arXiv subjects

Sebastian Haslebacher

Publications and source records attributed to Sebastian Haslebacher.

8 recordsLinked to original sources

arXiv2026

Faster Approximate Fixed Points of $\ell_\infty$-Contractions

We present a new algorithm for finding an $\epsilon$-approximate fixed point of an $\ell_\infty$-contracting function $f : [0, 1]^d \rightarrow [0, 1]^d$. Our algorithm is based on the query-efficient algorithm by Chen, Li, and Yannakakis (STOC 2024), but comes with an improved upper bound of $(\log \frac{1}{\epsilon})^{\mathcal{O}(d \log d)}$ on the overall runtime (while still being query-efficient). By combining this with a recent decomposition theorem for $\ell_\infty$-contracting functions, we then describe a second algorithm that finds an $\epsilon$-approximate fixed point in $(\log \frac{1}{\epsilon})^{\mathcal{O}(\sqrt{d} \log d)}$ queries and time. The key observation here is that decomposition theorems such as the one for $\ell_\infty$-contracting maps often allow a trade-off: If an algorithm's runtime is worse than its query complexity in terms of the dependency on the dimension $d$, then we can improve the runtime at the expense of weakening the query upper bound. By well-known reductions, our results imply a faster algorithm for $\epsilon$-approximately solving Shapley stochastic games.

cs.DS↗

arXiv2026

Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements

The famous Ham-Sandwich theorem states that any $d$ point sets in $\mathbb{R}^d$ can be simultaneously bisected by a single hyperplane. The $\alpha$-Ham-Sandwich theorem gives a sufficient condition for the existence of biased cuts, i.e., hyperplanes that do not cut off half but some prescribed fraction of each point set. We give two new proofs for this theorem. The first proof is completely combinatorial and highlights a strong connection between the $\alpha$-Ham-Sandwich theorem and Unique Sink Orientations of grids. The second proof uses point-hyperplane duality and the Poincar\'e-Miranda theorem and allows us to generalize the result to and beyond oriented matroids. For this we introduce a new concept of rainbow arrangements, generalizing colored pseudo-hyperplane arrangements. Along the way, we also show that the realizability problem for rainbow arrangements is $\exists \mathbb{R}$-complete, which also implies that the realizability problem for grid Unique Sink Orientations is $\exists \mathbb{R}$-complete.

math.CO↗

arXiv2025

A Levelset Algorithm for 3D-Tarski

We present a simple new algorithm for finding a Tarski fixed point of a monotone function $F : [N]^3 \rightarrow [N]^3$. Our algorithm runs in $O(\log^2 N)$ time and makes $O(\log^2 N)$ queries to $F$, matching the $\Omega(\log^2 N)$ query lower bound due to Etessami et al. as well as the existing state-of-the-art algorithm due to Fearnley et al.

cs.DS↗

arXiv2025

On Finding $\ell$-th Smallest Perfect Matchings

Given an undirected weighted graph $G$ and an integer $k$, Exact-Weight Perfect Matching (EWPM) is the problem of finding a perfect matching of weight exactly $k$ in $G$. In this paper, we study EWPM and its variants. The EWPM problem is famous, since in the case of unary encoded weights, Mulmuley, Vazirani, and Vazirani showed almost 40 years ago that the problem can be solved in randomized polynomial time. However, up to this date no derandomization is known. Our first result is a simple deterministic algorithm for EWPM that runs in time $n^{O(\ell)}$, where $\ell$ is the number of distinct weights that perfect matchings in $G$ can take. In fact, we show how to find an $\ell$-th smallest perfect matching in any weighted graph (even if the weights are encoded in binary, in which case EWPM in general is known to be NP-complete) in time $n^{O(\ell)}$ for any integer $\ell$. Similar next-to-optimal variants have also been studied recently for the shortest path problem. For our second result, we extend the list of problems that are known to be equivalent to EWPM. We show that EWPM is equivalent under a weight-preserving reduction to the Exact Cycle Sum problem (ECS) in undirected graphs with a conservative (i.e. no negative cycles) weight function. To the best of our knowledge, we are the first to study this problem. As a consequence, the latter problem is contained in RP if the weights are encoded in unary. Finally, we identify a special case of EWPM, called BCPM, which was recently studied by El Maalouly, Steiner and Wulf. We show that BCPM is equivalent under a weight-preserving transformation to another problem recently studied by Schlotter and Seb\H{o} as well as Geelen and Kapadia: the Shortest Odd Cycle problem (SOC) in undirected graphs with conservative weights.

