Lotsize optimization leading to a pp-median problem with cardinalities

We consider the problem of approximating the branch and size dependent demand of a fashion discounter with many branches by a distributing process being based on the branch delivery restricted to integral multiples of lots from a small set of available lot-types. We propose a formalized model which arises from a practical cooperation with an industry partner. Besides an integer linear programming formulation and a primal heuristic for this problem we also consider a more abstract version which we relate to several other classical optimization problems like the p-median problem, the facility location problem or the matching problem.Comment: 14 page

Diversity-aware kk-median : Clustering with fair center representation

We introduce a novel problem for diversity-aware clustering. We assume that the potential cluster centers belong to a set of groups defined by protected attributes, such as ethnicity, gender, etc. We then ask to find a minimum-cost clustering of the data into $k$ clusters so that a specified minimum number of cluster centers are chosen from each group. We thus require that all groups are represented in the clustering solution as cluster centers, according to specified requirements. More precisely, we are given a set of clients $C$, a set of facilities \pazocal{F}, a collection $\mathcal{F}=\{F_1,\dots,F_t\}$ of facility groups F_i \subseteq \pazocal{F}, budget $k$, and a set of lower-bound thresholds $R=\{r_1,\dots,r_t\}$, one for each group in $\mathcal{F}$. The \emph{diversity-aware $k$-median problem} asks to find a set $S$ of $k$ facilities in \pazocal{F} such that $|S \cap F_i| \geq r_i$, that is, at least $r_i$ centers in $S$ are from group $F_i$, and the $k$-median cost $\sum_{c \in C} \min_{s \in S} d(c,s)$ is minimized. We show that in the general case where the facility groups may overlap, the diversity-aware $k$-median problem is \np-hard, fixed-parameter intractable, and inapproximable to any multiplicative factor. On the other hand, when the facility groups are disjoint, approximation algorithms can be obtained by reduction to the \emph{matroid median} and \emph{red-blue median} problems. Experimentally, we evaluate our approximation methods for the tractable cases, and present a relaxation-based heuristic for the theoretically intractable case, which can provide high-quality and efficient solutions for real-world datasets.Comment: To appear in ECML-PKDD 202

Solution Methods for the \u3cem\u3ep\u3c/em\u3e-Median Problem: An Annotated Bibliography

The p-median problem is a graph theory problem that was originally designed for, and has been extensively applied to, facility location. In this bibliography, we summarize the literature on solution methods for the uncapacitated and capacitated p-median problem on a graph or network

Fast and Robust Techniques for the Euclidean p-Median Problem with Uniform Weights

We discuss new solution techniques for the p-median problem, with the goal being to improve the solution time and quality of current techniques. In particular, we hybridize the discrete Lloyd algorithm and the vertex substitution heuristic. We also compare three starting point techniques and present a new solution method that provides consistently good results when appropriately initialized

Grasp and tabu search for redesigning web communities

Web topologies are commonly characterised by hierarchical structures and highly unbalanced compositions, as illustrated by the difference of centrality and connectivity as to their elements. The major interest of the problem addressed in this paper lies in restructuring web communities to reduce these initial disequilibria so as to democratise information access or even for the purpose of preserving contents distributed on the Internet. Discussion of this issue thus leads to a hub location problem, formalised by network and integer programming models. Due to its highly complex nature, a GRASP and a tabu search heuristics were developed to find good quality feasible solutions to the problem. The set of test instances includes web communities obtained by crawling the web and using epistemic boundaries, as well as other randomly generated communities, built with specific network analysis software. The experiment demonstrated that the metaheuristics produced low costs and balanced structures, at least for the lower dimension web communities considered. All the redesigned web communities are more closely connected than before and the average distance among their elements reduced

Advances in Graph-Cut Optimization: Multi-Surface Models, Label Costs, and Hierarchical Costs

Computer vision is full of problems that are elegantly expressed in terms of mathematical optimization, or energy minimization. This is particularly true of low-level inference problems such as cleaning up noisy signals, clustering and classifying data, or estimating 3D points from images. Energies let us state each problem as a clear, precise objective function. Minimizing the correct energy would, hypothetically, yield a good solution to the corresponding problem. Unfortunately, even for low-level problems we are confronted by energies that are computationally hard—often NP-hard—to minimize. As a consequence, a rather large portion of computer vision research is dedicated to proposing better energies and better algorithms for energies. This dissertation presents work along the same line, specifically new energies and algorithms based on graph cuts. We present three distinct contributions. First we consider biomedical segmentation where the object of interest comprises multiple distinct regions of uncertain shape (e.g. blood vessels, airways, bone tissue). We show that this common yet difficult scenario can be modeled as an energy over multiple interacting surfaces, and can be globally optimized by a single graph cut. Second, we introduce multi-label energies with label costs and provide algorithms to minimize them. We show how label costs are useful for clustering and robust estimation problems in vision. Third, we characterize a class of energies with hierarchical costs and propose a novel hierarchical fusion algorithm with improved approximation guarantees. Hierarchical costs are natural for modeling an array of difficult problems, e.g. segmentation with hierarchical context, simultaneous estimation of motions and homographies, or detecting hierarchies of patterns