
In the simplest game-theoretic formulation of Schelling's model of segregation on graphs, agents of two different types each select their own vertex in a given graph so as to maximize the fraction of agents of their type in their occupied neighborhood. Two ways of modeling agent movement here are either to allow two agents to swap their vertices or to allow an agent to jump to a free vertex.
The contributions of this paper are twofold. First, we prove that deciding the existence of a swap-equilibrium and a jump-equilibrium in this simplest model of Schelling games is NP-hard, thereby answering questions left open by Agarwal et al. [AAAI '20] and Elkind et al. [IJCAI '19]. Second, we introduce two measures for the robustness of equilibria in Schelling games in terms of the minimum number of edges or the minimum number of vertices that need to be deleted to make an equilibrium unstable. We prove tight lower and upper bounds on the edge- and vertex-robustness of swap-equilibria in Schelling games on different graph classes.

Equilibria in Schelling Games: Computational Hardness and Robustness

Following up on purely theoretical work, we contribute further theoretical insights into adapting stable two-sided matchings to change. Moreover, we perform extensive empirical studies hinting at numerous practically useful properties. Our theoretical extensions include the study of new problems (that is, incremental variants of Almost Stable Marriage and Hospital Residents), focusing on their (parameterized) computational complexity and the equivalence of various change types (thus simplifying algorithmic and complexity-theoretic studies for various natural change scenarios). Our experimental findings reveal, for instance, that allowing the new matching to be blocked by a few pairs significantly decreases the difference between the old and the new matching.

Theory of and Experiments on Minimally Invasive Stability Preservation in Changing Two-Sided Matching Markets

Given an initial resource allocation, where some agents may envy others or where a different distribution of resources might lead to higher social welfare, our goal is to improve the allocation without reassigning resources. We consider a sharing concept allowing resources being shared with social network neighbors of the resource owners. To this end, we introduce a formal model that allows a central authority to compute an optimal sharing between neighbors based on an initial allocation. Advocating this point of view, we focus on the most basic scenario where a resource may be shared by two neighbors in a social network and each agent can participate in a bounded number of sharings. We present algorithms for optimizing utilitarian and egalitarian social welfare of allocations and for reducing the number of envious agents. In particular, we examine the computational complexity with respect to several natural parameters. Furthermore, we study cases with restricted social network structures and, among others, devise polynomial-time algorithms in path- and tree-like (hierarchical) social networks.

On Improving Resource Allocations by Sharing

The classic Cluster Editing problem (also known as Correlation Clustering) asks to transform a given graph into a disjoint union of cliques (clusters) by a small number of edge modifications. When applied to vertex-colored graphs (the colors representing subgroups), standard algorithms for the NP-hard Cluster Editing problem may yield  solutions that are biased towards subgroups of data (e.g., demographic groups), measured in the number of modifications incident to the members of the subgroups.
We propose a modification fairness constraint which ensures that the number of edits incident to each subgroup is proportional to its size. To start with, we study Modification-Fair Cluster Editing for graphs with two vertex colors. We show that the problem is NP-hard even if one may only insert edges within a subgroup; note that in the classic "non-fair" setting, this case is trivially polynomial-time solvable. However, in the more general editing form, the modification-fair variant remains fixed-parameter tractable with respect to the number of edge edits. We complement these and further theoretical results with an empirical analysis of our model on real-world social networks where we find that the price of modification-fairness is surprisingly low, that is, the cost of optimal modification-fair differs from the cost of optimal "non-fair" solutions only by a small percentage.

Modification-Fair Cluster Editing

The area of computational social choice (COMSOC) analyzes collective decision problems from an algorithmic perspective. So far, the main focus in this area lied on analyzing problems where a single preference relation for each agent is given and a single solution reflecting all agents' preferences needs to be found. However, this modeling is often not rich enough to capture the changing and ambivalent nature of real-world problems. We will argue that one possibility to incorporate such aspects is to allow for multiple preference profiles in the input and multiple solutions in the output. We systematically review different types of arising settings, point out how classical problems and solution concepts can be generalized, and identify several research challenges.

Broadening the Research Agenda for Computational Social Choice: Multiple Preference Profiles and Multiple Solutions (Blue Sky Ideas Track)

The envy-free, Pareto-efficient allocation of indivisible goods leads to computationally hard problems. There is a big variety of modeling issues, such as agent-specific utility functions or (high numbers of) different types of goods. In recent work, Bredereck et al. [ACM EC~2019] addressed this topic by showing (theoretical) fixed-parameter tractability results for "high-multiplicity fair allocation", exploiting parameters such as number of agents or maximum absolute   utility values. To this end, they used a number of tools from (theoretical) integer linear programming. We "engineer" their work towards practical usefulness, thereby being able to solve all real-world instances from the state-of-art online platform "spliddit.org for provably fair solutions". Besides providing the foundations for a fast tool for fair allocations, we also offer a flexible framework with the possibility to relax fairness or efficiency demands so to, e.g., allow tradeoffs between fairness and social welfare. Moreover, our framework provides ways to interpret and explain "solution paths" which makes it possible to perform further explorations in cases when no envy-free and efficient allocations exist.

High-Multiplicity Fair Allocation Made More Practical

We introduce a natural but seemingly yet unstudied generalization of the problem of scheduling jobs on a single machine so as to minimize the number of tardy jobs. Our generalization lies in simultaneously considering several instances of the problem at once. In particular, we have n clients over a period of m days, where each client has a single job with its own processing time and deadline per day. Our goal is to provide a schedule for each of the m days, so that each client is guaranteed to have their job meet its deadline in at least k <= m days. This corresponds to an equitable schedule where each client is guaranteed a minimal level of service throughout the period of m days. We provide a thorough analysis of the computational complexity of three main variants of this problem, identifying both efficient algorithms and worst-case intractability results.

Equitable Scheduling on a Single Machine

The NP-hard Material Consumption Scheduling Problem and closely related problems have been thoroughly studied since the 1980's. Roughly speaking, the problem deals with minimizing the makespan when scheduling jobs that consume non-renewable resources. We focus on the single-machine case without preemption: from time to time, the resources of the machine are (partially) replenished, thus allowing for meeting a necessary pre-condition for processing further jobs, each of which having individual resource demands. We initiate a systematic exploration of the parameterized computational complexity landscape of the problem, providing parameterized tractability as well as intractability results. Doing so, we mainly investigate how parameters related to the resource supplies influence the computational complexity. Thereby, we get a deepened understanding of this fundamental scheduling problem.

Rolf Niedermeier

8

9

Stay up to date with the latest Underline news!

PRESENTATIONS

CONFERENCES

COMPANY

RESOURCES