Approximating the multi-level bottleneck assignment problem.

We consider the multi-level bottleneck assignment problem (MBA). This problem is described in the recent book 'Assignment Problems' by Burkard et al. (2009) on pages 188-189. One of the applications described there concerns bus driver scheduling.We view the problem as a special case of a bottleneck m-dimensional multi-index assignment problem. We give approximation algorithms and inapproximability results, depending upon the completeness of the underlying graph. Keywords: bottleneck problem; multidimensional assignment; approximation; computational complexity; efficient algorithm.Bottleneck problem; Multidimensional assignment; Approximation; Computational complexity; Efficient algorithm;

The Traveling Salesman Problem Under Squared Euclidean Distances

Let $P$ be a set of points in $\mathbb{R}^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by TSP($d,\alpha$). We design a 5-approximation algorithm for TSP(2,2) and generalize this result to obtain an approximation factor of $3^{\alpha-1}+\sqrt{6}^{\alpha}/3$ for $d=2$ and all $\alpha\ge2$. We also study the variant Rev-TSP of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-TSP$(2,\alpha)$ with $\alpha\ge2$, and we show that Rev-TSP$(d, \alpha)$ is APX-hard if $d\ge3$ and $\alpha>1$. The APX-hardness proof carries over to TSP$(d, \alpha)$ for the same parameter ranges.Comment: 12 pages, 4 figures. (v2) Minor linguistic change

Bounding Stochastic Dependence, Complete Mixability of Matrices, and Multidimensional Bottleneck Assignment Problems

We call a matrix completely mixable if the entries in its columns can be permuted so that all row sums are equal. If it is not completely mixable, we want to determine the smallest maximal and largest minimal row sum attainable. These values provide a discrete approximation of of minimum variance problems for discrete distributions, a problem motivated by the question how to estimate the $\alpha$-quantile of an aggregate random variable with unknown dependence structure given the marginals of the constituent random variables. We relate this problem to the multidimensional bottleneck assignment problem and show that there exists a polynomial $2$-approximation algorithm if the matrix has only $3$ columns. In general, deciding complete mixability is $\mathcal{NP}$-complete. In particular the swapping algorithm of Puccetti et al. is not an exact method unless $\mathcal{NP}\subseteq\mathcal{ZPP}$. For a fixed number of columns it remains $\mathcal{NP}$-complete, but there exists a PTAS. The problem can be solved in pseudopolynomial time for a fixed number of rows, and even in polynomial time if all columns furthermore contain entries from the same multiset

Multi-Dimensional Commodity Covering for Tariff Selection in Transportation

In this paper, we study a multi-dimensional commodity covering problem, which we encountered as a subproblem in optimizing large scale transportation networks in logistics. The problem asks for a selection of containers for transporting a given set of commodities, each commodity having different extensions of properties such as weight or volume. Each container can be selected multiple times and is specified by a fixed charge and capacities in the relevant properties. The task is to find a cost minimal collection of containers and a feasible assignment of the demand to all selected containers. From theoretical point of view, by exploring similarities to the well known SetCover problem, we derive NP-hardness and see that the non-approximability result known for set cover also carries over to our problem. For practical applications we need very fast heuristics to be integrated into a meta-heuristic framework that - depending on the context - either provide feasible near optimal solutions or only estimate the cost value of an optimal solution. We develop and analyze a flexible family of greedy algorithms that meet these challenges. In order to find best-performing configurations for different requirements of the meta-heuristic framework, we provide an extensive computational study on random and real world instance sets obtained from our project partner 4flow AG. We outline a trade-off between running times and solution quality and conclude that the proposed methods achieve the accuracy and efficiency necessary for serving as a key ingredient in more complex meta-heuristics enabling the optimization of large-scale networks

Models for the optimization of promotion campaigns: exact and heuristic algorithms.

This paper presents an optimization model for the selection of sets of clients that will receive an offer for one or more products during a promotion campaign. The complexity of the problem makes it very difficult to produce optimal solutions using standard optimization methods. We propose an alternative set covering formulation and develop a branch-and-price algorithm to solve it. We also describe five heuristics to approximate an optimal solution. Two of these heuristics are algorithms based on restricted versions of the basic formulation, the third is a successive exact k-item knapsack procedure. A heuristic inspired by the Next-Product-To-Buy model and a depth-first branch-and-price heuristic are also presented. Finally, we perform extensive computational experiments for the two formulations as well as for the five heuristics.Promotion campaign; Minimum quantity commitment; Integer programming; Branch-and-price algorithm; Non-approximability; Heuristics; Business-to-business; Business-to-consumer;

Minimum Scan Cover and Variants - Theory and Experiments

We consider a spectrum of geometric optimization problems motivated by contexts such as satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs, we are given a graph G that is embedded in Euclidean space. The edges of G need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices need to face each other; changing the heading of a vertex incurs some cost in terms of energy or rotation time that is proportional to the corresponding rotation angle. Our goal is to compute schedules that minimize the following objective functions: (i) in Minimum Makespan Scan Cover (MSC-MS), this is the time until all edges are scanned; (ii) in Minimum Total Energy Scan Cover (MSC-TE), the sum of all rotation angles; (iii) in Minimum Bottleneck Energy Scan Cover (MSC-BE), the maximum total rotation angle at one vertex. Previous theoretical work on MSC-MS revealed a close connection to graph coloring and the cut cover problem, leading to hardness and approximability results. In this paper, we present polynomial-time algorithms for 1D instances of MSC-TE and MSC-BE, but NP-hardness proofs for bipartite 2D instances. For bipartite graphs in 2D, we also give 2-approximation algorithms for both MSC-TE and MSC-BE. Most importantly, we provide a comprehensive study of practical methods for all three problems. We compare three different mixed-integer programming and two constraint programming approaches, and show how to compute provably optimal solutions for geometric instances with up to 300 edges. Additionally, we compare the performance of different meta-heuristics for even larger instances