Solution discovery via reconfiguration for problems in P

In the recently introduced framework of solution discovery via reconfiguration [Fellows et al., ECAI 2023], we are given an initial configuration of $k$ tokens on a graph and the question is whether we can transform this configuration into a feasible solution (for some problem) via a bounded number $b$ of small modification steps. In this work, we study solution discovery variants of polynomial-time solvable problems, namely Spanning Tree Discovery, Shortest Path Discovery, Matching Discovery, and Vertex/Edge Cut Discovery in the unrestricted token addition/removal model, the token jumping model, and the token sliding model. In the unrestricted token addition/removal model, we show that all four discovery variants remain in P. For the toking jumping model we also prove containment in P, except for Vertex/Edge Cut Discovery, for which we prove NP-completeness. Finally, in the token sliding model, almost all considered problems become NP-complete, the exception being Spanning Tree Discovery, which remains polynomial-time solvable. We then study the parameterized complexity of the NP-complete problems and provide a full classification of tractability with respect to the parameters solution size (number of tokens) $k$ and transformation budget (number of steps) $b$. Along the way, we observe strong connections between the solution discovery variants of our base problems and their (weighted) rainbow variants as well as their red-blue variants with cardinality constraints

Parametrisierte Algorithmen für Ganzzahlige Lineare Programme und deren Anwendungen für Zuweisungsprobleme

This thesis is concerned with solving NP-hard problems. We consider two prominent strategies of coping with such computationally hard questions efficiently. The first approach aims to design approximation algorithms, that is, we are content to find good, but non-optimal solutions in polynomial time. The second strategy is called Fixed-Parameter Tractability (FPT) and considers parameters of the instance to capture the hardness of the problem and by that, obtain efficient algorithms with respect to the remaining input. This thesis employs both strategies jointly to develop efficient approximation and exact algorithms using parameterization and modeling the problem as structured integer linear programs (ILPs), which can be solved in FPT. In the first part of this work, we concentrate on these well-structured ILPs. On the one hand, we develop an efficient algorithm for block-structured integer linear programs called n-fold ILPs. On the other hand, we investigate the similarly block-structured 2-stage stochastic ILPs and prove conditional lower bounds regarding the running time of any algorithm solving them that match the best known upper bounds. We also prove the tightness of certain structural parameters called sensitivity and proximity for ILPs which arise from combinatorial questions such as allocation problems. The second part utilizes n-fold ILPs and structural properties to add to and improve upon known results for Scheduling and Bin Packing problems. We design exact FPT algorithms for the Scheduling With Clique Incompatibilities, Bin Packing, and Multiple Knapsack problems. Further, we provide constant-factor approximation algorithms and polynomial time approximation schemes (PTAS) for the Class Constraint Scheduling problems. Broadening our scope, we also investigate this problem and the closely related Cardinality Constraint Scheduling problem in the online setting and derive lower bounds for the approximation ratios as well as a PTAS for them. Altogether, this thesis contributes to the knowledge about structured ILPs, proves their limits and reaffirms their usefulness for a plethora of allocation problems. In doing so, various new and improved algorithms with respect to the running time or approximation quality emerge