Discrete Optimization最新文献

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.

In this paper we study the rank of polytopes contained in the 0-1 cube with respect to $t$-branch split cuts and $t$-dimensional lattice cuts for a fixed positive integer $t$. These inequalities are the same as split cuts when $t=1$ and generalize split cuts when $t>1$. For polytopes contained in the $n$-dimensional 0-1 cube, the work of Balas implies that the split rank can be at most $n$, and this bound is tight as Cornuéjols and Li gave an example with split rank $n$. All known examples with high split rank – i.e., at least $cn$ for some positive constant $c<1$ – are defined by exponentially many (as a function of $n$) linear inequalities. For any fixed integer $t>0$, we give a family of polytopes contained in ${[0,1]}^n$ for sufficiently large $n$ such that each polytope has empty integer hull, is defined by $O\left(n\right)$ inequalities, and has rank $\Omega \left(n\right)$ with respect to $t$-dimensional lattice cuts. Therefore the split rank of these polytopes is $\Omega \left(n\right)$. It was shown earlier that there exist generalized branch-and-bound proofs, with logarithmic depth, of the nonexistence of integer points in these polytopes. Therefore, our lower bound results on split rank show an exponential separation between the depth of branch-and-bound proofs and split rank.

Given a connected graph and a subset of its vertices referred to as the sources, the minimum broadcast time problem asks for the shortest time necessary for communicating a message from the sources to all other vertices in the graph. Information exchange is possible only between neighbors, and each vertex can transmit the message to at most one neighbor at a time. Since early works on complexity theory, the problem has been known to be NP-hard. Contributions from the current text to the understanding of the minimum broadcast time problem are threefold. Through considerations of the shortest distances between sources and other vertices, a new lower bound on the broadcast time is derived. Analytical expressions of this bound are given in the single source instances of several graph classes. Fast procedures for computing upper bounds are studied next, including both the construction of feasible solutions, and the improvement of existing ones. Finally, with a focus on a new stable-set interpretation of the problem, integer programming formulations are studied, and for their theoretical interest, associated facet-defining valid inequalities are given. The computational performance of the novel methodology is evaluated in numerical experiments applied to standard benchmark instances and to instances larger than those studied in preceding recent works.