The Networked Common Goods Game

We introduce a new class of games called the networked common goods game (NCGG), which generalizes the well-known common goods game. We focus on a fairly general subclass of the game where each agent's utility functions are the same across all goods the agent is entitled to and satisfy certain natural properties (diminishing return and smoothness). We give a comprehensive set of technical results listed as follows. * We show the optimization problem faced by a single agent can be solved efficiently in this subclass. The discrete version of the problem is however NP-hard but admits an fully polynomial time approximation scheme (FPTAS). * We show uniqueness results of pure strategy Nash equilibrium of NCGG, and that the equilibrium is fully characterized by the structure of the network and independent of the choices and combinations of agent utility functions. * We show NCGG is a potential game, and give an implementation of best/better response Nash dynamics that lead to fast convergence to an $\epsilon$-approximate pure strategy Nash equilibrium. * Lastly, we show the price of anarchy of NCGG can be as large as $\Omega(n^{1-\epsilon})$ (for any $\epsilon>0$), which means selfish behavior in NCGG can lead to extremely inefficient social outcomes

Knapsack Problems with Side Constraints

The thesis considers a specific class of resource allocation problems in Combinatorial Optimization: the Knapsack Problems. These are paradigmatic NP-hard problems where a set of items with given profits and weights is available. The aim is to select a subset of the items in order to maximize the total profit without exceeding a known knapsack capacity. In the classical 0-1 Knapsack Problem (KP), each item can be picked at most once. The focus of the thesis is on four generalizations of KP involving side constraints beyond the capacity bound. More precisely, we provide solution approaches and insights for the following problems: The Knapsack Problem with Setups; the Collapsing Knapsack Problem; the Penalized Knapsack Problem; the Incremental Knapsack Problem. These problems reveal challenging research topics with many real-life applications. The scientific contributions we provide are both from a theoretical and a practical perspective. On the one hand, we give insights into structural elements and properties of the problems and derive a series of approximation results for some of them. On the other hand, we offer valuable solution approaches for direct applications of practical interest or when the problems considered arise as sub-problems in broader contexts

Fair Assortment Planning

Many online platforms, ranging from online retail stores to social media platforms, employ algorithms to optimize their offered assortment of items (e.g., products and contents). These algorithms tend to prioritize the platforms' short-term goals by solely featuring items with the highest popularity or revenue. However, this practice can then lead to undesirable outcomes for the rest of the items, making them leave the platform, and in turn hurting the platform's long-term goals. Motivated by that, we introduce and study a fair assortment planning problem, which requires any two items with similar quality/merits to be offered similar outcomes. We show that the problem can be formulated as a linear program (LP), called (FAIR), that optimizes over the distribution of all feasible assortments. To find a near-optimal solution to (FAIR), we propose a framework based on the Ellipsoid method, which requires a polynomial-time separation oracle to the dual of the LP. We show that finding an optimal separation oracle to the dual problem is an NP-complete problem, and hence we propose a series of approximate separation oracles, which then result in a $1/2$-approx. algorithm and a PTAS for the original Problem (FAIR). The approximate separation oracles are designed by (i) showing the separation oracle to the dual of the LP is equivalent to solving an infinite series of parameterized knapsack problems, and (ii) taking advantage of the structure of the parameterized knapsack problems. Finally, we conduct a case study using the MovieLens dataset, which demonstrates the efficacy of our algorithms and further sheds light on the price of fairness.Comment: 86 pages, 7 figure

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