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Klinz and Woeginger (1995) prove that the minimum cost quickest flow problem is NP-hard. On the other hand, the quickest minimum cost flow problem can be solved efficiently via a straightforward reduction to the quickest flow problem without costs. More generally, we show how the quickest minimum cost transshipment problem can be reduced to the efficiently solvable quickest transshipment problem, thus adding another mosaic tile to the rich complexity landscape of flows over time.

The Moore-Hodgson Algorithm minimizes the number of late jobs on a single machine. That is, it finds an optimal schedule for the classical problem $1~|\;|~\sum{U_j}$. Several proofs of the correctness of this algorithm have been published. We present a new short proof., Comment: 3 pages

In this paper, we study the transient optimization of gas networks, focusing in particular on maximizing the storage capacity of the network. We include nonlinear gas physics and active elements such as valves and compressors, which due to their switching lead to discrete decisions. The former is described by a model derived from the Euler equations that is given by a coupled system of nonlinear parabolic partial differential equations ( $${\text{PDEs}}$$ ). We tackle the resulting mathematical optimization problem by a first-discretize-then-optimize approach. To this end, we introduce a new discretization of the underlying system of parabolic $${\text{PDEs}}$$ and prove well-posedness for the resulting nonlinear discretized system. Endowed with this discretization, we model the problem of maximizing the storage capacity as a non-convex mixed-integer nonlinear problem ( $${\text{MINLP}}$$ ). For the numerical solution of the $${\text{MINLP}}$$ , we algorithmically extend a well-known relaxation approach that has already been used very successfully in the field of stationary gas network optimization. This method allows us to solve the problem to global optimality by iteratively solving a series of mixed-integer problems. Finally, we present two case studies that illustrate the applicability of our approach.

We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are APX-hard to approximate and present constant-factor approximation algorithms based upon three different algorithmic techniques: (1) a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation whose approximation ratio equals the golden ratio; (2) a greedy strategy; (3) a randomized rounding method leading to an approximation algorithm for the more general case with multiple convex quadratic constraints. We further show that a combination of the first two strategies can be used to yield a monotone algorithm leading to a strategyproof mechanism for a game-theoretic variant of the problem. Finally, we present a computational study of the empirical approximation of the three algorithms for problem instances arising in the context of real-world gas transport networks.

We study a simplistic model of instationary gas flows consisting of a sequence of k stationary gas flows. We present efficiently solvable cases and NP-hardness results, establishing complexity gaps between stationary and instationary gas flows (already for k = 2 ) as well as between instationary gas s - t -flows and instationary gas b -flows.

In this paper, the notion of stability is extended to network flows over time. As a useful device in our proofs, we present an elegant preflow-push variant of the Gale-Shapley algorithm that operates directly on the given network and computes stable flows in pseudo-polynomial time, both in the static flow and the flow over time case. We show periodical properties of stable flows over time on networks with an infinite time horizon. Finally, we discuss the influence of storage at vertices, with different results depending on the priority of the corresponding holdover edges.

We present a e/(e1)-approximation algorithm for the nonpreemptive scheduling problem to minimize the total weighted completion time of jobs on a single machine subject to release dates and precedence constraints. The previously best known approximation algorithm dates back to 1997; its performance guarantee can be made arbitrarily close to the Euler constant e (Schulz and Skutella, 1997).

Multi-commodity flows over time exhibit the non-intuitive property that letting flow wait can allow us to send flow faster overall. Fleischer and Skutella (IPCO~2002) show that the speed-up through storage is at most a factor of~$2$, and that there are instances where the speed-up is as large as a factor of~$4/3$. We close this gap by presenting a family of instances for which the speed-up factor through storage converges to~$2$.

Earliest arrival flows model a central aspect of evacuation planning: In a dangerous situation, as many individuals as possible should be rescued at any point in time . Unfortunately, given a network with multiple sinks, flows over time satisfying this condition do not always exist. We analyze the special case of flows over time with zero transit times and characterize which networks always allow for earliest arrival flows.

Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an $O(\sqrt{m})$ -approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP.

Motivated by a famous open question on the single-source unsplittable minimum cost flow problem, we present a new approximation result for the relaxation of the problem where, for a given number k, each commodity must be routed along at most k paths.

