arXiv subjects

Céline Swennenhuis

Publications and source records attributed to Céline Swennenhuis.

2 recordsLinked to original sources

arXiv2021

On the Parameterized Complexity of the Connected Flow and Many Visits TSP Problem

We study a variant of Min Cost Flow in which the flow needs to be connected. Specifically, in the Connected Flow problem one is given a directed graph $G$, along with a set of demand vertices $D \subseteq V(G)$ with demands $\mathsf{dem}: D \rightarrow \mathbb{N}$, and costs and capacities for each edge. The goal is to find a minimum cost flow that satisfies the demands, respects the capacities and induces a (strongly) connected subgraph. This generalizes previously studied problems like the (Many Visits) TSP. We study the parameterized complexity of Connected Flow parameterized by $|D|$, the treewidth $tw$ and by vertex cover size $k$ of $G$ and provide: (i) $\mathsf{NP}$-completeness already for the case $|D|=2$ with only unit demands and capacities and no edge costs, and fixed-parameter tractability if there are no capacities, (ii) a fixed-parameter tractable $\mathcal{O}^{\star}(k^{\mathcal{O}(k)})$ time algorithm for the general case, and a kernel of size polynomial in $k$ for the special case of Many Visits TSP, (iii) a $|V(G)|^{\mathcal{O}(tw)}$ time algorithm and a matching $|V(G)|^{o(tw)}$ time conditional lower bound conditioned on the Exponential Time Hypothesis. To achieve some of our results, we significantly extend an approach by Kowalik et al.~[ESA'20].

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arXiv2020

Parameterized Complexity of Partial Scheduling

We study a natural variant of scheduling that we call \emph{partial scheduling}: In this variant an instance of a scheduling problem along with an integer $k$ is given and one seeks an optimal schedule where not all, but only $k$ jobs, have to be processed. Specifically, we aim to determine the fine-grained parameterized complexity of partial scheduling problems parameterized by $k$ for all variants of scheduling problems that minimize the makespan and involve unit/arbitrary processing times, identical/unrelated parallel machines, release/due dates, and precedence constraints. That is, we investigate whether algorithms with runtimes of the type $f(k)n^{\mathcal{O}(1)}$ or $n^{\mathcal{O}(f(k))}$ exist for a function $f$ that is as small as possible. Our contribution is two-fold: First, we categorize each variant to be either in $\mathsf{P}$, $\mathsf{NP}$-complete and fixed-parameter tractable by $k$, or $\mathsf{W}[1]$-hard parameterized by $k$. Second, for many interesting cases we further investigate the run time on a finer scale and obtain run times that are (almost) optimal assuming the Exponential Time Hypothesis. As one of our main technical contributions, we give an $\mathcal{O}(8^kk(|V|+|E|))$ time algorithm to solve instances of partial scheduling problems minimizing the makespan with unit length jobs, precedence constraints and release dates, where $G=(V,E)$ is the graph with precedence constraints.

cs.DS↗