## Final Report Summary - METAKER (New Directions in Meta-Kernelization)

This project initially set aim at furthering the applicability of meta-kernelization, a powerful yet rare tool in the area of parameterized complexity. However, as it turns out, extending the usage of meta-kernelization to other domains seems rather limited, and I was only able to obtain superficial results in this context. The project therefore deviated towards other goals, namely exploring graph-cut problems, and exploring the application of parameterized complexity techniques to scheduling problems.

Most notably, this project provided quite an extensive picture in terms of the parameterized complexity of three graph-cut problems: Steiner Multicut, Critical Node Cut, and Length-Bound Edge Cut. In the domain of scheduling we studied the Single Machine Minimum Weighted Tardy Jobs problem, as well as the Just-In-Time Flow-Shop Scheduling problem, and provided one of the first attempts at a concise parameterized analysis in the area of scheduling.

Graph-Cut problems:

The Steiner Multicut problem asks, given a graph G, an integer k, and vertex subsets T1,...,Tt (the terminal sets), whether there are k vertices (or edges in the edge variant of the problem) in G whose removal disconnects each Ti. This problem is the natural generalization of two classical graph cut problems, Multicut and Multiway Cut. We considered four natural parameterizations for this problem: The size of the cut k, the number of terminal sets t, the maximum size of a terminal set p, and the treewidth w of the input graph G. Our analysis for Steiner Multicut was quite extensive, and we provided a complete dichotomy of the parameterized complexity of the problem under these parameters. That is, for any combination of k, t, p, and w as constant, parameter, or unbounded, we proved either that the (vertex or edge version of the) problem is fixed-parameter tractable or that it is

hard in the parameterized complexity sense. Some quite sophisticated tools were used in obtaining this dichotomy, including the technique of randomized contractions, the notion of important separators, and treewidth reductions.

Another graph-cut problem considered in the project is the Critical Node Cut problem: Given a graph G and two integers k and x, are there k vertices in the graph whose removal leaves at most x connected pairs (pairs in the same connected component) in the remaining graph. Here, we focused on for natural parameters for the problem: Parameters k and x, parameter y which equals the number of connected pairs in G minus x, and parameter w which denotes the treewidth of G. Similar to the case of Steiner Multicut, we were able to obtain almost a complete dichotomy of fixed parameter tractability and kernelization regarding these parameters and their combination. The most technical results here is the hardness result showing that the problem is not fixed-parameter tractable for parameter w + k.

Finally, the Length-Bound Edge Cut problem asks, given a graph G, two distinguish vertices s and t, and two integers k and x, whether on can delete at most k edges so that the shortest path between s and t in the resulting graph is of length at least x. We showed that this problem does not admit polynomial-size kernels for parameter k + x, answering a previously open question. This was done via a novel construction which we dubbed "fractal compositions". Using these fractal compositions, we were also able to show that other problems do not admit polynomial kernels, including Minimum Diameter Edge Deletion and Directed Small Cycle Traversal.

Scheduling problems:

In the Single Machine Minimum Weighted Tardy Jobs problem we are given a set of jobs, each with its own processing time, due date, and weight, and the goal is to schedule all jobs so that the total weight of tardy jobs (jobs completing after their deadline) is minimized. Our analysis focused on the case where one or more of three natural parameters is either constant or is taken as a parameter. These three parameters are the number of different due dates, processing times, and weights in our set of input jobs. We showed that the problem belongs to the class of fixed parameter tractable problems when combining any two of these three parameters. We also showed that the problem is polynomial-time solvable when the latter two parameters are constant, complementing Karp’s result who showed that the problem is NP-hard already for a single due date.

In the Just-In-Time Flow-Shop Scheduling problem we are given of jobs that are to be scheduled on a set of m machines which are arranged in a flow-shop machine setting; that is, each job has to be processed on each machine, and the order of machines on which each job is processed is the same among all jobs. The goal is to minimize the number of jobs that do not complete exactly on their due date. We provided several positive and negative results for the problem when the number of different due dates is taken as a parameter.

Most notably, this project provided quite an extensive picture in terms of the parameterized complexity of three graph-cut problems: Steiner Multicut, Critical Node Cut, and Length-Bound Edge Cut. In the domain of scheduling we studied the Single Machine Minimum Weighted Tardy Jobs problem, as well as the Just-In-Time Flow-Shop Scheduling problem, and provided one of the first attempts at a concise parameterized analysis in the area of scheduling.

Graph-Cut problems:

The Steiner Multicut problem asks, given a graph G, an integer k, and vertex subsets T1,...,Tt (the terminal sets), whether there are k vertices (or edges in the edge variant of the problem) in G whose removal disconnects each Ti. This problem is the natural generalization of two classical graph cut problems, Multicut and Multiway Cut. We considered four natural parameterizations for this problem: The size of the cut k, the number of terminal sets t, the maximum size of a terminal set p, and the treewidth w of the input graph G. Our analysis for Steiner Multicut was quite extensive, and we provided a complete dichotomy of the parameterized complexity of the problem under these parameters. That is, for any combination of k, t, p, and w as constant, parameter, or unbounded, we proved either that the (vertex or edge version of the) problem is fixed-parameter tractable or that it is

hard in the parameterized complexity sense. Some quite sophisticated tools were used in obtaining this dichotomy, including the technique of randomized contractions, the notion of important separators, and treewidth reductions.

Another graph-cut problem considered in the project is the Critical Node Cut problem: Given a graph G and two integers k and x, are there k vertices in the graph whose removal leaves at most x connected pairs (pairs in the same connected component) in the remaining graph. Here, we focused on for natural parameters for the problem: Parameters k and x, parameter y which equals the number of connected pairs in G minus x, and parameter w which denotes the treewidth of G. Similar to the case of Steiner Multicut, we were able to obtain almost a complete dichotomy of fixed parameter tractability and kernelization regarding these parameters and their combination. The most technical results here is the hardness result showing that the problem is not fixed-parameter tractable for parameter w + k.

Finally, the Length-Bound Edge Cut problem asks, given a graph G, two distinguish vertices s and t, and two integers k and x, whether on can delete at most k edges so that the shortest path between s and t in the resulting graph is of length at least x. We showed that this problem does not admit polynomial-size kernels for parameter k + x, answering a previously open question. This was done via a novel construction which we dubbed "fractal compositions". Using these fractal compositions, we were also able to show that other problems do not admit polynomial kernels, including Minimum Diameter Edge Deletion and Directed Small Cycle Traversal.

Scheduling problems:

In the Single Machine Minimum Weighted Tardy Jobs problem we are given a set of jobs, each with its own processing time, due date, and weight, and the goal is to schedule all jobs so that the total weight of tardy jobs (jobs completing after their deadline) is minimized. Our analysis focused on the case where one or more of three natural parameters is either constant or is taken as a parameter. These three parameters are the number of different due dates, processing times, and weights in our set of input jobs. We showed that the problem belongs to the class of fixed parameter tractable problems when combining any two of these three parameters. We also showed that the problem is polynomial-time solvable when the latter two parameters are constant, complementing Karp’s result who showed that the problem is NP-hard already for a single due date.

In the Just-In-Time Flow-Shop Scheduling problem we are given of jobs that are to be scheduled on a set of m machines which are arranged in a flow-shop machine setting; that is, each job has to be processed on each machine, and the order of machines on which each job is processed is the same among all jobs. The goal is to minimize the number of jobs that do not complete exactly on their due date. We provided several positive and negative results for the problem when the number of different due dates is taken as a parameter.