Discrete Optimization最新文献

The routing and spectrum assignment problem in modern flexgrid elastic optical networks asks for assigning to given demands a route in an optical network and a channel within an optical frequency spectrum so that the channels of two demands are disjoint whenever their routes share a link in the optical network. This problem can be modeled in two phases: firstly, a selection of paths in the network and, secondly, an interval coloring problem in the edge intersection graph of these paths. The interval chromatic number equals the smallest size of a spectrum such that a proper interval coloring is possible, the weighted clique number is a natural lower bound. Graphs where both parameters coincide for all possible non-negative integral weights are called superperfect. Therefore, the occurrence of non-superperfect edge intersection graphs of routing paths can provoke the need of larger spectral resources. In this work, we examine the question which minimal non-superperfect graphs can occur in the edge intersection graphs of routing paths in different underlying networks: when the network is a path, a tree, a cycle, or a sparse planar graph with small maximum degree. We show that for any possible network (even if it is restricted to a path) the resulting edge intersection graphs are not necessarily superperfect. We close with a discussion of possible consequences and of some lines of future research.

A packing in a graph is a set of vertices that are mutually distance at least 3 apart. By using optimization and linear programming to help analyze the greedy algorithm, we improve on a result of Favaron and show that every connected cubic graph of order $n$ has a packing of size at least $\frac{17}{132}n-O\left(1\right)$.

We develop theoretical foundations and practical algorithms for vehicle routing with time-dependent travel times. We also provide new benchmark instances and experimental results.

First, we study basic operations on piecewise linear arrival time functions. In particular, we devise a faster algorithm to compute the pointwise minimum of a set of piecewise linear functions and a monotonicity-preserving variant of the Imai–Iri algorithm to approximate an arrival time function with fewer breakpoints.

Next, we show how to evaluate insertion and deletion operations in tours efficiently and update the underlying data structure faster than previously known when a tour changes. Evaluating a tour also requires a scheduling step which is non-trivial in the presence of time windows and time-dependent travel times. We show how to perform this in linear time.

Based on these results, we develop a local search heuristic to solve real-world vehicle routing problems with various constraints efficiently and report experimental results on classical benchmarks. Since most of these do not have time-dependent travel times, we generate and publish new benchmark instances that are based on real-world data. This data also demonstrates the importance of considering time-dependent travel times in instances with tight time windows.

We address in this paper the two-machine job shop scheduling problem with a single server that sets up the jobs before they get processed on the machines. The server is only needed during the set-up and becomes free at the end of this phase. Moreover, the set-ups are non-anticipatory and the set-up times are sequence-independent. We seek a schedule that minimizes the overall completion time, also called the makespan. We propose several lower bounds to the problem and prove the $NP$-hardness in the strong sense of two restricted cases. In addition, we present a linear time algorithm for a special case. In order to solve the general problem, we develop a genetic and simulated annealing algorithms that use feasibility guaranteed procedures. An experimental study is carried out to analyze the performance of these meta-heuristic methods.

Consider an undirected network with traversal times on its edges and a set of commodities with connection requests from sources to destinations and release dates. The non-stop disjoint trajectories problem is to find trajectories that fulfill all requests, such that the commodities never meet. In this extension to the $\mathrm{NP}$-complete disjoint paths problem, trajectories must satisfy a non-stop condition, which disallows waiting at vertices or along arcs. This problem variant appears, for example, when disjoint aircraft trajectories shall be determined or in bufferless packet routing. We study the border of tractability for feasibility and optimization problems on three graph classes that are frequently used where space and time are discretized simultaneously: the path, the grid, and the mesh. We show that if all commodities have a common release date, feasibility can be decided in polynomial time on paths. For the unbounded mesh and unit-costs, we show how to construct optimal trajectories. In contrast, if commodities have individual release intervals and turns are forbidden, then even feasibility is $\mathrm{NP}$-complete for the path. For the mesh and arbitrary edge costs, with individual release dates and turning abilities of commodities restricted to at most 90°, we show that optimization and approximation are not fixed-parameter tractable.