cs.DS↗

arXiv2025

Query-Efficient Fixpoints of $\ell_p$-Contractions

We prove that an $\epsilon$-approximate fixpoint of a map $f:[0,1]^d\rightarrow [0,1]^d$ can be found with $\mathcal{O}(d^2(\log\frac{1}{\epsilon} + \log\frac{1}{1-\lambda}))$ queries to $f$ if $f$ is $\lambda$-contracting with respect to an $\ell_p$-metric for some $p\in [1,\infty)\cup\{\infty\}$. This generalizes a recent result of Chen, Li, and Yannakakis [STOC'24] from the $\ell_\infty$-case to all $\ell_p$-metrics. Previously, all query upper bounds for $p\in [1,\infty) \setminus \{2\}$ were either exponential in $d$, $\log\frac{1}{\epsilon}$, or $\log\frac{1}{1-\lambda}$. Chen, Li, and Yannakakis also show how to ensure that all queries to $f$ lie on a discrete grid of limited granularity in the $\ell_\infty$-case. We provide such a rounding for the $\ell_1$-case, placing an appropriately defined version of the $\ell_1$-case in $\textsf{FP}^{dt}$. To prove our results, we introduce the notion of $\ell_p$-halfspaces and generalize the classical centerpoint theorem from discrete geometry: for any $p \in [1, \infty) \cup \{\infty\}$ and any mass distribution (or point set), we prove that there exists a centerpoint $c$ such that every $\ell_p$-halfspace defined by $c$ and a normal vector contains at least a $\frac{1}{d+1}$-fraction of the mass (or points).

cs.CC↗

arXiv2025

ARRIVAL: Recursive Framework & $\ell_1$-Contraction

ARRIVAL is the problem of deciding which out of two possible destinations will be reached first by a token that moves deterministically along the edges of a directed graph, according to so-called switching rules. It is known to lie in NP $\cap$ CoNP, but not known to lie in P. The state-of-the-art algorithm due to G\"artner et al. (ICALP `21) runs in time $2^{O(\sqrt{n} \log n)}$ on an $n$-vertex graph. We prove that ARRIVAL can be solved in time $2^{O(k \log^2 n)}$ on $n$-vertex graphs of treewidth $k$. Our algorithm is derived by adapting a simple recursive algorithm for a generalization of ARRIVAL called G-ARRIVAL. This simple recursive algorithm acts as a framework from which we can also rederive the subexponential upper bound of G\"artner et al. Our second result is a reduction from G-ARRIVAL to the problem of finding an approximate fixed point of an $\ell_1$-contracting function $f : [0, 1]^n \rightarrow [0, 1]^n$. Finding such fixed points is a well-studied problem in the case of the $\ell_2$-metric and the $\ell_\infty$-metric, but little is known about the $\ell_1$-case. Both of our results highlight parallels between ARRIVAL and the Simple Stochastic Games (SSG) problem. Concretely, Chatterjee et al. (SODA `23) gave an algorithm for SSG parameterized by treewidth that achieves a similar bound as we do for ARRIVAL, and SSG is known to reduce to $\ell_\infty$-contraction.

cs.DS↗

arXiv2024

On the Exact Matching Problem in Dense Graphs

In the Exact Matching problem, we are given a graph whose edges are colored red or blue and the task is to decide for a given integer k, if there is a perfect matching with exactly k red edges. Since 1987 it is known that the Exact Matching Problem can be solved in randomized polynomial time. Despite numerous efforts, it is still not known today whether a deterministic polynomial-time algorithm exists as well. In this paper, we make substantial progress by solving the problem for a multitude of different classes of dense graphs. We solve the Exact Matching problem in deterministic polynomial time for complete r-partite graphs, for unit interval graphs, for bipartite unit interval graphs, for graphs of bounded neighborhood diversity, for chain graphs, and for graphs without a complete bipartite t-hole. We solve the problem in quasi-polynomial time for Erd\H{o}s-R\'enyi random graphs G(n, 1/2). We also reprove an earlier result for bounded independence number/bipartite independence number. We use two main tools to obtain these results: A local search algorithm as well as a generalization of an earlier result by Karzanov.

cs.CC↗

arXiv2021

A Subexponential Algorithm for ARRIVAL

The ARRIVAL problem is to decide the fate of a train moving along the edges of a directed graph, according to a simple (deterministic) pseudorandom walk. The problem is in $NP \cap coNP$ but not known to be in $P$. The currently best algorithms have runtime $2^{Θ(n)}$ where $n$ is the number of vertices. This is not much better than just performing the pseudorandom walk. We develop a subexponential algorithm with runtime $2^{O(\sqrt{n}\log n)}$. We also give a polynomial-time algorithm if the graph is almost acyclic. Both results are derived from a new general approach to solve ARRIVAL instances.

cs.DS↗