We analyze teh performance fo evolutionary algorithms on various matroid optimization problems that encompass a vast number of efficiently solvable as well as NP-hard combinatorial optimization problems (including many well-known examples such as minimum spanning tree and maximum bipartite matching). We obtain very promising bounds on the expected running time and quality of the computed solution. Our results establish a better theoretical understanding of why randomized search heuristics yield empirically good results for many real-world problems., Reihe CI; 225-07

Due to the importance of robustness in many real-world optimization problems, the field of robust optimization has gained a lot of attention over the past decade. We concentrate on maximum flow problems and introduce a novel robust optimization model which, compared to known models from the literature, features several advantageous properties: (i) We consider a general class of path-based flow problems which can be used to model a large variety of network routing problems (and other packing problems). (ii) We aim at solutions that are robust against targeted attacks by a potent adversary who may attack any flow path of his choice on any edge of the network. (iii) In contrast to previous robust maximum flow models, for which no efficient algorithms are known, optimal robust flows for the most important basic variants of our model can be found in polynomial time. We also consider generalizations where the flow player can spend a budget to protect the network against the interdictor. Here, we show that the problem can be solved efficiently when the interdiction costs are determined by the flow player from scratch. However, the problem becomes hard to approximate when the flow player has to improve an initial protection infrastructure.

In traditional multi-commodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However, no restriction is imposed on the number of paths used for delivering each commodity; it is thus feasible to spread the flow over a large number of different paths. Motivated by routing problems arising in real-life applications, e.g., telecommunication, unsplittable flows have moved into the focus of research. Here, the demand of each commodity may not be split but has to be sent along a single path. In this paper a generalization of this problem is studied. In the considered flow model, a commodity can be split into a bounded number of chunks which can then be routed on different paths. In contrast to classical (splittable) flows and unsplittable flows, the single-commodity case of this problem is already NP-hard and even hard to approximate. We present approximation algorithms for the single- and multi-commodity case and point out strong connections to unsplittable flows. Moreover, results on the hardness of approximation are presented. In particular, we show that some of our approximation results are in fact best possible, unless P = NP.

In this paper, we consider the generalized min-sum set cover problem, introduced by Azar, Gamzu, and Yin (STOC 2009). Bansal, Gupta, and Krishnaswamy (SODA 2010) give a 485-approximation algorithm for the problem. We are able to alter their algorithm and analysis to obtain a 28-approximation algorithm, improving the performance guarantee by an order of magnitude. We use concepts from α -point scheduling to obtain our improvements.

We study flows over time in networks with transit times on the arcs. Transit times describe how long it takes to traverse an arc. A flow over time specifies for each arc a time-dependent flow rate that must always be bounded by the arc's capacity. Only recently, Melkonian introduced an alternative model where so-called bridge capacities bound the total amount of flow traveling along an arc, at any point of time. The contribution of this paper is twofold. Firstly, we introduce a common generalization of both the classical flow over time model and Melkonian's model. Secondly, we present a non-trivial extension of an FPTAS by Fleischer and Skutella to our new flow model. Prior to this, no approximation algorithm was known for Melkonian's model.

We study the minimum s-t-cut problem in graphs with costs on the edges in the context of evolutionary algorithms. Minimum cut problems belong to the class of basic network optimization problems that occur as crucial subproblems in many real-world optimization problems and have a variety of applications in several different areas. We prove that there exist instances of the minimum s-t-cut problem that cannot be solved by standard single-objective evolutionary algorithms in reasonable time. On the other hand, we develop a bicriteria approach based on the famous MaxFlow-MinCut Theorem that enables evolutionary algorithms to find an optimum solution in expected polynomial time., Reihe CI; 242-08

The VPN Tree Routing Conjecture states that there always exists an optimal solution to the symmetric Virtual Private Network Design (sVPND) problem where the paths between all terminals form a tree. Only recently, Hurkens, Keijsper, and Stougie gave a proof of this conjecture for the special case of ring networks. Their proof is based on a dual pair of linear programs and is somewhat in- volved. We present a short proof of a slightly stronger conjecture which might also turn out to be useful for proving the VPN Tree Routing Conjecture for general networks., Comment: Accepted for publication in Oper. Res. Lett.; referee's comments incorporated

Algorithmica, 47 (3), ISSN:0178-4617, ISSN:1432-0541

We consider an optimization problem arising in the design of controllers for OLED displays. Our objective is to minimize amplitude of the electrical current through the diodes which has a direct impact on the lifetime of such a display. Modeling the problem in mathematical terms yields a class of network flow problems where we group the arcs and pay in each group only for the arc carrying the maximum flow. We develop (fully) combinatorial approximation heuristics suitable for being implemented in the hardware of a control device that drives an OLED display.

http://www.sciencedirect.com; Given a complete graph on~$n$ nodes with metric edge costs, the minimum-cost k-hop spanning tree ($k$HMST) problem asks for a spanning tree of minimum total cost such that the longest root-leaf-path in the tree has at most $k$ edges. We present an algorithm that computes such a tree of total expected cost $O(log n)$ times that of a minimum-cost $k$-hop spanning-tree.