Airlines solve many different optimization problems and combine the resulting solutions to ensure smooth, minimum-cost operations. Crucial problems are the Fleet Assignment, which assigns aircraft types to flights of a given schedule, and the Tail Assignment, which determines individual flight sequences to be performed by single aircraft. In order to find a cost-optimal solution, many airlines use mathematical optimization models. For these to be effective, the available data and forecasts must reflect the situation as accurately as possible. However, especially in times of a pandemic, the underlying plan is subject to severe uncertainties: Staff and demand uncertainties can even lead to flight cancellations or result in entire aircraft having to be grounded. Therefore, it is advantageous for airlines to protect their mathematical models against uncertainties in the input parameters. In this work, two computational tractable and cost-efficient robust models and solution approaches are developed: First, we set up a novel mixed integer model for the integrated fleet and tail assignment, which ensures that as few subsequent flights as possible have to be canceled in the event of initial flight cancellations. We then solve this model using a procedure that ensures that the costs of the solution remain considerably low. Our second model is an extended fleet assignment model that allows us to compensate for entire aircraft cancellations in the best possible way, taking into account rescheduling options. We demonstrate the effectiveness of both approaches by conducting an extensive computational study based on real flight schedules of a major German airline. It turns out that both models deliver stable, cost-efficient solutions within less than ten minutes, which significantly reduce follow-up costs in the case uncertainties arise.

We investigate a market without money in which every agent offers indivisible goods in multiple copies, in exchange for goods of other agents. The exchange must be balanced in the sense that each agent should receive a quantity of good(s) equal to the one she transfers to others. We describe the market in graph-theoretic terms hence we use the notion of circulations to describe a balanced exchange of goods. Each agent has strict preferences over the agents from which she will receive goods and an upper bound on the quantity of each transaction, while a positive integer weight reflects the social importance of each unit exchanged. In this paper, we propose a simple variant of the Top Trading Cycles mechanism that finds a Pareto optimal circulation. We then offer necessary and sufficient conditions for a circulation to be Pareto optimal and, as a consequence, a easy recognition procedure. Last, we show that finding a maximum weight Pareto optimal circulation is NP-hard but becomes polynomial if weights are concordant with preferences.

Circuits and extended formulations are classical concepts in linear programming theory. The circuits of a polyhedron are the elementary difference vectors between feasible points and include all edge directions. We study the connection between the circuits of a polyhedron $P$ and those of an extended formulation of $P$, i.e., a description of a polyhedron $Q$ that linearly projects onto $P$. It is well known that the edge directions of $P$ are images of edge directions of $Q$. We show that this ‘inheritance’ under taking projections does not extend to the set of circuits, and that this non-inheritance is quite generic behavior. We provide counterexamples with a provably minimal number of facets, vertices, and extreme rays, including relevant polytopes from clustering, and show that the difference in the number of circuits that are inherited and those that are not can be exponentially large in the dimension. We further prove that counterexamples exist for any fixed linear projection map, unless the map is injective. Finally, we characterize those polyhedra $P$ whose circuits are inherited from all polyhedra $Q$ that linearly project onto $P$. Conversely, we prove that every polyhedron $Q$ satisfying mild assumptions can be projected in such a way that the image polyhedron $P$ has a circuit with no preimage among the circuits of $Q$. Our proofs build on standard constructions such as homogenization and disjunctive programming.

We present necessary and sufficient conditions when a certain greedy object selection algorithm gives optimal results. Our approach covers known results for the Unbounded Knapsack Problem and Change Making Problem and gives new theoretical results for a variety of packing problems. We also provide connections between packing problems and certain bidirectional capacity installation problems on networks.

We introduce the parametric matroid one-interdiction problem. Given a matroid, each element of its ground set is associated with a weight that depends linearly on a real parameter from a given parameter interval. The goal is to find, for each parameter value, one element that, when being removed, maximizes the weight of a minimum weight basis. The complexity of this problem can be measured by the number of slope changes of the piecewise linear function mapping the parameter to the weight of the optimal solution of the parametric matroid one-interdiction problem. We provide two polynomial upper bounds as well as a lower bound on the number of these slope changes. Using these, we develop algorithms that require a polynomial number of independence tests and analyse their running time in the special case of graphical matroids.