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## COMSOC-MPMS: Computational Social Choice: Multiple Preference Profiles and Multiple Solutions

Funded by Deutsche Forschungsgemeinschaft (Project COMSOC-MPMS, NI 369/22), started November 2021.

Synopsis:

The area of computational social choice (COMSOC) is concerned with the analysis of collective decision problems from an algorithmicperspective. 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 not rich enough to capture the changing and ambivalent nature of real-world problems. To be able to model such aspects, in this project, we relax the classical paradigm which assumes that we are given only a single preference profile for which a single solution needs to be found. Instead, we allow for multiple preference profiles in the input and/or for multiple separate solutions in the output. We identify various conceptually fundamentally different settings that naturally arise in this context, which we plan to systematically study in this project. Our ultimate goal is to create a toolbox of concepts, axiomatic properties, and algorithms that can be used to model complex real-world problems involving aspects like time and multimodality.

People at TU Berlin: Niclas Böhmer; Head: Rolf Niedermeier

## DiPa: The Art of Difference Parameterization in Multivariate Algorithmics

Funded by Deutsche Forschungsgemeinschaft (Project DiPa, NI 369/21), started March 2021.

Synopsis:

Further developing and analyzing analytical tools such as ``difference parameterization'' (a core example is the established framework of above-guarantee parameterization), we aim at achieving (practically) more meaningful statements about theperformance of exact (fixed-parameter) algorithms for NP-hard problems. To this end, our focus is on employing search trees and data reduction techniques as core algorithmic tools, and methods from parameterized complexity analysis as core theoretical tools.Our key lines of attack concern- to study difference parameters through lower bounds achieved via packing of combinatorial structures; - to study difference parameters gained through approximation algorithms, also including linear programming; - to study difference parameters resulting through systematic exploration of structural parameter hierarchies; and - to study linear combinations of parameters, thus generalizing the concept of difference parameters.The utmost goal is to significantly push the “art of problem parameterization'' towards the next level, particularly making the performance of branch & bound (search tree) algorithms better explainable on the one side, and to potentially improve algorithmic approaches (by exploring and exploiting further refined parameterizations) on the other side. Our studies shall mainly focus on NP-hard graph problems.

People: Tomohiro Koana; Head: Rolf Niedermeier

## MaMu: Matching under Preferences: Multimodal Views and Domain Restrictions

Funded by Deutsche Forschungsgemeinschaft (Project MaMu, NI 369/19) and the National Natural Science Foundation of China, started April 2018.

Synopsis:

Given the great relevance of matching under preferences, the goals of this joint research project are basically two-fold. First, we want to investigate how restrictions on the allowed preferences can improve the efficiency for several solving algorithms. Doing so, we want to extend and improve results from the literature. Second, we plan to study stable matching for multi-layer preferences, a new direction we introduce here. Indeed, it is very well-motivated by the fact of having multi-modal data available (from different sources). To this end, after carefully introducing the relevant notions it is of central interest to investigate how classical concepts and algorithms can be transferred to this more general scenario. Finally, in a third part we plan to study further models, going beyond the two mentioned main parts. In particular, here we aim at touching on issues such as “incremental scenarios” or game-theoretic aspects.

The projects brings together the expertise of the Chinese and the German research groups. Both sides have actively contributed to several topics related to Computational Social Choice and, in particular, to restricted preferences and issues of preference aggregation. Moreover, both groups are heavily involved into parameterized complexity analysis and algorithms for hard combinatorial problems in general. Based on this common ground, the proposed project builds on (partially joint) previous experiences and joins forces in attacking a fundamental issue of Computational Social Choice and closely related areas; matching under preferences, as we will demonstrate, both being fundamental and being unexplored in several directions, allows for numerous future innovations (conceptually and methodologically) we want to provide.

People at TU Berlin: Niclas Böhmer; Head: Rolf Niedermeier

People at Shandong University: Jiong Guo and Hong Liu; Head: Daming Zhu

## TORE: Trade-offs in Parameterized Data Reduction

Funded by Deutsche Forschungsgemeinschaft (Project TORE, NI 369/18) and Russian Foundation for Basic Research, starting in March 2018.

