Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines

The task of scheduling jobs to machines while minimizing the total makespan, the sum of weighted completion times, or a norm of the load vector, are among the oldest and most fundamental tasks in combinatorial optimization. Since all of these problems are in general NP-hard, much attention has been given to the regime where there is only a small number k of job types, but possibly the number of jobs n is large; this is the few job types, high-multiplicity regime. Despite many positive results, the hardness boundary of this regime was not understood until now. We show that makespan minimization on uniformly related machines (Q|HM|C_max) is NP-hard already with 6 job types, and that the related Cutting Stock problem is NP-hard already with 8 item types. For the more general unrelated machines model (R|HM|C_max), we show that if either the largest job size p_max, or the number of jobs n are polynomially bounded in the instance size |I|, there are algorithms with complexity |I|^poly(k). Our main result is that this is unlikely to be improved, because Q||C_max is W[1]-hard parameterized by k already when n, p_max, and the numbers describing the speeds are polynomial in |I|; the same holds for R|HM|C_max (without speeds) when the job sizes matrix has rank 2. Our positive and negative results also extend to the objectives ??-norm minimization of the load vector and, partially, sum of weighted completion times ? w_j C_j. Along the way, we answer affirmatively the question whether makespan minimization on identical machines (P||C_max) is fixed-parameter tractable parameterized by k, extending our understanding of this fundamental problem. Together with our hardness results for Q||C_max this implies that the complexity of P|HM|C_max is the only remaining open case

Tree Drawings with Columns

Our goal is to visualize an additional data dimension of a tree with multifaceted data through superimposition on vertical strips, which we call columns. Specifically, we extend upward drawings of unordered rooted trees where vertices have assigned heights by mapping each vertex to a column. Under an orthogonal drawing style and with every subtree within a column drawn planar, we consider different natural variants concerning the arrangement of subtrees within a column. We show that minimizing the number of crossings in such a drawing can be achieved in fixed-parameter tractable (FPT) time in the maximum vertex degree $\Delta$ for the most restrictive variant, while becoming NP-hard (even to approximate) already for a slightly relaxed variant. However, we provide an FPT algorithm in the number of crossings plus $\Delta$, and an FPT-approximation algorithm in $\Delta$ via a reduction to feedback arc set.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Layered Drawing of Undirected Graphs with Generalized Port Constraints

The aim of this research is a practical method to draw cable plans of complex machines. Such plans consist of electronic components and cables connecting specific ports of the components. Since the machines are configured for each client individually, cable plans need to be drawn automatically. The drawings must be well readable so that technicians can use them to debug the machines. In order to model plug sockets, we introduce port groups; within a group, ports can change their position (which we use to improve the aesthetics of the layout), but together the ports of a group must form a contiguous block. We approach the problem of drawing such cable plans by extending the well-known Sugiyama framework such that it incorporates ports and port groups. Since the framework assumes directed graphs, we propose several ways to orient the edges of the given undirected graph. We compare these methods experimentally, both on real-world data and synthetic data that carefully simulates real-world data. We measure the aesthetics of the resulting drawings by counting bends and crossings. Using these metrics, we compare our approach to Kieler [JVLC 2014], a library for drawing graphs in the presence of port constraints.Comment: Appears in the Proceedings of the 28th International Symposium on Graph Drawing and Network Visualization (GD 2020

Nanoporous Morphogenesis in Amorphous Carbon Layers: Experiments and Modeling on Energetic Ion Induced Self‐Organization

Nanoporous amorphous carbon constitutes a highly relevant material for a multitude of applications ranging from energy to environmental and biomedical systems. In the present work, it is demonstrated experimentally how energetic ions can be utilized to tailor porosity of thin sputter deposited amorphous carbon films. The physical mechanisms underlying self-organized nanoporous morphogenesis are unraveled by employing extensive molecular dynamics and phase field models across different length scales. It is demonstrated that pore formation is a defect induced phenomenon, in which vacancies cluster in a spinodal decomposition type of self-organization process, while interstitials are absorbed by the amorphous matrix, leading to additional volume increase and radiation induced viscous flow. The proposed modeling framework is capable to reproduce and predict the experimental observations from first principles and thus opens the venue for computer assisted design of nanoporous frameworks

The Complexity of Finding Tangles

We study the following combinatorial problem. Given a set of $n$ y-monotone curves, which we call wires, a tangle determines the order of the wires on a number of horizontal layers such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset $L$ of swaps (that is, unordered pairs of wires) and an initial order of the wires, a tangle realizes $L$ if each pair of wires changes its order exactly as many times as specified by $L$. Finding a tangle that realizes a given multiset of swaps and uses the least number of layers is known to be NP-hard. We show that it is even NP-hard to decide if a realizing tangle exists

Coloring and Recognizing Directed Interval Graphs

A \emph{mixed interval graph} is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are defined by the geometry of intervals. In a proper coloring of a mixed interval graph $G$, an interval $u$ receives a lower (different) color than an interval $v$ if $G$ contains arc $(u,v)$ (edge $\{u,v\}$). Coloring of mixed graphs has applications, for example, in scheduling with precedence constraints; see a survey by Sotskov [Mathematics, 2020]. For coloring general mixed interval graphs, we present a $\min \{\omega(G), \lambda(G)+1 \}$-approximation algorithm, where $\omega(G)$ is the size of a largest clique and $\lambda(G)$ is the length of a longest directed path in $G$. For the subclass of \emph{bidirectional interval graphs} (introduced recently for an application in graph drawing), we show that optimal coloring is NP-hard. This was known for general mixed interval graphs. We introduce a new natural class of mixed interval graphs, which we call \emph{containment interval graphs}. In such a graph, there is an arc $(u,v)$ if interval $u$ contains interval $v$, and there is an edge $\{u,v\}$ if $u$ and $v$ overlap. We show that these graphs can be recognized in polynomial time, that coloring them with the minimum number of colors is NP-hard, and that there is a 2-approximation algorithm for coloring.Comment: To appear in Proc. ISAAC 202

Recognizing Stick Graphs with and without Length Constraints

Stick graphs are intersection graphs of horizontal and vertical line segments that all touch a line of slope -1 and lie above this line. De Luca et al. [GD'18] considered the recognition problem of stick graphs when no order is given (STICK), when the order of either one of the two sets is given (STICK_A), and when the order of both sets is given (STICK_AB). They showed how to solve STICK_AB efficiently. In this paper, we improve the running time of their algorithm, and we solve STICK_A efficiently. Further, we consider variants of these problems where the lengths of the sticks are given as input. We show that these variants of STICK, STICK_A, and STICK_AB are all NP-complete. On the positive side, we give an efficient solution for STICK_AB with fixed stick lengths if there are no isolated vertices