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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.

Potential-based flows provide a simple yet realistic mathematical model of transport in many real-world infrastructure networks such as, e.g., gas or water networks, where the flow along each edge depends on the difference of the potentials at its end nodes. We call a network topology robust if the maximal node potential needed to satisfy a set of demands never increases when demands are decreased. This notion of robustness is motivated by infrastructure networks where users first make reservations for certain demands that may be larger than the actual flows sent later on. In these networks, node potentials correspond to physical quantities such as pressures or hydraulic heads and must be guaranteed to lie within a fixed range, even if the actual amounts are smaller than the previously reserved demands. Our main results are a precise characterization of robust network topologies for the case of point-to-point demands via forbidden node-labeled graph minors, as well as an efficient algorithm for testing robustness.

In a digraph with a source and several destination nodes with associated demands, an unsplittable flow routes each demand along a single path from the common source to its destination. Given some flow x that is not necessarily unsplittable but satisfies all demands, it is a natural question to ask for an unsplittable flow y that does not deviate from x by too much, i.e., $$y_a\approx x_a$$ y a ≈ x a for all arcs a. Twenty years ago, in a landmark paper, Dinitz et al. (Combinatorica 19:17–41, 1999) proved that there exists an unsplittable flow y such that $$y_a\le x_a+d_{\max }$$ y a ≤ x a + d max for all arcs a, where $$d_{\max }$$ d max denotes the maximum demand value. Our first contribution is a considerably simpler one-page proof for this classical result, based upon an entirely new approach. Secondly, using a subtle variant of this approach, we obtain a new result: There is an unsplittable flow y such that $$y_a\ge x_a-d_{\max }$$ y a ≥ x a - d max for all arcs a. Finally, building upon an iterative rounding technique previously introduced by Kolliopoulos and Stein (SIAM J Comput 31:919–946, 2002) and Skutella (Math Program 91:493–514, 2002), we prove existence of an unsplittable flow that simultaneously satisfies the upper and lower bounds for the special case when demands are integer multiples of each other. For arbitrary demand values, we prove the weaker simultaneous bounds $$x_a/2-d_{\max }\le y_a\le 2x_a+d_{\max }$$ x a / 2 - d max ≤ y a ≤ 2 x a + d max for all arcs a.

We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are $$\mathsf {APX}$$ APX -hard to approximate and present constant-factor approximation algorithms based upon two different algorithmic techniques: a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation, and a greedy strategy. We further show that a combination of these techniques 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 these algorithms for problem instances arising in the context of real-world gas transport networks.

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

Potential-based flows are an extension of classical network flows in which the flow on an arc is determined by the difference of the potentials of its incident nodes. Such flows are unique and arise, for example, in energy networks. Two important algorithmic problems are to determine whether there exists a feasible flow and to maximize the flow between two designated nodes. We show that these problems can be solved for the single source and sink case by reducing the network to a single arc. However, if we additionally consider switches that allow to force the flow to 0 and decouple the potentials, these problems are NP-hard. Nevertheless, for particular series-parallel networks, one can use algorithms for the subset sum problem. Moreover, applying network presolving based on generalized series-parallel structures allows to significantly reduce the size of realistic energy networks.

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 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.

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~$2$.

Network flows over time form a fascinating area of research. They model the temporal dynamics of network flow problems occurring in a wide variety of applications. Research in this area has been pursued in two different and mainly independent directions with respect to time modeling: discrete and continuous time models. In this paper we deploy measure theory in order to introduce a general model of network flows over time combining both discrete and continuous aspects into a single model. Here, the flow on each arc is modeled as a Borel measure on the real line (time axis) which assigns to each suitable subset a real value, interpreted as the amount of flow entering the arc over the subset. We focus on the maximum flow problem formulated in a network where capacities on arcs are also given as Borel measures and storage might be allowed at the nodes of the network. We generalize the concept of cuts to the case of these Borel Flows and extend the famous MaxFlow-MinCut Theorem.

The natural linear programming formulation of the maximum s-t-flow problem in path variables has a dual linear program whose underlying polyhedron is the dominant $${{\ensuremath{P_{s-t-{\rm cut}}}^{\uparrow}}}$$of the s-t-cut polytope. We present a complete characterization of $${{\ensuremath{P_{s-t-{\rm cut}}}^{\uparrow}}}$$with respect to vertices, facets, and adjacency.

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.

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.

