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We study the existence of approximate pure Nash equilibria ($\alpha$-PNE) in weighted atomic congestion games with polynomial cost functions of maximum degree $d$. Previously it was known that $d$-approximate equilibria always exist, while nonexistence was established only for small constants, namely for $1.153$-PNE. We improve significantly upon this gap, proving that such games in general do not have $\tilde{\Theta}(\sqrt{d})$-approximate PNE, which provides the first super-constant lower bound. Furthermore, we provide a black-box gap-introducing method of combining such nonexistence results with a specific circuit gadget, in order to derive NP-completeness of the decision version of the problem. In particular, deploying this technique we are able to show that deciding whether a weighted congestion game has an $\tilde{O}(\sqrt{d})$-PNE is NP-complete. Previous hardness results were known only for the special case of exact equilibria and arbitrary cost functions. The circuit gadget is of independent interest and it allows us to also prove hardness for a variety of problems related to the complexity of PNE in congestion games. For example, we demonstrate that the question of existence of $\alpha$-PNE in which a certain set of players plays a specific strategy profile is NP-hard for any $\alpha < 3^{d/2}$, even for unweighted congestion games. Finally, we study the existence of approximate equilibria in weighted congestion games with general (nondecreasing) costs, as a function of the number of players $n$. We show that $n$-PNE always exist, matched by an almost tight nonexistence bound of $\tilde\Theta(n)$ which we can again transform into an NP-completeness proof for the decision problem.

Budget Minimization is a scheduling problem with precedence constraints, i.e., a scheduling problem on a partially ordered set of jobs $(N, \unlhd)$. A job $j \in N$ is available for scheduling, if all jobs $i \in N$ with $i \unlhd j$ are completed. Further, each job $j \in N$ is assigned real valued costs $c_{j}$, which can be negative or positive. A schedule is an ordering $j_{1}, \dots, j_{\vert N \vert}$ of all jobs in $N$. The budget of a schedule is the external investment needed to complete all jobs, i.e., it is $\max_{l \in \{0, \dots, \vert N \vert \} } \sum_{1 \le k \le l} c_{j_{k}}$. The goal is to find a schedule with minimum budget. Rafiey et al. (2015) showed that Budget Minimization is NP-hard following from a reduction from a molecular folding problem. We extend this result and prove that it is NP-hard to $\alpha(N)$-approximate the minimum budget even on bipartite partial orders. We present structural insights that lead to arguably simpler algorithms and extensions of the results by Rafiey et al. (2015). In particular, we show that there always exists an optimal solution that partitions the set of jobs and schedules each subset independently of the other jobs. We use this structural insight to derive polynomial-time algorithms that solve the problem to optimality on series-parallel and convex bipartite partial orders.

The $k$ disjoint shortest paths problem ($k$-DSPP) on a graph with $k$ source-sink pairs $(s_i, t_i)$ asks for the existence of $k$ pairwise edge- or vertex-disjoint shortest $s_i$-$t_i$-paths. It is known to be NP-complete if $k$ is part of the input. Restricting to $2$-DSPP with strictly positive lengths, it becomes solvable in polynomial time. We extend this result by allowing zero edge lengths and give a polynomial time algorithm based on dynamic programming for $2$-DSPP on undirected graphs with non-negative edge lengths., Comment: 6 pages, 4 figures

Transforming deterministic $\omega$-automata into deterministic parity automata is traditionally done using variants of appearance records. We present a more efficient variant of this approach, tailored to Rabin automata, and several optimizations applicable to all appearance records. We compare the methods experimentally and find out that our method produces smaller automata than previous approaches. Moreover, the experiments demonstrate the potential of our method for LTL synthesis, using LTL-to-Rabin translators. It leads to significantly smaller parity automata when compared to state-of-the-art approaches on complex formulae.

In this paper we report on the full classification of Dirichlet-Voronoi polyhedra and Delaunay subdivisions of five-dimensional translational lattices. We obtain a complete list of $110244$ affine types (L-types) of Delaunay subdivisions and it turns out that they are all combinatorially inequivalent, giving the same number of combinatorial types of Dirichlet-Voronoi polyhedra. Using a refinement of corresponding secondary cones, we obtain $181394$ contraction types. We report on details of our computer assisted enumeration, which we verified by three independent implementations and a topological mass formula check., Comment: 16 pages