the resulting robust problem for combinatorial problems with bounded uncertainty sets. We present a reformulation of the robust problem which decomposes it into polynomially many subproblems. In each subproblem one needs to find two solutions which are connected by a cost function which penalizes if the same element is part of both solutions. Using this reformulation, we show how the robust problem can be solved efficiently for the unconstrained combinatorial problem, the selection problem, and the minimum spanning tree problem. The robust problem corresponding to the shortest path problem turns out to be NP-complete on general graphs. However, for series-parallel graphs, the robust shortest path problem can be solved efficiently. Further, we show how approximation algorithms for the subproblem can be used to compute approximate solutions for the original problem.

In this work we study and investigate the minimum width annulus problem (MWAP), the circle center location or circle location problem (CLP) and the point center location or point location problem (PLP) on Rectilinear and Chebyshev planes as well as in networks. The relations between the problems have served as a basis for finding of elegant solution, algorithms for both new and well known problems. So, MWAP was formulated and investigated in Rectilinear space. In contrast to Euclidean metric, MWAP and PLP have at least one common optimal point. Therefore, MWAP on Rectilinear plane was solved in linear time with the help of PLP. Hence, the solution sequence was PLP-->MWAP. It was shown, that MWAP and CLP are equivalent. Thus, CLP can be also solved in linear time. The obtained results were analysed and transfered to Chebyshev metric. After that, the notions of circle, sphere and annulus in networks were introduced. It should be noted that the notion of a circle in a network is different from the notion of a cycle. An O(mn) time algorithm for solution of MWAP was constructed and implemented. The algorithm is based on the fact that the middle point of an edge represents an optimal solution of a local minimum width annulus on this edge. The resulting complexity is better than the complexity O(mn+n^2logn) in unweighted case of the fastest known algorithm for minimizing of the range function, which is mathematically equivalent to MWAP. MWAP in unweighted undirected networks was extended to the MWAP on subsets and to the restricted MWAP. Resulting problems were analysed and solved. Also the p–minimum width annulus problem was formulated and explored. This problem is NP–hard. However, the p–MWAP has been solved in polynomial O(m^2n^3p) time with a natural assumption, that each minimum width annulus covers all vertexes of a network having distances to the central point of annulus less than or equal to the radius of its outer circle. In contrast to the planar case MWAP in undirected unweighted networks have appeared to be a root problem among considered problems. During investigation of properties of circles in networks it was shown that the difference between planar and network circles is significant. This leads to the nonequivalence of CLP and MWAP in the general case. However, MWAP was effectively used in solution procedures for CLP giving the sequence MWAP-->CLP. The complexity of the developed and implemented algorithm is of order O(m^2n^2). It is important to mention that CLP in networks has been formulated for the first time in this work and differs from the well–studied location of cycles in networks. We have constructed an O(mn+n^2logn) algorithm for well–known PLP. The complexity of this algorithm is not worse than the complexity of the currently best algorithms. But the concept of the solution procedure is new – we use MWAP in order to solve PLP building the opposite to the planar case solution sequence MWAP-->PLP and this method has the following advantages: First, the lower bounds LB obtained in the solution procedure are proved to be in any case better than the strongest Halpern’s lower bound. Second, the developed algorithm is so simple that it can be easily applied to complex networks manually. Third, the empirical complexity of the algorithm is equal to O(mn). MWAP was extended to and explored in directed unweighted and weighted networks. The complexity bound O(n^2) of the developed algorithm for finding of the center of a minimum width annulus in the unweighted case does not depend on the number of edges in a network, because the problems can be solved in the order PLP-->MWAP. In the weighted case computational time is of order O(mn^2).