We study the incremental facility location problem, wherein we are given an instance of the uncapacitated facility location problem UFLP and seek an incremental sequence of opening facilities and an incremental sequence of serving customers along with their fixed assignments to facilities open in the partial sequence. We say that a sequence has a competitive ratio of k, if the cost of serving the first i¾? customers in the sequence is at most k times the optimal solution for serving any i¾? customers for all possible values of i¾?. We provide an incremental framework that computes a sequence with a competitive ratio of at most eight and a worst-case instance that provides a lower bound of three for any incremental sequence. We also present the results of our computational experiments carried out on a set of benchmark instances for the UFLP. The problem has applications in multistage network planning. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 654, 306-311 2015

More than forty years ago, Ford and Fulkerson studied maximum s-t-flows over time (also called "dynamic" flows) in networks with fixed transit times on the arcs and a fixed time horizon. Here, flow on arcs may change over time and transit times specify the amount of time it takes for flow to travel through a particular arc. Ford and Fulkerson proved that there always exists an optimal solution which sends flow on certain s-t-paths at a constant rate as long as there is enough time left for the flow along a path to arrive at the sink; a flow over time featuring this simple structure is called "temporally repeated." Although this result does not hold for the more general and also more realistic setting where transit times depend on the current flow situation, we show that there always exists a provably good temporally repeated solution. Moreover, such a solution can be determined very efficiently by only one minimum convex cost flow computation. Our results rest upon a new model of flow-dependent transit times. It is based on two assumption on the pace of flow on a particular arc. First, the pace of flow on an arc is assumed to be uniform for all flow units on an arc for each point in time. Second, this uniform pace is for each moment determined by the actual amount of flow on this arc. Finally, we show that the resulting flow-over-time problem is strongly NP-hard and cannot be approximated with arbitrary precision in polynomial time, unless P=NP.

We consider the problem of preemptively scheduling a set of n jobs on m (identical, uniformly related, or unrelated) parallel machines. The scheduler may reject a subset of the jobs and thereby incur job-dependent penalties for each rejected job, and he must construct a schedule for the remaining jobs so as to optimize the preemptive makespan on the m machines plus the sum of the penalties of the jobs rejected. We provide a complete classification of these scheduling problems with respect to complexity and approximability. Our main results are on the variant with an arbitrary number of unrelated machines. This variant is APX-hard, and we design a 1.58-approximation algorithm for it. All other considered variants are weakly NP-hard, and we provide fully polynomial time approximation schemes for them. Finally, we argue that our results for unrelated machines can be carried over to the corresponding preemptive open shop scheduling problem with rejection.

Flows over time are used to model many real-world logistic and routing problems. The networks underlying such problems-streets, tracks, etc.-are inherently undirected and directions are only imposed on them to reduce the danger of colliding vehicles and similar problems. Thus, the question arises, what influence the orientation of the network has on the network flow over time problem that is being solved on the oriented network. In the literature, this is also referred to as the contraflow or lane reversal problem. We introduce and analyze the price of orientation: How much flow is lost in any orientation of the network if the time horizon remains fixed? We prove that there is always an orientation where we can still send one-third of the flow and this bound is tight. For the special case of networks with a single source or sink, this fraction is half, which is again tight. We present more results of similar flavor and also show nonapproximability results for finding the best orientation for single and multicommodity maximum flows over time. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 663, 196-209 2015

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.

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

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.

In the single source unsplittable min-cost flow problem, commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given graph with edge capacities and costs; the demand of each commodity must be routed along a single path so that the total flow through any edge is at most its capacity. Moreover, the total cost must not exceed a given budget. This problem has been introduced by Kleinberg [7] and generalizes several NP-complete problems from various areas in combinatorial optimization such as packing, partitioning, scheduling, load balancing, and virtual-circuit routing. Kolliopoulos and Stein [9] and Dinitz, Garg, and Goemans [4] developed algorithms improving the first approximation results of Kleinberg for the problem of minimizing the violation of edge capacities and for other variants. However, known techniques do not seem to be capable of providing solutions without also violating the cost constraint. We give the first approximation results with hard cost constraints. Moreover, all our results dominate the best known bicriteria approximations. Finally, we provide results on the hardness of approximation for several variants of the problem.

We consider the problem of scheduling unrelated parallel machines subject to release dates so as to minimize the total weighted completion time of jobs. The main contribution of this paper is a provably good convex quadratic programming relaxation of strongly polynomial size for this problem. The best previously known approximation algorithms are based on LP relaxations in time- or interval-indexed variables. Those LP relaxations, however, suffer from a huge number of variables. As a result of the convex quadratic programming approach we can give a very simple and easy to analyze 2-approximation algorithm which can be further improved to performance guarantee 3/2 in the absence of release dates. We also consider preemptive scheduling problems and derive approximation algorithms and results on the power of preemption which improve upon the best previously known results for these settings. Finally, for the special case of two machines we introduce a more sophisticated semidefinite programming relaxation and apply the random hyperplane technique introduced by Goemans and Williamson for the MaxCut problem; this leads to an improved 1.2752-approximation.