## 90C27 Combinatorial optimization

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- Combinatorial Optimization (2)
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- Annulus (1)
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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.

A new solution approach for solving the 2-facility location problem in the plane with block norms
(2015)

Motivated by the time-dependent location problem over T time-periods introduced in
Maier and Hamacher (2015) we consider the special case of two time-steps, which was shown
to be equivalent to the static 2-facility location problem in the plane. Geometric optimality
conditions are stated for the median objective. When using block norms, these conditions
are used to derive a polygon grid inducing a subdivision of the plane based on normal cones,
yielding a new approach to solve the 2-facility location problem in polynomial time. Combinatorial algorithms for the 2-facility location problem based on geometric properties are
deduced and their complexities are analyzed. These methods differ from others as they are
completely working on geometric objects to derive the optimal solution set.

A single facility problem in the plane is considered, where an optimal location has to be
identified for each of finitely many time-steps with respect to time-dependent weights and
demand points. It is shown that the median objective can be reduced to a special case of the
static multifacility median problem such that results from the latter can be used to tackle the
dynamic location problem. When using block norms as distance measure between facilities,
a Finite Dominating Set (FDS) is derived. For the special case with only two time-steps, the
resulting algorithm is analyzed with respect to its worst-case complexity. Due to the relation
between dynamic location problems for T time periods and T-facility problems, this algorithm
can also be applied to the static 2-facility location problem.

We consider a variant of the generalized assignment problem (GAP) where the amount of space used in each bin is restricted to be either zero (if the bin is not opened) or above a given lower bound (a minimum quantity). We provide several complexity results for different versions of the problem and give polynomial time exact algorithms and approximation algorithms for restricted cases.
For the most general version of the problem, we show that it does not admit a polynomial time approximation algorithm (unless P=NP), even for the case of a single bin. This motivates to study dual approximation algorithms that compute solutions violating the bin capacities and minimum quantities by a constant factor. When the number of bins is fixed and the minimum quantity of each bin is at least a factor \(\delta>1\) larger than the largest size of an item in the bin, we show how to obtain a polynomial time dual approximation algorithm that computes a solution violating the minimum quantities and bin capacities by at most a factor \(1-\frac{1}{\delta}\) and \(1+\frac{1}{\delta}\), respectively, and whose profit is at least as large as the profit of the best solution that satisfies the minimum quantities and bin capacities strictly.
In particular, for \(\delta=2\), we obtain a polynomial time (1,2)-approximation algorithm.

We consider the maximum flow problem with minimum quantities (MFPMQ), which is a variant of the maximum flow problem where
the flow on each arc in the network is restricted to be either zero or above a given lower bound (a minimum quantity), which
may depend on the arc. This problem has recently been shown to be weakly NP-complete even on series-parallel graphs.
In this paper, we provide further complexity and approximability results for MFPMQ and several special cases.
We first show that it is strongly NP-hard to approximate MFPMQ on general graphs (and even bipartite graphs) within any positive factor.
On series-parallel graphs, however, we present a pseudo-polynomial time dynamic programming algorithm for the problem.
We then study the case that the minimum quantity is the same for each arc in the network and show that, under this restriction, the problem is still
weakly NP-complete on general graphs, but can be solved in strongly polynomial time on series-parallel graphs.
On general graphs, we present a \((2 - 1/\lambda) \)-approximation algorithm for this case, where \(\lambda\) denotes the common minimum quantity of all arcs.

In the context of inverse optimization, inverse versions of maximum flow and minimum cost flow problems have thoroughly been investigated. In these network flow problems there are two important problem parameters: flow capacities of the arcs and costs incurred by sending a unit flow on these arcs. Capacity changes for maximum flow problems and cost changes for minimum cost flow problems have been studied under several distance measures such as rectilinear, Chebyshev, and Hamming distances. This thesis also deals with inverse network flow problems and their counterparts tension problems under the aforementioned distance measures. The major goals are to enrich the inverse optimization theory by introducing new inverse network problems that have not yet been treated in the literature, and to extend the well-known combinatorial results of inverse network flows for more general classes of problems with inherent combinatorial properties such as matroid flows on regular matroids and monotropic programming. To accomplish the first objective, the inverse maximum flow problem under Chebyshev norm is analyzed and the capacity inverse minimum cost flow problem, in which only arc capacities are perturbed, is introduced. In this way, it is attempted to close the gap between the capacity perturbing inverse network problems and the cost perturbing ones. The foremost purpose of studying inverse tension problems on networks is to achieve a well-established generalization of the inverse network problems. Since tensions are duals of network flows, carrying the theoretical results of network flows over to tensions follows quite intuitively. Using this intuitive link between network flows and tensions, a generalization to matroid flows and monotropic programs is built gradually up.

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).

In an undirected graph G we associate costs and weights to each edge. The weight-constrained minimum spanning tree problem is to find a spanning tree of total edge weight at most a given value W and minimum total costs under this restriction. In this thesis a literature overview on this NP-hard problem, theoretical properties concerning the convex hull and the Lagrangian relaxation are given. We present also some in- and exclusion-test for this problem. We apply a ranking algorithm and the method of approximation through decomposition to our problem and design also a new branch and bound scheme. The numerical results show that this new solution approach performs better than the existing algorithms.

Multiobjective combinatorial optimization problems have received increasing attention in recent years. Nevertheless, many algorithms are still restricted to the bicriteria case. In this paper we propose a new algorithm for computing all Pareto optimal solutions. Our algorithm is based on the notion of level sets and level curves and contains as a subproblem the determination of K best solutions for a single objective combinatorial optimization problem. We apply the method to the Multiobjective Quadratic Assignment Problem (MOQAP). We present two algorithms for ranking QAP solutions and nally give computational results comparing the methods.