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Klaus Heeger

Publications and source records attributed to Klaus Heeger.

26 records · Page 2Linked to original sources
arXiv2021

Equitable Scheduling on a Single Machine

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 $k \le m$ 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.

cs.DM↗
arXiv2021

Multidimensional Stable Roommates with Master List

Since the early days of research in algorithms and complexity, the computation of stable matchings is a core topic. While in the classic setting the goal is to match up two agents (either from different "gender" (this is Stable Marriage) or "unrestricted" (this is Stable Roommates)), Knuth [1976] triggered the study of three- or multidimensional cases. Here, we focus on the study of Multidimensional Stable Roommates, known to be NP-hard since the early 1990's. Many NP-hardness results, however, rely on very general input instances that do not occur in at least some of the specific application scenarios. With the quest for identifying islands of tractability for Multidimensional Stable Roommates, we study the case of master lists. Here, as natural in applications where agents express their preferences based on "objective" scores, one roughly speaking assumes that all agent preferences are "derived from" a central master list, implying that the individual agent preferences shall be similar. Master lists have been frequently studied in the two-dimensional (classic) stable matching case, but seemingly almost never for the multidimensional case. This work, also relying on methods from parameterized algorithm design and complexity analysis, performs a first systematic study of Multidimensional Stable Roommates under the assumption of master lists.

cs.CC↗
arXiv2021

Parameterized Complexity of Stable Roommates with Ties and Incomplete Lists Through the Lens of Graph Parameters

We continue and extend previous work on the parameterized complexity analysis of the NP-hard Stable Roommates with Ties and Incomplete Lists problem, thereby strengthening earlier results both on the side of parameterized hardness as well as on the side of fixed-parameter tractability. Other than for its famous sister problem Stable Marriage which focuses on a bipartite scenario, Stable Roommates with Incomplete Lists allows for arbitrary acceptability graphs whose edges specify the possible matchings of each two agents (agents are represented by graph vertices). Herein, incomplete lists and ties reflect the fact that in realistic application scenarios the agents cannot bring all other agents into a linear order. Among our main contributions is to show that it is W[1]-hard to compute a maximum-cardinality stable matching for acceptability graphs of bounded treedepth, bounded tree-cut width, and bounded disjoint paths modulator number (these are each time the respective parameters). However, if we `only' ask for perfect stable matchings or the mere existence of a stable matching, then we obtain fixed-parameter tractability with respect to tree-cut width but not with respect to treedepth. On the positive side, we also provide fixed-parameter tractability results for the parameter feedback edge set number.

cs.CC↗
arXiv2021

A Fine-Grained View on Stable Many-To-One Matching Problems with Lower and Upper Quotas

In the Hospital Residents problem with lower and upper quotas ($HR-Q^U_L$), the goal is to find a stable matching of residents to hospitals where the number of residents matched to a hospital is either between its lower and upper quota or zero [Biró et al., TCS 2010]. We analyze this problem from a parameterized perspective using several natural parameters such as the number of hospitals and the number of residents. Moreover, we present a polynomial-time algorithm that finds a stable matching if it exists on instances with maximum lower quota two. Alongside $HR-Q^U_L$, we also consider two closely related models of independent interest, namely, the special case of $HR-Q^U_L$ where each hospital has only a lower quota but no upper quota and the variation of $HR-Q^U_L$ where hospitals do not have preferences over residents, which is also known as the House Allocation problem with lower and upper quotas. Lastly, we investigate how the parameterized complexity of these three models changes if preferences may contain ties.

cs.GT↗
arXiv2020

Multistage Graph Problems on a Global Budget

Time-evolving or temporal graphs gain more and more popularity when studying the behavior of complex networks. In this context, the multistage view on computational problems is among the most natural frameworks. Roughly speaking, herein one studies the different (time) layers of a temporal graph (effectively meaning that the edge set may change over time, but the vertex set remains unchanged), and one searches for a solution of a given graph problem for each layer. The twist in the multistage setting is that the solutions found must not differ too much between subsequent layers. We relax on this already established notion by introducing a global instead of the local budget view studied so far. More specifically, we allow for few disruptive changes between subsequent layers but request that overall, that is, summing over all layers, the degree of change is moderate. Studying several classical graph problems (both NP-hard and polynomial-time solvable ones) from a parameterized complexity angle, we encounter both fixed-parameter tractability and parameterized hardness results. Somewhat surprisingly, we find that sometimes the global multistage versions of NP-hard problems such as Vertex Cover turn out to be computationally more tractable than the ones of polynomial-time solvable problems such as Matching.

cs.CC↗
arXiv2019

Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters

In the presented paper we study the Length-Bounded Cut problem for special graph classes as well as from a parameterized-complexity viewpoint. Here, we are given a graph $G$, two vertices $s$ and $t$, and positive integers $β$ and $λ$. The task is to find a set of edges $F$ of size at most $β$ such that every $s$-$t$-path of length at most $λ$ in $G$ contains some edge in $F$. Bazgan et al. conjectured that Length-Bounded Cut admits a polynomial-time algorithm if the input graph $G$ is a~proper interval graph. We confirm this conjecture by showing a dynamic-programming based polynomial-time algorithm. We strengthen the W[1]-hardness result of Dvořák and Knop. Our reduction is shorter, seems simpler to describe, and the target of the reduction has stronger structural properties. Consequently, we give W[1]-hardness for the combined parameter pathwidth and maximum degree of the input graph. Finally, we prove that Length-Bounded Cut is W[1]-hard for the feedback vertex number. Both our hardness results complement known XP algorithms.

cs.DS↗
arXiv2019

The stable marriage problem with ties and restricted edges

In the stable marriage problem, a set of men and a set of women are given, each of whom has a strictly ordered preference list over the acceptable agents in the opposite class. A matching is called stable if it is not blocked by any pair of agents, who mutually prefer each other to their respective partner. Ties in the preferences allow for three different definitions for a stable matching: weak, strong and super-stability. Besides this, acceptable pairs in the instance can be restricted in their ability of blocking a matching or being part of it, which again generates three categories of restrictions on acceptable pairs. Forced pairs must be in a stable matching, forbidden pairs must not appear in it, and lastly, free pairs cannot block any matching. Our computational complexity study targets the existence of a stable solution for each of the three stability definitions, in the presence of each of the three types of restricted pairs. We solve all cases that were still open. As a byproduct, we also derive that the maximum size weakly stable matching problem is hard even in very dense graphs, which may be of independent interest.

cs.DM↗