Synopsis:

Kernelization is arguably one of the major contributions of parameterized complexity analysis to algorithm design. In this project, based on many years of experience with various facets of kernelization, we plan to further push the development of new aspects of kernelization, particularly focussing on various trade-off effects. In particular, we are interested in trade-offs of efficient kernelization time and effective kernel size with a number of quality measures for kernels. So we are interested in Pareto-optimal kernels, then making use of “chaining effects” by executing one kernelization after another. Further, we plan to systematically explore concepts such as partial kernelization, approximative kernelization, randomized kernelization, Turing kernelization etc., all with the ultimate goal to trade better running time and/or kernel size against some relaxation of the kernel concept. Doing so, we plan to contribute both theoretically (including new frameworks and lower bounds) and practically (up to algorithm engineering). Although our focus is on NP-hard problems, we also include polynomial-time solvable problems in our considerations.

People at TU Berlin: Till Fluschnik; Head: Rolf Niedermeier

People at Novosibirsk State University: Oxana Tsidulko; Head: René van Bevern

## MATE: Multivariate Algorithmics for Temporal Graph Problems

Funded by Deutsche Forschungsgemeinschaft (Project MATE, NI 369/17) started in February 2018.

Synopsis:

The focus of this project shall lie on temporal graphs: We are given a fixed vertex set and discrete time steps such that at each time step we may have another edge set. We aim to develop efficient (parameterized) algorithms for some of the most central problems related to temporal graphs. The topics we study herein are

• community detection and clustering, in particular with respect to their evolution over time,

• routing problems on temporal graphs or with temporal aspects,

• multi-layer graphs and preservation problems.

We aim to fill a gap in the current literature by concentrating mostly on exact combinatorial algorithms in the context of parameterized complexity. That is, we study the influence of small parameters on the complexity of our problems. To this end, we need to extend the toolbox of parameterized algorithmics with new approaches that deal with the temporal aspects of the given problems.

Using problems from the above contexts in case studies, we aim to identify key properties of temporal graphs that enable or hinder the development of efficient algorithms. That is, we aim to identify important parameters that exhibit the structure of temporal graphs and we want to study the ways in which this structure can be exploited to find efficient algorithms. In this way, we aim to contribute in equal parts to the theory of temporal graphs as well as to practical solutions for temporal graph problems.

People: Hendrik Molter, Malte Renken

Head: Rolf Niedermeier

## FPTinP: More Efficient Algorithms for Polynomial-Time Solvable Graph Problems

Funded by Deutsche Forschungsgemeinschaft (Project FPTinP, NI 369/16) started in December 2017.

Synopsis:

The central goal of this project is to extend the focus of parametrized algorithmics from NP-hard problems to polynomial-time solvable problems. This is motivated by the observation that polynomial-time algorithms with cubic running times are for many applications questionable at least; this is a fact that is particularly obvious in the context of Big Data.

In contrast, an algorithm with running in O(k^4 · n) time for some problem specific parameter can be much more desirable provided that k is much smaller than n in the application. Like in classical parameterized algorithmics, we want to focus on graph problems with polynomial-time algorithms using the rich arsenal of parameterized methods and techniques developed so far.

People: Tomohiro Koana

Head: Rolf Niedermeier

## AFFA: Algorithms for Fair Allocations

Funded by Deutsche Forschungsgemeinschaft (Project AFFA, NI 369/15 and BR5207/1) started in October 2016.

Synopsis:

The main focus of AFFA is the investigation of important computational problems in context of fair allocations with respect to their algorithmic tractability. The goals include basic problems such as the verification of the existence of fair allocations, finding the corresponding allocations, and the computational complexity of specific fixed protocols that find the allocations. Moreover, we want to analyze problems in context of manipulation, bribery, and external control. The underlying problems very often are NP-hard, which also motivates the application of parameterized, approximation, and heuristic approaches. In particular, we want to identify the limits of approximability in polynomial and FPT time. In addition, efficient reductions to established standard solving methods (like highly optimized SAT or ILP solvers) seem promising to find exact solutions for at least for some inputs. Finally, we also plan to test the developed algorithms with respect to their practical usefulness by implementations and experimental investigations and also want to make our software publicly available.

People at TU Berlin: Andrzej Kaczmarczyk; Head: Rolf Niedermeier

People at Humboldt-Universität zu Berlin: Robert Bredereck